Costaguanan

2011 NFL Playoffs: Are the Patriots Better Off as a #1 or #2 Seed?

2012-01-06T16:01:00.000-08:00

I'm a fan of the New England Patriots. During the past few weeks it became clear that the Patriots were going to be in the NFL playoffs as either the #1 or #2 seed. I've heard opinions expressed both ways as to which seed is preferable. Some argued that the #1 seed is better as it means home field advantage throughout the playoffs while others argued that the #2 seed was preferable as the Patriots would likely avoid playing the strong #5 seed in the 2nd round (Baltimore or Pittsburgh) followed by playing the #2 seed in the 3rd round (Baltimore or Pittsburgh). So do the Patriots have a better chance of making the Super Bowl being the #1 or #2 seed?

Some background on the NFL Playoffs. 12 teams make the playoffs, 6 in each conference. I'm focusing on the Patriots conference (the AFC), though this analysis can be applied to either conference or any future year (click here to use a tool to supply your own teams). The 6 teams in the AFC are seeded #1-#6. The #1 and #2 seeds are given a huge advantage by being given a bye in the 1st round thus automatically proceeding to 2nd round. In the first round, the #3 seed plays the #6 seed and the #4 seed plays the #5 seed. The winning teams advance to the 2nd round. In the 2nd round, the #1 seed plays the lowest seed still in the playoffs and the #2 seed plays the other team. In the 3rd round the two remaining teams play with the winner going to the Super Bowl. All games are played at the home stadium of the lower numbered seed.

The twist though is how the NFL determines the #1-#6 seeds. The #1-#4 seed are the 4 teams that won their division, ranked by the strength of their record. The #5 and #6 seeds are the two best teams that did not win their division. The problem though is that the #5 seed is often much better that the #3 or #4 seeds. That is the case this year where Pittsburgh is seeded #5 despite having a 12-4 Won-Loss record while Denver is seeded #4 with an 8-8 loss record.

Here are the seeds this year followed by their Won-Loss record:
#1 New England Patriots (13-3)
#2 Baltimore Ravens (12-4)
#3 Houston Texans (10-6)
#4 Denver Broncos (8-8)
#5 Pittsburgh Steelers (12-4)
#6 Cincinnati Bengals (9-7)

To calculate the odds for each team making the Super Bowl I made a few simplifications. If you would rather supply your own chances each team has of winning a game and calculate your own odds, click the link here. The assumptions:

There are three strong teams (New England, Baltimore, and Pittsburgh) and three weak teams (Houston, Denver, and Cincinnati)
When a strong team plays a weak team at the home of the strong team, the strong team wins 70% of the time. On the road, the strong team wins 65% of the time
When a strong team plays a strong team or a weak team plays a weak team, the home team wins 55% of the time.

The % chance of winning are arbitrary, hence the link to the tool that let's you supply your own odds. I felt that New England, Baltimore, and Pittsburgh were fairly evenly matched. The Vegas gambling lines put Houston and Cincinnati on equal footing, with Houston only being a favorite because the game is being played in Houston. The tool also allows you to input your own teams so it can be used in future playoffs or the NFC. The results:

Team	% of Making Super Bowl
New England	37.5%
Baltimore	33.7%
Houston	5.6%
Denver	3.8%
Pittsburgh	15.1%
Cincinnati	4.3%

To answer my original question, yes the New England Patriots do have a better chance of making the Super Bowl as the #1 seed. Paradoxically, Baltimore has a slightly better chance of making the 3rd round of the playoffs than New England does (66% to 65%). This is because New England is more likely to face Pittsburgh in the 2nd round. However because New England would play Baltimore in New England in the 3rd round--were the two teams to meet--New England has slightly better odds overall. New England has a 36% chance of playing Pittsburgh in the 2nd round and a 42% chance of playing Baltimore in the 3rd round.

NBA Amnesty Clause: Best Amnesty Targets

2011-12-10T12:04:00.000-08:00

The rules of the new CBA make my prior analysis of the amnesty clause mostly moot. As detailed in my last post, the new rules place a greater emphasis on saving the amnesty cut rather than using it right away. The amnesty provision is best used in order to either (1) clear cap space to make a free agent signing or (2) get under the luxury tax line--thus saving money and taking advantage of the rules favorable to teams under the tax line. Enough of the theorizing, what decision should each team make?

Boston Celtics

2011 Amnesty Targets to Consider: Jermaine O'Neal (1 year, $6m remaining)

The Celtics are over the Salary Cap and below the Luxury Tax line. The Celtics cannot get far enough under the cap to sign a coveted free agent without using the amnesty on a useful player. The Celtics could use it on O'Neal only if their other signings this off season push them over the Luxury Tax line and the team wants to save some money. The alternative is to save the amnesty for the future and use it on one of the only two players who are signed past 2012: Rajon Rondo and Paul Pierce. Saving the amnesty in case Paul Pierce's skills decline--he's 34 years old--or the Celtics want enormous cap space in the next couple years is the best move.

Best Decision: Save the amnesty

New Jersey Nets

2011 Amnesty Targets to Consider: Travis Outlaw (4 years, $28m)

Travis Outlaw is the obvious target with his awful play and large contract. However the Nets are under the minimum salary level so eliminating Outlaw would not save them any money. The best option is to cut Outlaw--either this year or prior to a future year--in conjunction with clearing up room for a major free agent signing or trade.

Best Decision: Amnesty Travis Outlaw and sign or trade for a major free agent

New York Knicks

2011 Amnesty Targets to Consider: Renaldo Balkman (2 years, $3.3m), Ronny Turiaf (1 year, $4.3m), Chauncey Billups (1 year, $14m)

The Knicks are well under the Luxury Tax line and cannot get far enough under the Salary Cap to make a major signing. Maybe save the amnesty in case Carmelo Anthony or Amare Stoudemire completely collapses in the future? Stoudemire has had some injury issues in his career.

Best Decision: According to reports, the Knicks will sign Tyson Chandler and either trade or amnesty Chauncey Billups (despite just picking up his $14m option) and Ronny Turiaf

Philadelphia 76ers

2011 Amnesty Targets to Consider: Elton Brand (2 years $35m), Andres Nocioni (1 year, $6.6m)

The 76ers are slightly below the Salary Cap line. Brand has the larger deal but was actually useful last year. Cutting Nocioni would put the 76ers about $9m under the Salary Cap. Cutting either one would make sense if a better free agent can then be signed in exchange. If the 76ers do not use it this year, their only other options are Brand next year or Andre Iguodala in the future (3 years, $42m)

Best Decision: Save the amnesty or cut Nocioni and sign a free agent

Toronto Raptors

2011 Amnesty Targets to Consider: Jose Calderon (2 years, $20m), Leandrinho Barbosa (1 year, $7.6m)

The Raptors are about $8m below the Salary Cap. Using the amnesty this year could clear up enough cap space to sign a major free agent. However the team was bad last year and there is no single free agent available this year that is good enough to turn the Raptors into a good team. Would cutting Calderon and using the cap space to sign Nene, Gasol, or Chandler really turn this 22 win team into something decent? If an amnesty is going to be used, Calderon is a somewhat useful player while Barbosa is more expendable. I'm also not sold on Andrea Bargnani (4 years, $42m) as a good player so saving the amnesty for him may be something to consider.

Best Decision: Save the amnesty

Chicago Bulls

2011 Amnesty Targets to Consider: None

A couple options here. Using it this year on Korver or Brewer--both productive players--might get Chicago slightly under the cap. A better option is to use the mid level to get the best free agent possible, and save the amnesty as insurance on one of the Bulls long term contracts (like Boozer, Noah, or Deng).

Best Decision: Save the amnesty

Cleveland Cavaliers

2011 Amnesty Targets to Consider: Baron Davis (2 years, $28.8m)

Baron Davis is the choice. Either use it this year or next depending on which free agents are available to sign. I lean towards using it next year as the Cavs may want a year to evaluate its young pieces and see what is available in the draft.

Best Decision: Save the amnesty

Detroit Pistons

2011 Amnesty Targets to Consider: Ben Gordon (3 years, $37.2m), Richard Hamilton (2 years, $25.3m), Charlie Villanueava (2 years, $24.1m)

I do not understand why they resigned Tayshaun Prince to a four year deal. In many of these write ups my advice is "amnesty X in order to sign a free agent." Detroit should just clean house. Too many overpaid veterans and not enough upside on this team. Trade Rip Hamilton or Ben Gordon and amnesty the other.

Best decision: Amnesty Ben Gordon or Rip Hamilton and trade the other

Indiana Pacers

2011 Amnesty Targets to Consider: James Posey (1 yr, $6.9m)

The Pacers are way under the cap this year and in the future. Other than James Posey there is not a viable target this year or in the future.

Best Decision: Save the amnesty

Milwaukee Bucks

2011 Amnesty Targets to Consider: Stephen Jackson (2 years, $19.3m), Beno Udrih (2 years, $15m), Drew Gooden (4 years, $26.2m)

Amnesty Stephen Jackson and sign a free agent this year or wait a year and amnesty Jackson, Beno, or Drew Gooden (yes they did just sign this guy last year).

Best Decision: Save the amnesty

Atlanta Hawks

2011 Amnesty Targets to Consider: Marvin Williams (3 years, $23.3m)

Man that Marvin Williams contract is bad and deserves the amnesty. It really does. However it does not help Atlanta's position this year and that Joe Johnson contract is immense. Save the amnesty in case that Joe Johnson contract becomes the next Rashard Lewis contract.

Best Decision: Save the amnesty

Charlotte Bobcats

2011 Amnesty Targets to Consider: DeSagana Diop (2 years, $14.3m)

DeSagana Diop is not a good NBA player.

Best Decision: DeSagana Diop

Miami Heat

2011 Amnesty Targets to Consider: Mike Miller (4 years, $24m), Joel Anthony (4 years, $15m), Udonis Haslem (4 years, $16.5m)

Lot of talk about using the amnesty on Mike Miller. Zach Lowe of si.com linked to an article calling it a necessity but I don't really see the need. I'd sign a mid level player this year--reports are it is going to be Shane Battier--then amnesty someone prior to next season and sign a new mid level player.

Best Decision: Save the amnesty

Orlando Magic

2011 Amnesty Targets to Consider: Gilbert Arenas (3 years, $62.3m), Hedo Turkoglu (3 years, $32.4m)

Gilbert Arenas is not the player he once was. Amnesty Arenas, get under the luxury line and under the cap. Dwight Howard, by most reports, is gone either this year or the next so it is time to start planning for the future. Using the amnesty on Arenas and trading Hedo Turkoglu with Howard is a good start.

Best Decision: Amnesty Gilbert Arenas

Washington Wizards

2011 Amnesty Targets to Consider: Rashard Lewis (2 years, $46m)

Time the dismissal of Rashard Lewis with the signing of a free agent. More likely to use it next year on Lewis than this year.

Best Decision: Rashard Lewis

Dallas Mavericks

2011 Amnesty Targets to Consider: Brandon Haywood (4 years, $34.8m), Shawn Marion (3 years, $25.9m)

I like what I hear about the Mavericks approach to this off season. Rather than over-pay Tyson Chandler for a long shot run at the title, the Mavs are staying avoiding free agency this year to clear up space to sign a major free agent next year. The amnesty could be used on Marion or Haywood prior to next season.

Best Decision: Save the amnesty

Houston Rockets

2011 Amnesty Targets to Consider: Hasheem Thatbeet (1 year, $5.1m)

Hasheem Thatbeet is the only target deserving of the amnesty on the Rockets. However there is not much benefit for this off season. The Rockets will have plenty of cap space next year.

Best Decision: Save the amnesty

Memphis Grizzlies

2011 Amnesty Targets to Consider: None

All the long term contracts on the Grizzlies are players the team has signed recently (Rudy Gay, Zach Randolph, and Mike Conley). All three may be slightly overpaid but none to the extent that they need to be cut this year. The amnesty will provide insurance for the next few years in case one of those players does not work out.

Best Decision: Save the amnesty

New Orleans Hornets

2011 Amnesty Targets to Consider: Emeka Okafor (3 years, $40.6m), Trevor Ariza (3 years, $21.7m)

The team has enough cap space this year and is unlikely to add significant salary due to the NBA owning the team. Next year though the under-performing Okafor and Ariza will be tempting targets for the new owner.

Best Decision: Save the amnesty

San Antonio Spurs

2011 Amnesty Targets to Consider: Richard Jefferson (3 years, $30.5m)

The Spurs used it already on Richard Jefferson. Good move. It gets the Spurs under the luxury tax line and Jefferson was not worth his salary.

Best Decision:

Denver Nuggets

2011 Amnesty Targets to Consider: Al Harrington (4 years, $26.7m)

The Nuggets are way under the cap. No need to use the amnesty this offseason. Down the line though getting rid of Harrington--he had an awful 2010-11 season--with the amnesty may not be such a bad idea.

Best Decision: Save the amnesty

Minnesota Timberwolves

2011 Amnesty Targets to Consider: Darko Milicic (3 years, $15.7m)

Free Darko.

Best Decision: Darko

Portland Trailblazers

2011 Amnesty Targets to Consider:

I was going to write that they should use it on Brandon Roy. However rumors are that he may retire and the Trailblazers can use the injured player clause of the CBA to get salary relief. That would be a year away though so the amnesty can be saved in case that does not go as planned.

Best Decision:

Oklahoma City Thunder

2011 Amnesty Targets to Consider: Kendrick Perkins (4 years, $32.6m)

Save it in case something goes wrong with Kendrick Perkins knees. The rest of the long term contracts on the team provide good value.

Best Decision: Save the amnesty

Utah Jazz

2011 Amnesty Targets to Consider: Mehmet Okur (1 year, $10.9m), Raja Bell (2 years, $6.7m), Devin Harris (2 years, $17.8m)

Amnesty Mehmet Okur this year and get a free agent or use it on Raja Bell or Devin Harris next year and target a free agent.

Best Decision: Save the amnesty

Sacramento Kings

2011 Amnesty Targets to Consider: None

Like the Nuggets, the Kings are so far under the cap that there is no point using the amnesty this year. Maybe amnesty John Salmons (3 years, $24m) down the line.

Best Decision: Save the amnesty

Phoenix Suns

2011 Amnesty Targets to Consider: Josh Childress (4 years, $27m)

Josh Childress was pretty bad. Due to the lack of the Suns interest in a getting a major free agent this year, saving the amnesty for Childress in the future (or in case a different player experiences a Childress like collapse) makes sense.

Best Decision: Save the amnesty

Golden State Warriors

2011 Amnesty Targets to Consider: Andris Biedrins (3 years, $27m)

Biedrins and get a free agent. Seems like a manner of when.

Best Decision: Amnesty Biedrins and sign a free agent. Warriors are already under the cap this year and the next two years so they have some flexibility with the timing.

Los Angeles Clippers

2011 Amnesty Targets to Consider: Mo Williams (2 years, $17m)

I'm not a huge fan of them signing Caron Butler for 3 years and $24m. The team has to use a lot of cap space over the next few years to keep its young promising players. Signing a veteran coming off a major knee injury limits the flexibility of a team with enough assets and potential cap space to make some major moves. Using the amnesty on Mo Williams this year or next year along with a free agent or a major trade is the best option.

Best Decision: Use the amnesty in conjunction with a major deal

Los Angeles Lakers

2011 Amnesty Targets to Consider: Luke Walton (2 years, $11.8m), Ron Artest (3 years, $21.8m)

Luke Walton seems like the obvious choice since he provides no value and the team is way over the luxury tax line. However the amount saved this year--$5m--is not worth using the amnesty over. If the Lakers wait a year or two they can use the amnesty on a larger contract to either help get a free agent or to get the team under the luxury tax line. For example, the Lakers are roughly $22m over the luxury tax. Let's say the Lakers were over the luxury tax by the same amount in 2013 except they were no longer contenders. At that point wouldn't it make business sense to use the amnesty on Kobe Bryant and save the team $50m to $80m in luxury tax savings? The wide spread in amount saved is due to the uncertainty in how the luxury tax is calculated, but the point remains; the potential to save that amount of cash and avoid the potential penalty for being above the luxury tax 4 out of 5 years is worth forgoing saving $5m on Luke Walton.

Best Decision: Save the amnesty

NBA Amnesty Clause Revisited

2011-12-03T12:54:00.000-08:00

With the outline of the new NBA Collective Bargaining Agreement agreed to here's another look at which teams should amnesty which players. The new amnesty rules are:

Each team can waive one player currently on the team and under contract prior to any season. It can be used only one time but can be used in any of the next six years of the CBA. The player must have been on that team's roster when the CBA was agreed to (ie you cannot trade for a player and then amnesty him or sign a new player and then amnesty him in a few years).
The amnesty'd player's entire salary does not count against the Luxury Tax or the Salary Cap. The team still must pay the player's entire salary.

In my next post I will get into which teams should cut which players and which teams should save their amnesty option for a later date. This post will discuss why teams would want to amnesty a player. Reasons to amnesty a player:

1.) Clear up Salary Cap space: Using the amnesty clause can clear up Salary Cap space, allowing the team to offer more money to free agents or to save some money. However there is a Salary Cap minimum of $49.3 million in the first two years of the deal. Teams must spend that minimum in salaries. If a team uses the amnesty on a player to get below that mark they will not be saving any money.

2.) Get under the Salary Cap: A team under the Salary Cap is able to submit a claim on amnesty'd players that other teams cut using the amnesty provision. A player cut with the amnesty clause does not become a free agent. Teams under the cap may submit an offer to pay any portion of that player's contract. The team that bids the highest gets the player. Teams under the cap can also use something caused the 'Mid-Level Exception for Room Teams' but that is not really worth much.

3.) Get under the Luxury Tax line: Getting under the Luxury Tax line saves a team money. In the first two years of the deal, teams pay $1 for every $1 the team is over the Luxury Tax line. In future years of the deal, the team can pay more than $3 for every $1 the team is over the tax line and that penalty becomes even more onerous if a team pays the tax 4 out of 5 years.

Teams under the Luxury Tax line can also offer a better Mid-Level Exception to lure in free agents. Non-tax paying teams can offer $20.45 million over 4 years while tax paying teams can only offer $9.25 million over 3 years. Non-tax paying teams can also offer the Bi-Annual Exception every other year ($3.9 million over 2 years). Non-taxpayers also get a 150% Traded Player Exception while tax paying teams only get a 125% exception. Teams under the Luxury Tax line may also receive preferential distribution of that tax ("Tax the rich and give to the slightly less rich!"), but that is still undecided.

There is some language in the CBA summary about certain teams not being more than $4 million above the Tax level that may benefit tax paying teams more than non-tax paying teams but that is unclear at this time.

4.) Save Money: Not only savings from not paying the Luxury Tax, but if another team claims the player that was amnesty'd that team has to pay whatever portion of the contract they agreed to.

Survivor Odds

2011-11-03T16:51:00.000-07:00

The TV show Survivor: South Pacific had a probability problem last night. The two tribes merged, meaning that the game shifted from two separate tribes into one individual tribe. However the tribes each had six members at the merge and appeared to be headed to a tie vote. The six members of Savaii tribe would vote for one of the Upolu members to leave the game and vice versa. The problem gets a little more interesting when the threat of hidden immunity idols, challenge idols, and the tie breaking rules of Survivor are considered.

For those of you unfamiliar with Survivor terminology, a short explanation. The challenge idol prevents the bearer from having votes cast against him. They cannot be voted out of the game. The hidden immunity idol can be cast after votes are cast but before the votes are tallied. Whoever gets the hidden immunity idol played on him has all votes cast against him rendered null, protecting him from getting voted out of the game. The tie breaking rule of Survivor is to "draw rocks." In the case of a tie vote, everyone votes again only this time the only people who can be voted for are the people who tied in the vote. If there is another tie, then everyone except for those people draw rocks. Whoever draws the one black rock is eliminated from the game.

Without those considerations, the probability is straightforward to calculate. A Savaii tribe member is voted out 50% of the time and an Upolu tribe member is voted out 50% of the time. Both sides had hidden immunity idols. Played optimally, the hidden immunity idols would cancel each other out and the odds remain at 50-50. However Savaii happened to win two challenge idols and--along with the hidden immunity idol--were able to shield three of the tribe members from votes. How does this change the odds?

First let's assume both teams play the optimal strategy. This didn't happen last night. The optimal strategy isn't to decide who to vote and who to shield hours before the vote. Doing this allows for the possibility of the other tribe either figuring out who the voting target was or the possibility of a tribe member betraying this information to the other tribe. Instead the voting target and the hidden immunity shield target should be chosen randomly as close to the voting time as possible. By randomly it does not necessarily have to be equally random--you can weigh certain alternatives more heavily than others--but it does have to be random and all possibilities have to have a chance. This is the optimum mixed strategy.

Now for the odds. There are four voting scenarios. Scenario One: both tribes play their hidden immunity idols but both tribes votes for someone not protected by the idol. They draw rocks to decide and someone is randomly sent home. Scenario Two: both tribes play their hidden immunity idols and both tribes successfully block the other tribe's vote. They draw rocks to decide and someone is randomly sent home. Scenario Three: both tribes play their hidden immunity idols. Savaii successfully shields the targeted member while Upolu fails to shield the correct member. Upolu loses a member. Scenario Four: same as the last scenario only Savaii fails to shield and loses a member while Upolu successfully shields a member.

Savaii has better odds because two of its members are always protected from elimination by having won challenge idols. In Scenario One and Two, only 3 Savaii members will have to draw rocks while 5 Upolu members will have to draw rocks. Scenario Three is also more likely to happen than Scenario Four. The odds of each Scenario:

  Chance of Occuring
Scenario 1 63%
Scenario 2 4%
Scenario 3 21%
Scenario 4 13%

And within each scenario, here are the chances of Savaii losing a member and Upolu losing a member.

  Savaii Loss   Upolu Loss
Scenario 1 23%   39%
Scenario 2 2%   3%
Scenario 3 0%   21%
Scenario 4 13%   0%
37.5% 62.5%

Podcasts I Like

2011-07-21T21:25:00.000-07:00

For some reason, someone decided to follow my music recommendations on GrooveShark.com. I guess if someone is going to listen to my music I might as well make some podcast recommendations too. I'll rank them in order of my personal enjoyment.

On The Media: NPR's show about all aspects of media. They have a variety of interesting stories and a unique angle on events large and small.

PTI: Sports podcast starring Tony Kornheiser and Mike Wilbon. Their chemistry is great. Provides a light, often amusing listen while I work out.

Stuff You Should Know: Josh and Chuck have great chemistry. Their sense of humor is right up my alley. The topic of each show is all sorts of random but almost always entertaining.

History of Rome: Mike Duncan is an amateur historian with a deep interest in the Roman Empire, which I find fascinating. I loved the earlier podcasts. The newer ones are good too but I wish he would diverge a little from following the succession of empires and talk about some other cultural and societal things occurring in the empire.

This American Life: The great show hosted by Ira Glass. Definitely the best podcast on the list. Awesome stories told in an awesome way. My only knock on it is sometimes the tone gets a little sanctimonious or smarmy and bugs the hell out of me.

After those five is a big drop in enjoyment for me but I still find time to listen to the following podcasts.

BS Report with Bill Simmons: I save these for the work day. Variety of guests and often pretty funny. Usually not too serious and I rarely learn anything but it takes my mind off of work.

Hardcore History with Dan Carlin: I loved his series on the eastern front of WWII and liked his series on the fall of the Roman republic. His cadence may not be for everyone but I eventually warmed up to it. Episodes are sporadic but enjoyable. I love how he will pause and then consider a moment or trend in history in a deep fashion. For example, it is one thing to say then the Romans executed 10,000 slaves that revolted. It's another thing to pause and force the audience to consider what that means practically and logistically.

Freakonomics: I have high hopes for this podcast. I feel like it is getting better and could become great. Some interesting topics and neat view points. I'd like them to have more Levitt on. Something about it is just slightly off and keeping me from putting it in the upper tier. For example the last episode was called Mouse in the Salad or something similar. They then spent 7 minutes (about 1/4 of the show) setting up an incident where--wait for it--Dubner was in a restaurant where someone else had a mouse in the salad. Shocking.

The Economist: I like their week ahead show and download some other random topics I might find interesting. I like the tone of the hosts and find much of the conversation intellectually stimulating.

NPR Sunday Puzzle: A neat listen. The types of puzzles are not really my favorite but I enjoy playing along.

Judge John Hodgman: This podcast can range from unbelievably funny to just so so. I like Hodgman's dry humor but it may not be for everyone.

The Sports Poscast with Joe Posnanski: A new entry. Frankly it's not very good now but I like Joe's blog and have high hopes that he can improve. He gets some high quality guests and is an entertaining guy so the foundation is there for a great podcast.

The rest are podcasts I have listened to at some point and will listen to from time to time.

Radiolab: I loved this show at first but have since soured on it. They dawdle a bit too much for me, have some clunker segments in between decent segments and I'm not sure how much I trust the accuracy of the info they give me. Not that I think they are inaccurate or lying I just think they do not nuance the info enough.

In Our Time with Melvin Bragg: A great podcast which probably should be higher on the list. Personally I just find that there are too many guests and the tensions between the guests bug me.

The Classic Tales Podcast: Worth a listen. I used to really enjoy this as I like classic literature. The narrator is hit and miss with his voices, but overall strong. Check out The Yellow Wallpaper and Three Short Stories by Oscar Wilde.

The Moth: Very entertaining stand up stories from a variety of people. I just don't trust that the veracity of the tales. Still entertaining though.

Stuff You Missed in History Class/TechStuff: Kind of meh. I find the TechStuff's host's puns great but not really worth the download.

First Thoughts The Challenge: Rivals

2011-06-27T20:38:00.000-07:00

I have yet to outgrow my odd fascination with the MTV reality/game show The Challenge, thus I must blog and analyze the silly show. The newest season premiered last week. Teams compete in challenges for money and prizes while trying to avoid being eliminated from the show. This season the twist is that the teams are made up of two "rivals" who dislike each other for some reason. I'll summarize the rules briefly and give my thoughts on optimal strategy in this post.

There are 14 teams, each with two players. 7 teams of men and 7 teams of women. Each episode has a reward challenge and an elimination challenge. Win a reward challenge and you get a prize. Win an elimination challenge and your team stays in the game, lose and you are eliminated from the show. The final 3 male teams and the final 3 female teams go to the final challenge. The winning male and female teams in the final challenge gets $100,000, the second place teams get $50,000 and the third place teams get a copy of George Plimpton's Video Falconry. The elimination challenges alternate between having two female team participate and two male teams. The team that performs the worst in the reward challenge automatically gets sent into the elimination challenge. The other team is voted into the elimination challenge. If you win a reward challenge you are immune from being voted into the elimination challenge.

The goal is to reach the final challenge and win money. The best way to accomplish this is to avoid having to play in elimination challenges. Each elimination challenge is essentially a coin flip so teams can maximize their chances of winning money by avoiding elimination challenges. Having a strong voting alliance allows a team to avoid being voted into the elimination challenge. A strong voting alliance also allows teams to vote strong non-alliance teams into the elimination challenge. The best outcome for a team is to reach the final challenge with two weak teams as only the 1st and 2nd place finishes get cash prizes. As there are 14 teams, a voting alliance of 7 teams is enough to control who gets voted into elimination challenges. In practice a unified alliance of 5 or 6 teams is likely enough to control the voting.

However there is a bit of a problem. If a team allies itself with weak teams, then it runs the risk of its allies being eliminated. Weak teams are more likely to lose reward challenges--and get sent into the elimination challenge--and lose elimination challenges. An alliance with weak teams risks losing its voting majority. Ally with strong teams though, and you will have a harder time in the final challenge and a harder time not losing the reward challenge. The optimal strategy to me is to ally with strong teams of the opposite sex. Since the teams are of the opposite sex, they do not compete against your team in the final challenge, elimination challenges, or reward challenges. Those teams are also less likely to get eliminated than the weaker teams so your voting alliance will be more stable. That will then allow you to ally yourself with some weak teams of your own sex.

If such a voting alliance can be formed, the strategy is simple. Do not lose the reward challenge and hope someone not in your alliance loses the reward challenge. Try to vote the strongest team not in your alliance into the elimination challenge and hope that they lose. This will maximize the expected winnings in the final challenge.

I am interested in seeing what happens as teams begin to be eliminated. There will likely be a tension between teams in an alliance wanting to stick together to control the votes while at the same time wanting to vote the strongest ally into the elimination challenge--in the hope that the strong team will be eliminated and not compete in the final challenge. I might explore this idea in my next post about when it makes sense to abandon an alliance and vote a strong ally into an elimination challenge.

Random Thoughts & Predictions:

If I had to wager on which teams would win it, I would say Laurel & Cara Maria and Kenny & Wes look to be the best bets.
I'm guessing that CT & Adam--who may be the strongest team--will be undone by the lack of a social game. I'm guessing Kenny & Wes, Johnny & Tyler, and Evan & Nehemiah will form an alliance with some female teams and control the voting.

NBA Amnesty Clause: Worst Contracts

2011-06-10T16:18:00.000-07:00

If the NBA Amnesty Clause is to be re-enacted, the NBA owners and players' union have to agree to it. I believe the players would largely be in favor of it. Eliminated contracts still get paid in full and teams would then have more money to spend; either from having more salary cap room or from not having to pay luxury tax. However there are many NBA teams that would not benefit from the Amnesty Clause. The following 10 teams hardly stand to benefit from the Amnesty Clause: Boston Celtics, New York Knicks, Toronto Raptors, Indiana Pacers, Chicago Bulls, Miami Heat, Houston Rockets, Memphis Grizzlies, Oklahoma City, Sacramento Kings. I doubt the owners of those 10 teams would like to see the Amnesty Clause enacted. If those teams agree to the Amnesty Clause they would be agreeing to give their opponents a large competitive advantage. Also, if the luxury tax remains those teams receiving luxury tax money would stand to earn even less money if the teams over the cap could cut some of their luxury tax causing contracts.

*Clever Segway*. Here are the top 5 best uses of the Amnesty Clause:

5. Travis Outlaw, New Jersey Nets. 3 years and $21m remaining on his contract. The dollar amount is small compared to some of the other potential targets, but Outlaw has been awful these past two seasons (under 1 WS each season, 8.8 PER last year).

4. Rashard Lewis, Washington Wizards. 2 years and $46m remaining on his contract. Yes, he only has 2 years remaining on his deal but $46m for his lack of production on a rebuilding team is a ridiculous waste of cap space.

3. Ben Gordon, Detroit Pistons. 3 years and $37m remaining on his contract. I am not a huge fan of volume scorers and neither is PER or WS. If Detroit cuts Gordon's deal and trades Richard Hamilton, then the Pistons could get a great start on rebuilding.

2. Brandon Roy, Portland Trailblazers. 3 years and $49m remaining on his contract. Not Roy's fault as his pre-knee injury play was deserving of this contract. After multiple surgeries and the scary rumors about the condition of his knees, the Trailblazers could use the Amnesty Clause to move on.

1. Gilbert Arenas, Orlando Magic. 3 years and $62m remaining on his contract. Arenas has been awful since his knee injuries and arrest on gun charges. If the Magic jettison him and move another deal in a trade, they could have some cap space.

Honorable Mentions: Brandon Heywood, Dallas Mavericks (4 years, $35m), Richard Jefferson, San Antonio Spurs ( 3 years, $30m). Cutting either deal could give these contending teams a chance to reload for another title run.

NBA Amnesty Clause: Atlantic & Central Divisions

2011-06-06T17:05:00.000-07:00

Wrapping up hypothetical targets of the NBA Amnesty, clause let's take a look at the Atlantic and Central Divisions. For some background see my original post on the clause.

Atlantic Division

Boston Celtics

The only large contracts are Rondo and Pierce. Rondo is an All Star on a good contract.
Pierce may not be worth his salary after this upcoming season but still provides value, particularly to this Celtics team which which is built for next season. Honestly, the only contracts to consider were small one year contracts to Shaq and Avery Bradley. However Shaq retired. That leaves Bradley but he's only 20 years old and has a chance to improve.

Contract to Eliminate: None

New Jersey Nets

The awfulness of Travis Outlaw's contract (3 years, $21m) is testing the the limits of my crude $ per Win Share projection formula. I estimate that the Nets will be paying an estimated $20m per win. What an awful signing for a rebuilding team.

Contract to Eliminate: Travis Outlaw

New York Knicks

Amare Stoudemire, Carmelo Anthony, and Ronaldo Balkman are the only players with contracts lasting more than one season. Balkman has only played in 21 games the past two years but his contract is only a 2 years and $3.3m.

Contract to Eliminate: None/Balkman

Philadelphia 76'ers

Elton Brand (2 years, $35m) has a big contract but he had a very good season this year with 9.4 Win Shares. Cutting it would clear up a bunch of cap space. Worth considering if there is a free agent Philly wants to target.

Contract to Eliminate: None (maybe Brand)

Toronto Raptors

Linas Kleiza (3 years, $14m) had an awful 2010-2011 season but it was his first year in the league and he's only 26.

Contract to Eliminate: None/Kleiza

Central Division

Chicago

Boozer (4 year, $61m), Noah (5 year, $61m), and Deng (3 year, $40m) are the candidates. I would be hesitant to eliminate any of their contracts unless it could get the Bulls under the cap enough to sign a better player. If that is a possibility, eliminating Deng's contract may make sense but Deng is still only 26. The Bulls had the best record in the league this year. Since the team is this close to the championship, it does not make sense to jettison a useful player and lessen their title chances for the possibility to have some extra cap space a few years down the line.

Contract to Eliminate: None

Cleveland

Baron Davis is the leading candidate with 2 years and $29m remaining. The Clippers had to trade their first round pick in order to get the Cavs to take him. He has a low expected Win Shares over the remainder of his contract and the Cavs are rumoured to be considering drafting a point guard to replace him with their #1 overall pick. Varejao (4 years, $35m) is worth considering if you believe he will be unable to return from injuries and his contract will be untradeable in the future.

Contract to Eliminate: Baron Davis

Detroit Pistons

Ug they have some bad contracts. The high price of mediocrity indeed. Ben Gordon (3 years, $37m) and Richard Hamilton (2 years, $25m) are worthy choices. My predictor has Detroit paying Gordon $9m per Win Share and Hamilton $12m per Win Share over the remainder of their contracts. I go with eliminating Gordon's deal as it has more years on it.

Contract to Eliminate: Ben Gordon

Indiana Pacers

Danny Granger (3 years and $39m) and Dahntay Jones (2 years, $6m) are the only contracts that last beyond the 2011-2012 season. Granger is a good player and a tradeable asset. Maybe Jones but Indiana will be well below the cap either way.

Contract to Eliminate: None

Milwaukee Bucks

John Salmons has 4 years and $31m left on his contract as he enters his age 32 season. His PER has been below average the past two years. His Win Shares have been solid but he only had .9 Offensive Win Shares this season. I still don't get why the Bucks signed the mediocre Salmons and well travelled Drew Gooden to such long contracts last year. We might have a lock out next year because dumb teams like the Bucks and Pistons choose to overpay mediocre journeyman and still miss the playoffs.

Contract to Eliminate: John Salmons

NBA Amnesty Clause: Southeast Division

2011-06-05T18:16:00.000-07:00

Next up for discussion, the contracts of the Southeast Division. See my first post on the Amnesty Clause for a little background.

Atlanta Hawks

Joe Johnson has a huge contract (5 years and $107m), is over 30 and is coming off a very average year (16.4 PER, 4.2 Win Shares). That contract looks bad now and could become an Arenas like disaster. After Johnson, Marvin Williams and Zaza Pachulia are possible candiates. Both have below average PER the past two years. Williams has 3 years and $23m remaining and Pachulia has 2 years and $10m. I would lean towards eliminating Williams' larger contract though I would give Joe Johnson's deal some consideration.

Contract to Eliminate: Marvin Williams

Charlotte Bobcats

Diop has averaged a 6.6 PER the past 2 years, had a negative Win Share in 2010, and only played in 43 games the past 2 years. His contract is the biggest waste but he only has 2 years and $14m remaining. Tyrus Thomas has 4 years and $33m remaining but his PER was a decent 18.2 last year and he is only 24. I would only eliminate his contract if you had to clear cap space 3 and 4 years from now or if you had major concerns about his ongoing injuries.

Contract to Eliminate: DeSagana Diop

Miami Heat

The Heat are projected to be over the Salary Cap through the 2015-2016 season. Bosh, Lebron, and Wade's contracts make up the majority of that but all three are on reasonable deals. Eliminating anyone else's contract would not be enough to bring the Heat below the cap. The Heat would lose a player and be unable to replace them with anyone other than a league minimum player. Miller, Anthony, and Haslem had sub par seasons but there is no sense in eliminating anyone who can contribute to the team's upcoming title chances when the replacement would likely be worse.

Contract to Eliminate: None

Orlando Magic

Yes it Gilbert Arenas. Of course it is Gilbert Arenas (3 years, $62m). If there is an Amnesty Clause, I would nominate calling it the 'Arenas Rule' as a homage to the 'Allan Houston' rule the first Amnesty Clause was nicknamed. Honorable mention to Hedo Turkoglu (3 years, $34m). The Magic are pretty much overpaying everyone on their roster with the exception of Dwight Howard.

Contract to Eliminate: Gilbert Arenas

Washington Magic

Rashard Lewis is due $46m over the next two years. His stats and skills have deteriorated the past two years, including a poor 12 PER in the 2010-11 season. He is on the wrong side of 30 and unlikely to start playing up to his contract.

Contract to Eliminate: Rashard Lewis

NBA Amnesty Clause: Southwest Division

2011-06-03T20:26:00.000-07:00

The analysis of a potential NBA Amnesty Clause now turns to the Western Conference Southwest Division. See my first post on this for some background. Basically I am looking for the worst contract on each team, with "worst" being decided by criteria particular to what the Amnesty Clause will likely be.

Dallas

Brandon Heywood has 4 years and $35m left on his contract. He is 32 and had a 11.7 PER this past year. His is the worst contract on the team. However if Tyson Chandler leaves as a free agent after this year and no suitable replacement can be found then Dallas may be stuck with Heywood. Marion has 3 years and $26m remaining on his contract entering his year 33 season. The ideal outcome for Dallas is to win the title and sign Chandler to a reasonable contract using the cap space saved by jettisoning Heywood's contract.

Contract to Eliminate: Brandon Heywood

Houston

Maybe Brad Miller (2 years, $10m, age 35) or Luis Scola (4 years, $39m, age 31) but both had above average PER and Win Shares last year. I would probably not eliminate either contract. Houston will be under the cap next year and Scola is a good enough player to contribute or if need be his contract can be traded.

Contract to Eliminate: None

Memphis

The Grizzlies overpaid Rudy Gay, Mike Conley, and Zach Randolph but not to the extent that it makes sense to eliminate his contract at this time. By overpay I don't necessarily mean pay more than their production makes them worth--though that may be the case--I meant they paid more than they had to. In all three of the cases the Grizzlies were only bidding against themselves and bid too high.

Contract to Eliminate: None

New Orleans

Trevor Ariza (3 years, $21m) has a contributed -.3 Offensive Win Shares the past two seasons while producing a well below average PER. Emeka Okafor is overpaid (3 years, $41m) but a slightly above average player.

Contract to Eliminate: Trevor Ariza

San Antonio Spurs

Richard Jeffersion (3 years, $30m, age 31) is the most likely target. Poor PER the past two years. Rumor is that the Spurs agreed to sign Jefferson to that awful deal in exchange for him declining his $15m player option after the end of last season. Despite that bad contract the Spurs are still in a great salary cap position. As I mentioned in a prior post, I am impressed by how the Spurs can put together such a competitive team while still having lots of future cap flexibility.

Contract to Eliminate: Richard Jefferson

NBA Amnesty Clause: Northwest Division

2011-06-02T18:54:00.000-07:00

Continuing the analysis of the Amnesty Clause on NBA contracts, which contracts in the Northwest Division are most fitting for elimination?

Denver

Al Harrington (4 years, $28m, 31 years old) is the only candidate. He had a below average PER and Win Shares this year. Denver has a lot of cap after next season. Unless Denver believes Harrington can be traded or Denver is striving for a couple years of mediocrity, it probably makes sense to eliminate his contract. There is some value in being a mediocre team making the playoffs as a 7 or 8 seed, but not enough in my opinion to forgo the chance to clear some major cap space.

Contract to Eliminate: Al Harrington

Minnesota

The honor goes to Darko Milicic, the perennial awful player. His contract is 3 years and $16m. Nikola Pekovic (2 years, $9m) and Martell Webster (2 years, $11m) also provided below average production in the 2010-2011 season.

Contract to Eliminate: Darko Milicic

Portland Trailblazers

Brandon Roy (3 years, $49m) is coming off a terrible year and consecutive seasons of major knee injuries. Reports are his knee condition is non-corrective. Ideally for the Trailblazers, he would volunteer to retire and see a doctor in order for the team to gain a medical exemption. That would clear the Blazers' cap space and allow them to use the Amnesty Clause elsewhere. However the key word there is volunteer. I see no reason for Roy to volunteer to end his career. He scored 18 points in a playoff game last month so he is not completely done. He may be able to salvage his career and become a decent role player. That would be great for him but still leave the Blazers in a tough position.

Contract to Eliminate: Brandon Roy

Oklahoma City

Kendrick Perkins was recently signed to a large contract and delivered poor results. The Kendrick Perkins deal may turn out to be a bad contract but there is reason to hope that he can recover from his knee injury and return to being a productive player. Thabo Sefolosha (3 years, $11m) has an awful PER the past two years but his Win Shares have him as a solid 4 per year. He started for the Thunder as they went to the Western Conference Finals so he is not a stiff. The best course is to not eliminate any contract.

Contract to Eliminate: None

Utah Jazz

Raja Bell (2 years, $7m, age 34) is the worst contract on the team. Devin Harris and Al Jefferson both have larger two year contracts but had good PER the past two seasons. The only reason to eliminate either of those two players is if the Jazz think they can make a run at a major free agent this off-season.

Contract to Eliminate: Raja Bell

NBA Amnesty Clause

2011-05-31T18:27:00.000-07:00

The current Collective Bargaining Agreement (CBA) between the NBA and the NBA Players' Union is set to expire this year. There has been talk of a new agreement including an Amnesty Clause like the last CBA did. The last amnesty provision allowed teams to eliminate one contract. The teams still had to pay the entire amount on the contract and the contract still counted against the salary cap. The benefit to the team is that the salary no longer counted towards the luxury tax. The luxury tax in the NBA is a penalty for team salaries that are too high. If the team salary is above the luxury tax line, the team has to pay a $1 tax for each $1 it is above the tax line. Using the Amnesty Clause a team could eliminate a $20m contract from their luxury tax calculations and save $20m. If the NBA brings back the Amnesty Clause, which player contracts are the likeliest targets?

As the exact details of the new Amnesty Clause--and whether it will even exist--are unknown I am going to make a few assumptions.

Assumption #1: There will be a hard salary cap. The last CBA has a "soft" cap. Teams could go over the salary cap by taking advantage of certain exemptions. Under a hard cap scenario, no team can go over the salary cap (or at least there will be much fewer exemptions that allow a team to). Under the hard cap scenario, an eliminated contract would no longer count against the salary cap. I have no insider information about whether there will or will not be a hard cap, it is just my opinion about what I believe the likeliest outcome will be from what I have read.

Assumption #2: Teams will keep a bad contract if they believe they can win a championship, and teams will eliminate a bad contract--even if keeping the contract will make the team better--if they do not have a reasonable shot at winning the championship. For example, a 55+ win team like the Miami Heat will keep a bad contract if it only makes the team slightly better (since the Heat are good enough to win a championship) while a mediocre team like the Milwaukee Bucks will eliminate a contract since they do not have a reasonable shot at winning a championship (even if the player they cut will make them a few wins per year better).

Assumption #3: Clearing cap space is often worth doing even if it makes the team worse in the short term. This was evident last year when teams like the Knicks, Bulls, and Heat basically gave away serviceable players and draft picks in order to clear cap space. They became a worse team in the short term in order to maximize cap space. The cap space--in theory--could then be used to sign or trade for coveted players.

I will split my analysis of which player contracts should get the Amnesty Clause into multiple posts. I will shun tradition by starting in the Western Conference Pacific Division. All salary data is from HoopsHype.com . All PER and Win Shares (WS) data is from Basketball-Reference.com. I looked at each players' contract, PER and WS from the past two seasons, their current age, and the team's current salary cap situation when doing my analysis. I also made a rough tool based on prior WS and future age that calculated how much $ per WS the team was projected to pay that player over the life of the contract.

Golden State Warriors

Andris Biedrins (3 years, $27m), Monta Ellis (3 years, $33m), and David Lee (5 years, $69m) have the largest contracts on the team. All three are young but Biedrens is the only one with below average PER and Win Shares the past two seasons. I would consider eliminating Biedrens unless the Warriors believe he will improve enough to justify his contract.

Contract to Eliminate: Andris Biedrins

Los Angeles Clippers

Ryan Gomes (2 years, $8m) and Mo Williams (2 years, $17m) are both coming off of bad seasons. Gomes is more useless but Williams has the larger deal. I lean towards eliminating Gomes' contract hoping that Williams will return to respectability but do not feel strongly either way.

Contract to Eliminate: Ryan Gomes

Los Angeles Lakers

If the salary cap remains at its current level, the Lakers are already over the cap through the 2013-2014 season. Eliminating a contract would not give the Lakers more cap space; it would simply lessen the amount the team is over the cap. Eliminating a contract would either be a money saving move--if the luxury tax remains--or used to clear up a roster spot. Ron Artest (3 years, $21m, age 31), Derek Fisher (2 years, $7m, age 37), Steve Blake (3 years, $12m, age 31), and Luke Walton (2 years, $12m, age 31) would be decent choices to save money. All had PER well below average each of the past two seasons. However as Walton is the least useful of the bunch and the Lakers have a chance to win a championship, his contract is the best to eliminate. On a side note, Odom may be the only decent contract on the Lakers.

Contract to Eliminate: Luke Walton

Phoenix Suns

Josh Childress (4 years, $27m) and Channing Frye (4 years, $25m) are both coming off of seaons with below average PER. Frye had a decent Win Shares (4.7) while Childress did not (1.4) and only played 54 games. I'd lean towards showing Childress the door but didn't Phoenix just sign the guy?

Contract to Eliminate: Josh Childress

Sacramento Kings

Beno Udrih and Franciso Garcia are the only King's players with more than one year on their contract. Udrih was around average the past two years while Garcia had a poor PER and mediocre Win Shares the past two years.

Contract to Eliminate: Francisco Garcia

More Memory Game

2011-05-15T14:25:00.000-07:00

This post is an attempt to answer the question I posed in a prior post about the Memory Game. If you are not familiar with the Memory Game, you can read that post to get familiar with the rules and terminology. The question:

With the second part of your turn, if you do not have a 100% chance of getting a pair is it better to flip an already known tile or try to flip an unknown tile?

As I explained in my prior post on Memory, there are two ways to score. The first method of scoring is to flip over your first tile and already know the position of its match on the board. The second method of scoring is to flip over your first tile, not know the position of its match, and then randomly guess and find its match. My question asks should you forgo the second method of scoring pairs?

Last post I mentioned how I created a table that gives the % chance of a player scoring a pair based on how many tiles are left in the game (n) and how many tiles are known at the beginning of the turn (m). By analyzing that table it quickly became clear that it is much more likely to score a pair by the first method than by the second method. If m=0 then you are more likely to score by the second method--as it is impossible to score by the first method--and if m=1 you are equally likely to score a pair by either method. For all other possible values of m and n, you are much more likely to score by using the first method; 68% - 98% of pairs happen by means of the first method. As m increases the higher the likelihood is that you will score using the first method.

Intuitively it makes sense to flip one tile--and if no match is known--pass your turn (not try to score using the second method) in some circumstances. Suppose there are 10 tiles on the board. 3 have already been flipped and are known. Player 1 (P1) flips an unknown tile. If that tile does not match with one of the known tiles, he can then flip one of the six unknown tiles. Let's consider his possible outcomes:

1/6 Chance: Flips a match to his existing tile and scores a pair (Favorable Result)
2/6 Chance: Flips an unknown card that does not match one of the known tiles (Neutral Result)
3/6 Chance: Flips a match to one of the known tiles (Unfavorable Result)

P1 has a 1/6 chance of getting that pair. However P1 has a 3/6 chance of flipping a match to one of the known tiles. If he flips one of those tiles, Player 2 (P2) will proceed to score a pair with 100% certainty.

P1 flips the first tile. If no match is known at that time he has two choices: try to score using the second method or pass his turn (flip an already known tile which will not result in a match). If he uses the second method, he will either flip a Favorable Result, Neutral Result, or Unfavorable Result. The chance of a Favorable Result is less likely than the chance of an Unfavorable Result if m is greater than 1. The chance of a Favorable Result = 1/(n-m-1) while the chance of an Unfavorable Result = m/(n-m-1). Basically by attempting to score using the 2nd method, you are more likely to give your opponent a pair than to score a pair yourself.

A few other considerations. The Neutral Result --while maybe not immediately harmful to P1--does increase m by 1 for P2, thus giving him a higher chance of scoring a pair. This is partially offset by P1 possibly also having a higher m on his turn but in general raising m is more helpful for your opponent than for you. If you begin the turn in a situation where m = n/2 - 1 it may be helpful to try to score using the second method. If m = n/2-1 at the beginning of the turn and you fail to score using the first method, your opponent will begin his turn with a 100% chance of scoring anyway. However by trying the second method, even if you fail to score you will have a 100% chance of scoring on your next turn as you will begin your next turn with m = n/2.

Memory Game

2011-05-01T09:33:00.000-07:00

In the seventh episode of this season (22nd) of Survivor, two contestants played the Memory Game to see who would remain on the show. In the Memory Game you score points by flipping over pairs of the same card. Whoever identifies the most pairs wins. The Survivor version had 20 tiles (10 unique pairs) laid face down. The contestants took turns flipping over two tiles per turn. If you flipped over identical tiles (a pair) you received a point and those two tiles were removed from the game. If the contestant flipped over non-identical tiles, the tiles were returned to their face down position. The players alternated turns regardless of whether they scored a pair or not. First to get five pairs won the game. Watching the game, I had two questions:

Is it better to go first or second?
With the second part of your turn, if you do not have a 100% chance of getting a pair is it better to flip an already known tile or try to flip an unknown tile?

This post is about the first question. The second question I may take up in another post. Unfortunately I was unable to completely answer the first question but laid the groundwork to solving it (if I had some programming skills it would be easy to finish).

I approached the problem by starting with the simplest case: 2 tiles (1 unique pair). Obviously you want to go first in this situation. Flip over two tiles and you win. For the next case 4 tiles, I had to develop a few rules.

If there is an odd number of pairs, whoever gets the majority of pairs wins.
There are no ties. If there is an even number of pairs, whoever gets half the pairs first wins.
All players are assumed to have perfect memory (ie once a tile is flipped over, they never forget its position).
A player must flip over two tiles per turn.

Rule 2. means that whoever gets the first pair in a 4 tile game wins. In this case, the person going second has large advantage. The person who goes first only wins 33% of the time. The person who goes second wins 67% of the time. Why is this so? In Memory there are two ways to get a pair. The first method of scoring is to flip over your first tile and already know the position of its match on the board. The second method of scoring is to flip over your first tile, not know the position of its match, and then have to randomly guess and find its match. Player 1 (P1) cannot score a pair the first way as all the tiles are unknown. That leaves only the second way. P1 flips over a tile and that leaves three remaining unknown tiles, any of which can be a match for the first tile. Thus P1 has a 1/3 chance of getting the first pair and winning. If P1 does not get a match, then Player 2 (P2) knows two of the tiles before his turn starts and is guaranteed to be able to find a match on his turn through the first method of scoring. This was one of my first findings:

Finding #1: Let n be the number of tiles left in the game. Let m be the number of known tiles (at the beginning of the players turn). If m is greater than or equal to n/2, then a player will score a pair 100% of the time.

I am not explaining it well but it makes sense if you think about it a bit. In the 4 tile game where P1 does not score on his first turn, P2 starts his turn in an n = 4, m = 2 game (there are 4 tiles left and 2 of them are known). P2 begins his turn by flipping over one of the unknown tiles. Whichever one he selects, he already knows the position of its match. If you know exactly half the tiles (m = n/2) on the board then the next unknown tile you flip, you will know exactly where its match is.

Finding #2: The chance of scoring a pair with the second method is equal to 1/(n-m-1).

Remember that the second method is a player flips over an unknown tile, does not know its match and then randomly has to guess for its match from the remaining unknown tiles. The numerator is 1 because you know that one and only one of the unknown tiles is the match for your flip. The denominator is the amount of remaining unknown tiles (total number of tile less known tiles (at the start of the turn) less 1 (the tile you just flipped over).

Finding #3: The chance of scoring a pair with the first method is equal to either 100% ("Solved Pair") or m/(n-m).

A Solved Pair is when your opponent accidentally finds you a match with their second flipped tile. P1 flips an unknown tile and then tries to score a pair using the second method. If the second tile P1 flips matches one of the m known tiles then P2 has a Solved Pair. P2 now knows the position of both tiles in the pair and flips them, thus getting a match 100% of the time. If a player does not have a solved pair then he flips an unknown tile. The player is choosing between the n-m unknown tiles and hoping to find the second half of the known tiles. Thus he is searching for the m second half's out of the n-m unknown tiles: m /(n-m).

Using those findings, I found that P1 has a 47% chance of winning a n = 6 game and a 54% chance of winning a n = 8 game. However as n grows, it gets more and more complicated to figure out the chance of winning. There are many different ways a game can unfold and the calculations are too tedious for me to want to do by hand so I stopped at the n =8 game.

For example in the n = 10 game, a player can win by scoring the first 3 pairs or by scoring the first 2 pairs and then the 5th pair, etc. Then to further complicate things, there are a multitude of different ways that a player win within each of the scoring outcomes. Like in the P1 gets the first three pairs game there are many different was P1 can accomplish this (ie get 3 pairs in a row on his first three turns or get 3 pairs in three of his first four turns where missed on his first turn, or gets 3 pairs in his first four turns when he missed on his second turn etc). As you can see it involves more rote computation than I would like to do. However a computer program or someone with some experience with Markov chains could come up with an a solution.

I was able to make a table that tells the players chance of scoring a pair on that turn. For any n tile game with m tiles known, I can tell you what % chance a player has of scoring a pair on that turn. A program could take that table and use it as a reference as it calculates the various ways a game can unfold. The computer would then have a bunch of saved states that it can reference when calculating the % chance each player has of winning the game at that time. I completed a couple of states for the heck of it and some clear patterns and rules emerge. The calculations are straightforward and could be programmed.

Unfortunately I did not solve my initial problem and can only guess at whether or not it is better to go first or second. For an n = 20 game, I would speculate that each player would have around a 50% chance of winning regardless if they went first or second. The number of ways to victory grows at such a large rate as n increases, that the importance of the first turn would mean less and less. The roughly 5% chance P1 has of making a pair in his first turn is counter-balanced by the increased odds P2 had of making a pair if P1 fails to make a pair.

NBA Salary Commitments

2011-04-17T08:59:00.000-07:00

As I sit around killing time until I can play some pick up basketball at the park, I'm taking a look at the salary data available at HoopsHype.com. NBA salaries amuse. Every year we get to see the winner's curse in action. They gave how much to Joe Johnson? Why on earth would you sign Hedo Turkoglu to a five year deal (and how the heck was he traded twice within the first 18 months of his contract)? I took a look at each team and made some tables and graphs of each team's salary commitment.

Here is the link to the spreadsheet I created. HoopsHype.com counts any guaranteed money and any Player Options as committed salary. Team options and qualifying offers do not count. Year 2011 is the 2011-2012 season.

Here's the link to the picture in its entirety. Miami has the highest salary commitments for the next five years and then Atlanta to a lesser extent. On the other end of the spectrum, Sacramento, LA Clippers and Utah have no salary commitments after the next two years. To summarize:

After three seasons, half the league will have less than $10 million in salary commitments. After four seasons, about 85% of the league has less than $10 million committed. As bad as your team's salary situation is now, in three to four years your team will have a fresh start.

I took a look at each teams' salary commitments and ranked them from the most onerous commitment (Miami Heat) to the least onerous (Sacramento Kings). This is highly subjective. I do not take into account cap space, which is different than committed salary. I tend to value teams with less years of committed salary over teams with less total dollars committed. I'd rather spend $86 million over two years like the Utah Jazz than have $83 million committed over 3 years like the New Jersey Nets. Most importantly, I do not take into account who that money is committed to. I do not distinguish between 5 years of Lebron James (yes please) versus 5 years of Joe Johnson (ug). Maybe in a subsequent post I'll delve into that a bit.

From Cleveland to Sacramento, the teams have lots of flexibility. They basically have a fresh start after two years. I'm impressed by how San Antonio is able to be so good now and still maintain so much future flexibility. San Antonio had the best record in its conference this year. The other contenders--and all the other playoff teams except Denver--have large salary commitments to the near term.

More Fangraphs.com Org Rankings

2011-03-31T13:59:00.000-07:00

To see the data in spreadsheet form, click here. You can see how teams changed in versus last years ranking and where they scored in each of the four component categories. Toronto, San Fransisco, Cincinatti, and the White Sox moved up 10 or more places while Milwaukee, Seattle, Cleveland, and Arizona were the big downgrades. I didn't like how Future Talent was compiled and have some reservations about Financial Resources (perhaps a topic for another post).

A quick summary of each of the component categories:

Below is a graph of team score vs. rank. One thing to remember when disputing the # of the rank is how close it is to surrounding team scores. Rankings #16 - #20 are separated by less than a point. If you're upset about Baltimore at #16 but wouldn't really mind them at #20 it's not really worth complaining about.

I can't say I'm a huge fan of how the Future Talent rankings were compiled:

Fangraphs.com Organization Rankings -- Star Plots

2011-03-31T11:13:00.000-07:00

Fangraphs.com has an annual series of posts where they rank each Major League Baseball organization on the following criteria: Present Talent (30%), Financial Resources (30%), Baseball Operations (25%), and Future Talent (15%). The writers grade each team in each of the categories probably on a scale of 1-100 (I believe; no ranking is below 62 or above 95). I took the rankings for each category and made some star plots.

This plot scales each score based on its relation to the min and max in that category. For example, Houston has the minimum in Future Talent (65 score) so the plot reflects a 0 on that part of the graph. Kansas City has the max score (95) so the plot reaches to the maximum distance on that part of the graph. The teams are ordered by their overall score from first to last.

Now what if you want to see how the full score--not the scaled score--relate to each other? It is not very informative. Next plot shows the full score for each category (ie a 65 is 65% of the distance from the min to the max along that part of the graph). Since each score is pretty much in the same 65-95 range, the graph is not very helpful as all the plots look largely the same.

This final plot tries to do what the last plot does but accounts for the natural clump in the data. I subtracted 50 (warning! arbitrary manipulation of data) from each score and then scaled accordingly.

The Challenge: Cutthroat

2010-10-16T18:50:00.000-07:00

After two episodes the MTV reality show The Challenge: Cutthroat has not had much to analyze strategically. The only major choices faced so far have been picking the teams and choosing who gets sent into the elimination round.

The basic structure of the game is as follows. Three team captains were chosen based on their weakness in the initial challenge. Each captain selects players until the contestants are divided into three equal teams of 8 players. The three teams then compete against each other in challenges. The winning team gets $20k in their bank account--which can only be redeemed if that team wins the show's final challenge--and the losing teams select one male and one female contestant to go into the elimination round. In the elimination round, the two males face off and the two females face off in separate competitions. The loser of each competition goes home.

The first important decision faced was by the three captains. They had to decide who to choose to be on their team. At that time the captains did not realize how people would be eliminated. However the show has featured voting and elimination rounds for the past few seasons so it was not a stretch to imagine this happening again. In hindsight it makes the most sense to choose players you can form an alliance with. Controlling the voting allows you to keep yourself out of the elimination round. Stronger players give you the chance to win more challenges but weaker players are more likely to get voted into the elimination rounds and thus give you some protection from going home. The show is notoriously hard on first-time players. It may have made sense for one of the newer captains to stack their team with first-time or newer players to prevent the captain from being at a disadvantage.

When voting for the elimination round, the optimal strategy it is pretty straightforward. Ally yourself with enough people in your team so you form a majority. At each vote, vote in someone who is not in your majority. It is preferable to keep the stronger players around. However with the challenges being three-way competitions and testing very random skills, even with a strong team your chances of winning are not that great. Teams have to figure they will lose multiple times and need to plan accordingly.

The beginning elimination rounds featured some weaker players. In hindsight it may have made sense to vote in a stronger player to eliminate the other teams' weaker players and preserve your weaker players (who can then become fodder in later rounds). The downside to this strategy is that if both teams think this way, then the initial elimination rounds feature stronger players and teams risk losing their strong players. Sort of similar to the game theory version of chicken.

Bill Simmons and the NFC West

2010-10-09T01:32:00.000-07:00

In his weekly football column, Bill Simmons of ESPN.com harshly criticizes the NFC West. His main point of evidence is the NFC's west record since 2002 (go-go arbitrary end points). He posts this table in his article:

DIVISION RECORDS SINCE 2002

Division	W	L	T	Pct.
AFC East	273	247	0	.525
AFC North	260	258	2	.502
AFC South	287	233	0	.552
AFC West	255	265	0	.490
NFC East	279	240	1	.538
NFC North	240	280	0	.462
NFC South	266	253	1	.513
NFC West	218	302	0	.419

He could have justifiably manipulated the data a bit further to prove his point. Each division plays 12 inter-division games each year. Against itself, each division goes 6-6 in those games. That weights the numbers in the above table towards .500 . If he had subtracted those 96+ games (12 games a year x 8 seasons = 96 games) included in each of the division stats, the table would bolster his argument slightly. To whit:

I made two assumptions. Simmons includes the first two weeks of the current NFL season. I assume that no division games had been played yet, which was not the case. I also assumed that the 4 ties that happened in the past 8 seasons were not division games (at least one of them wasn't and perhaps both). There you have it. the NFC West winning percentage drops to .401. Eyeballing it it looks significant. 6 of the 8 divisions are within .050 of .500 and the NFC west is .052 away from the next weakest team. To formally prove that the .401 is statistically significant, a t-test could be set up assuming .500 as the mean. The standard deviation would have to be sampled from historical data which would necessitate some assumptions.

Aging Curves in Basketball

2010-08-28T19:53:00.000-07:00

The NBA offseason raised a few questions. In order to answer these questions I need to do a little research and figure out a few things. One of these things is an aging curve for basketball players. What is their prime? What rate do they decline after that prime? Luckily this question has been asked and answered in baseball so I can use their methodology and apply it to basketball.

For data, I went to www.basketball-reference.com and pulled the advanced data for players from the 2000-2001 season through the 2009-2010 season (the seasons from the last collective bargaining agreement). My methodology follows the work of TangoTiger and MGL, particularly this article written by MGL. I will try to briefly summarize the methodology. MGL offers a much clearer explanation so I recommend reading that article first.

The stat I relied on was Win Shares (WS) per 2500 minutes. The 2500 minutes is an arbitrary figure. I compared each player's WS -per-2500 minutes with their next season season to arrive at a difference in WS between the two seasons. I then averaged the delta to arrive at a rough win curve. For example Lebron James had a WS-per-2500 of 4 in is age 19 season and 10 in his age 20. 10 - 4 = 6 for a delta of 6. This delta is then compared to all the deltas for people in their age 19 to age 20 seasons.

It is a little more subtle than that. One concern is survivor bias. Survivor bias (read the MGL article) is the idea that the sample is biased because the people who survived to play another year are a biased sample. Say there are two players who have a WS of around 2 and that due to random factors they can fall up to 3 wins shares around that average. Player 1 gets lucky and gets 3 WS worth of random factors that help him and Player 2 gets unlucky and has -3 worth of WS befall him. Despite Player 1 and Player 2 being the same skill level, Player 1 has a 5 WS season and Player 2 has a -1 WS season. Player 2 is cut and does not play next season. Player 1 does get to play another season but is likely to regress from his lucky season of 5 WS to his mean WS of 2. This survivor bias influences the sample. To prevent this, players who do not have a next season have to have one projected for them.

I followed the MGL method for projecting seasons for players who did not have a next season. I averaged their last three years of WS per 2500 and regressed that with the league average WS per 2500 for someone that age; less -.5 WS. The -.5 is another arbitrary figure. This is a conservative prediction for a future season.

Another issue, and one I struggled with, is how to weigh the data points. Should I include all players, even those who played less than 5 minutes a game? If so should I weigh the data points of players who played more higher than those who played less? In the end I included all players and weighed each of the data points by the amount of time that they played. For example if a player played 3000 minutes and year one and 2500 minutes in year two and another player played 400 minutes in year one and 1200 minutes in year two, then the former player's results would be carry more weight. The results:

Age 25 is the peak of the graph. There is not much difference between Ages 24 - 28. Players tended to improve until Age 24, then plateau until 28 or 29 before declining by about .5 WS per year. For some perspective, WS in general peak in the Age 26-28 years.

I am not really thrilled with my process or results. Some of my assumptions are a bit too arbitrary for my liking. To feel more comfortable with my results I would have to research this further, taking a closer look at players year-by-year and in different segments to get a better feel for the data. However for the purposes of my original problem, the decline of -.5 WS per 2500 minutes per year after the age of 28 will suffice.

The World Cup Approaches...

2010-05-30T10:43:00.000-07:00

Another World Cup tradition: players complaining about the new ball. The Adidas Jabulani replaces the much maligned 2006 Teamgeist and the hated 2002 Fevernova. Goalkeepers complained about the 1994 Questra ball and I'm sure soccer players wished many unkind things towards the 1998 Tricolore ball but I can't find a link. I'm not sure why but I find the sky-is-falling stories about the new balls amusing.

Fresh Meat II: Episode 7

2010-05-22T12:57:00.000-07:00

Continuing my strange fascination with the MTV show Fresh Meat II, here's a post about the latest episode. Wes's coalition officially reached its end as the two remaining teams in it where forced into the elimination challenge. Wes's team lost and left the game. We are left with 6 teams. 4 eams are in Kenny's alliance and 2 are on the outskirts. The last 4 teams go to the finals where they compete for cash money. 1st place in the final challenge wins $200k, 2nd place $60k, 3rd place $40k, and 4th place receives nothing. The goal is to not only make it to the finals, but to maximize your expected winnings in the finals. Strategically, how should the six remaining teams behave to optimize their expected payout?

The six remaining teams can be roughly classified as belonging to one of three groups. The first group is the two strong teams in Kenny's alliance (Kenny & Laurel and Pete & Gillian), the second group is the weaker teams in Kenny's alliance (Jenn & Noor and Ryan & Theresa), and the third group is the two teams not in Kenny's alliance (Landon & Carlie and Eve & Luke).

The two strong teams in Kenny's alliance are in the best position. They are likely to be in the remaining four teams, they are strong at the elimination challenges so they are likely to win them if sent in, and once in the finals stand a good chance of getting 1st or 2nd place. Since they are in such a strong position, I recommend a conservative strategy. Consolidate your alliance, try to eliminate the teams outside of your alliance, and avoid being sent into an elimination challenge. The only way these two teams could increase their chances of winning more money would be to try to eliminate the other strong team.

This would involve betraying your alliance and creating some fallout. This would only be a recommended strategy if the strong team wins the final reward challenge (when there are 5 teams remaining). At that point being in the alliance does not matter since the winner of the final reward challenge automatically makes the finals. The strong team could chose the other strong team to go to the elimination challenge and hope that that team loses. The downside to this strategy is it involves an emotional cost--as you betray the people who helped you reach this point in the game. More tangentially, if you come back on a future show other players may have trouble trusting you. If Kenny's team or Pete's team wins the final reward challenge it will be interesting if they consider this strategy.

The third group is the two teams on the outside of the alliance. The most likely outcome is that these two teams are voted into the next elimination challenge. The winner of the elimination challenge stays on the show and will then probably have to go into the next elimination challenge. That's only a 25% (50% x 50% = 25%) chance of making the finals. The teams on the outside of the alliance need to win reward challenges to avoid the elimination challenge or try to get the two weak teams in Kenny's alliance to defect from Kenny's alliance and join them.

The two weak teams in Kenny's alliance are in the most interesting position. Yes they have a great chance of reaching the finals (I'd estimate about an 85% chance). However, once in the finals they do not have a good chance of making much money. The strong teams in their alliance are more likely to win 1st and 2nd place. If the two weak teams stick together and join the two teams on the outside of Kenny's alliance, they could vote the strong teams into the elimination challenge. However this would require the two teams outside of Kenny's alliance being trustworthy (clearly they are not) and the strong teams would have to be prevented from winning the two remaining reward challenges (which will likely not happen).

This leaves the weak teams in a somewhat poor position. When there are 5 teams left, one of them will likely have to go into the elimination challenge against the remaining team not in Kenny's alliance. Then in the finals the weak teams will have to compete against the strong teams. I ran three models (see below) where I estimated each teams chance of winning 1st, 2nd, 3rd, or 4th place depending on their competition. Note that eliminating both strong teams, leads to the weak teams winning an estimated $20k more than competing against the strong teams. Eliminating just one of the strong teams leads to an estimated winnings of $10k more. Also note that a strong team stands to win $10k more on average if the other strong team is eliminated prior to the finals.

Other Thoughts on Episode 7:
Landon's team won the reward challenge and thus got to choose one of the teams sent into the elimination challenge. I'm surprised Landon didn't use that to his advantage and try to work himself into Kenny's alliance. Instead he remains on the outside of the alliance in a very poor position. Landon gets to choose the order the teams compete in the next reward challenge. He should maximize his chances of winning the reward challenge by placing the strongest teams first (ie Kenny and Pete's teams) or use the threat of that to try to work his way into Kenny's alliance.

The Challenge: Fresh Meat II Episode 6

2010-05-14T22:13:00.000-07:00

A few weeks ago I wrote a post saying that Kenny's alliance had basically no chance to retake the majority (0.2%). The Kenny's alliance was down 7 teams to 3. The alliance would have to win 4 reward challenges and 3 elimination challenges in order to take back the majority. 3 reward challenge victories and 3 elimination challenge victories later, Kenny's alliance has seized control of the game and is in a dominant position. What happened? Why was I so wrong in underestimating the chances of Kenny's alliance?

One of the obvious places I was wrong was in my assumption that all the teams in the reward challenges had an equal shot of winning. While that may have seemed like a valid assumption when the challenges were seemingly random like trying to grab a huge rubber ball while blindfolded or trying to hold your breath underwater, it turns out Kenny's team had a huge advantage in the reward challenges. Kenny and Laurel are a very strong team. The other assumption I made was that Wes's coalition would behave rationally. I thought Wes' coalition would try to maintain their numbers advantage and keep voting a team from Kenny's alliance into the elimination challenge. I assumed the two teams in the elimination would each have a 50% of winning. That was not the case. Wes's coalition came from the Ignatius J. Riley school of dumbass-ery.

Wes's coalition made some terrible choices. In two of the elimination challenges they voted one of their own teams into the elimination challenge, making it so it was a 100% chance that a team from their coalition would go home. In the other elimination challenge Wes's coalition set it up so their weakest team went against one of the stronger teams from Kenny's alliance. Rather than trying to protect their teams from the elimination round, Wes's coalition sacrificed them.

Some specific examples of the stupidity of Wes's coalition:

Episode 4: Wes's coalition is up 7 teams to 3. Kenny and Laurel win the reward challenge and vote a team from Wes's coalition into the elimination challenge. Wes's coalition decides to vote one of their members into the elimination challenge because the team had a member with a hurt leg. Strategically this makes no sense. Wes's coalition has nothing to gain from such a match up. 100% chance of losing a team. Wes's coalition loses a team.
Episode 5: Wes's coalition is up 6 teams to 3. Kenny and Laurel win the reward challenge and vote a team for Wes's coalition into the elimination challenge. Wes's coalition strikes a deal with a member of Kenny's alliance (Ryan) basically saying we won't vote you in if you help convince Kenny which team to put into the elimination challenge. Wes also views Ryan's team as the weakest team remaining and wants to keep Ryan around as insurance. If Wes gets voted into the elimination round, he'll vote Ryan in since it's a weak team. Wes's coalition values this relationship and insurance more than trying to keep one of their coalition teams in the game. Wes's coalition votes the strongest team from Kenny's alliance into the elimination challenge. Wes's coalition loses the elimination challenge and another member. In the process, Wes appears to have betrayed his good friend and weakened loyalty in his coalition.
Episode 6: Wes's coalition is up 5 teams to 3. Kenny's alliance wins the reward challenge. Wes is afraid he will be selected into the elimination challenge. He betrays his coalition and helps set it up so two of his coalition teams will go into the elimination challenge. Wes loses a coalition member, alienates the coalition member he betrayed, and one of his coalition team defects. Kenny's alliance is now up 4 teams to 2 with 1 team's allegiance unknown (the betrayed former member of Wes's coalition).

Wes made three general mistakes that lead to his coalition collapsing. First, he did not try to maximize his coalition's chances in elimination challenges, secondly he valued hidden alliances with teams more than the actual bonds he had with his coalition members, and thirdly he was too afraid to compete in the elimination challenge. He played too cute of a game. Hidden double agents, insurance teams, minimizing his chance to get thrown into the elimination round... these strategies aren't worth a damn if you give up your numbers advantage and destroy your coalition pursuing them. The most important part of the game is in having a numbers advantage. Wes did not value the numbers advantage and now it is going to cost him.

Fresh Meat II: The Challenge Episode #4

2010-05-03T21:19:00.000-07:00

My earlier posts on the MTV reality show Fresh Meat II: The Challenge recommended two strategies for the competing factions in the show: one side should tighten the bounds in its group and keep loyalty strong while the other side should use the threat throwing a team into the elimination challenge as a way to get teams to defect to their faction. However the factions got the strategies backwards. I recommended Kenny's alliance use the threat of being sent into the elimination challenge to get someone to defect to his alliance. Wes's coalition had the superior numbers. It was in Wes's coalition's best interests to keep loyalty strong within the group. Instead Kenny let someone in his alliance choose who was going into the elimination challenge, thus trying to boost loyalty withing the group while Wes's coalition used the threat of being thrown into the elimination round to get a team to defect to his coalition.

I'm not sure why Kenny--once again--did not use the threat of sending someone into the elimination round as a tool to get a team to defect to his alliance. He seems smart enough. Maybe his arrogance and strong results in the reward challenges--more on this later--are deterring him.

I thought the strategy used by Wes's coalition--using the threat of the elimination challenge to get a team to defect--was somewhat valid but not for the reason's Eve suggested. Eve thought it was a good idea as it would peel a team away from Kenny's alliance. As I wrote in an earlier post, Kenny's alliance already had too few teams to be much of a threat. Wes' coalition didn't need to get bigger; it just needed to stay together. Eve's action undermined the coalition and pissed off two teams (CJ and Landon's teams) which had the show ending with those teams considering defecting to Kenny's alliance. However the move was valid in that it kept Kenny from choosing Eve's team to go into the elimination challenge. Removing that risk and uncertainty was a worthwhile goal for Eve to pursue. As long as she felt that the risk of her losing the elimination challenge was greater than the risk of her coalition falling apart, it was a good move. If that was not the case though then she was just plain scared.

Back to Kenny and Laurel's prowess at the reward challenges. They certainly look like a strong team. Laurel has performed much better than the rest of the females in the challenges (and she may be the best looking one too). In my earlier posts, my calculations assumed that each team had an equal shot of winning the reward challenge. Now that Kenny's team has won 3 out of the 4 reward challenges is that assumption still valid?

Let's use the binomial distribution to test the assumption. The binomial distribution forces the probability for each trial to be set at a fixed probability p. Even though the number of teams changes after each elimination challenge as a team is sent home and thus p changes for each reward challenge (ie if you are 1 of 2 teams you have a 50% chance of winning but if you are 1 of 10 teams you only have a 10% chance of winning). I am going to fix the probability p at 9%. The number of teams in the reward challenges has been 12 (p = 8.3%), 11 ( p = 9.1%), and 10 (p = 10%) so 9% is about right. Here's the binomial theorem:

p = .09; n is the number of reward challenges so far which is 4; k is the number of times Kenny's team has won the reward challenge. I am going to calculate the theorem for k = 3 and k = 4 since if Kenny's team had won all 4 challenges I would also be doing this little test. Popping this into an spreadsheet, the probability of Kenny's team winning 3 or 4 challenges out of 4 challenges--assuming they only had a .09 chance of winning a reward challenge--is a tiny 0.3%.

The math is a little more complicated than the above calculation. If we had chosen one particular team apriori of the show's first episode and done the above calculation than the 0.3% number would have been correct. However there were 12 teams to start the show so any one of them could have been the team to win 3 out of the first 4 challenges. What are the odds that none of them would have won 3 out of the first 4 challenges? I believe we can use the Bonferroni Inequality to get an approximation:
1 - (1 - 0.3)^12 = 3.2%
Assuming that all the teams had an equal chance of winning the reward challenge, 3.2% of the time 1 of the 12 teams would win 3 or 4 out of the first 4 challenges. 0.3% and 3.2%. Either way it is safe to conclude that Kenny's team has a better than equal chance of winning the reward challenges and thus may assumption in earlier posts was incorrect.

Even with Kenny's team having a better chance than I had assumed at winning the reward challenges, his alliance was still in a very unfavorable position. His alliance had to win 3 elimination challenges in a row just to even the amount of teams in each faction. I calculated the probability of that happening at a low 12.5%. Still unfavorable, just not as dire as I had thought.

On an aside is there a distribution that allows the p to change after each trial and would work better than the binomial in this case?

NBA #1 Seeds that Struggle in the First Round

2010-04-27T20:39:00.000-07:00

A post on Hoops Analyst talks about the Los Angeles Lakers - Oklahoma Thunder NBA playoff first round series. The Lakers are the highly rated #1 seed and the Thunder are the lowest rated #8 seed. The two teams are playing a best-of-seven series. The author asks if a #1 seed that is struggling against a #8 seed is "a bad indicator for their success in the playoffs"? The post then looks at each of the #1 seeds that struggled against #8 seeds and concludes "I would venture that there is little correlation between a round one struggle and overall team weakness." What does the data say?

I looked at each #1 seed since 1984--the year the playoffs expanded to 16 teams--and for each #1 seed that won its first round series I looked at how many games, series, and championships the teams went onto win. By comparing the teams that struggled in the first round versus the teams that did not, we can see if first round #1 teams that struggle is a bad indicator for future success. The data supports the author's conclusion. There is little correlation between a round one struggle and a lack of future success in the playoffs. I imported the data into R, ran some regressions, and some t-tests and Wilcox-tests. There was nothing statistically significant between #1 seeds that struggled in the opening round and a lack of future success.

I also made some pivot tables in excel. First is the full data from 1984 to 2009 then the data from 1984 to 2002 and then the data from 2003 to 2009 (in 2003 the first round of the playoffs changed from a best-of-5 to a best-of-7 game series). From 1984 to 2009 teams that did not struggle in the first round, won one more game on average per year than teams that did struggle in the first round. Note that from 1984 to 2002, teams that struggled in the first round went on to win the championship at at a much lesser % (14%) than teams that did not struggle in the first round (45%). From 2003 to 2009 the data does not show anything.

The Challenge: Fresh Meat II Wes's Coalition

2010-04-25T21:36:00.000-07:00

With Wes's coalition in such a dominate position, the 7 teams in it have to start moving beyond the threat of Kenny's alliance (3 teams) and strategizing for the end game. In particular two things: avoiding the elimination round and being one of the final four teams that gets to compete for the cash prize. As long as Kenny's alliance remains, Wes's coalition will keep choosing them to go into the elimination challenge. Wes's coalition could be left with 7 teams or it could lose a couple elimination challenges and be left with 5 teams when Kenny's alliance is down to one team. At that point Wes's coalition will have to turn on itself.

Wes's first priority is to keep the teams in the coalition together so those on the outside do not defect to Kenny's alliance. At the same time, the teams in Wes's coalition want to form a union of 4 or 5 teams. This union would then try to control the game. If the union wins a reward challenge, it will select one of the non-union teams into the elimination challenge and vote another non-union team into the elimination challenge. The most important thing is to be one of the teams in the union. It also helps if the teams in your union are good at reward challenges but bad at the final round (which would be an unlikely combination). Besides, if a team in your union is likely to be good in the final round you can turn on them when there are 5 teams left and vote them into the elimination challenge.

So that's the balancing act. Form a strong union while at the same time keep the teams non-union teams from defecting. If you are a non-union team you would have to win at least one and possibly four elimination challenges just to survive to the final round. What is the probability of a non-union team making it to the final round?

Let's start with the easier case of a union team making it to the final round. To simplify things, let's say there are 4 union teams, they win all the reward challenges and control the voting thus making it so only non-union teams go into the elimination challenges, and that each team in an elimination challenge has a 50% chance of winning it. When there are 5 teams left one of the union teams will have to go into the elimination challenge against the non-union team. Under such a scenario, the union team has a 87.5% chance of making it to the final round: 3/4*1+1/4*1/2 = 87.5% (Chance of not being selected for the elimination round * chance of making it too the final round if not selected to the elimination challenge + chance of being selected for the elimination challenge * chance of winning the elimination challenge)

Now the case of a non-union team making it to the final round. Some assumptions: each of the non-union teams are equally likely to be selected to the elimination challenge and each of the teams in an elimination challenge is equally likely win. Let's say there are 8 teams left. 7 teams in Wes's coalition and a remaining team in Kenny's alliance. Let's say the union is 4 teams and the non-union is 3 teams. Kenny's alliance team is always voted into the elimination challenge as long as it remains and one of the non-union teams is randomly selected into the challenge. Under such assumptions a non-union team has a 14.6% chance of making it to the final round.

87.5% > 14.6% It is much better to be in the union than not in the union. In real dollar terms--I believe the teams in the final round will be competing for $300,000--being in the union is worth about $55,000 more in expected earnings.

Fresh Meat II: The Future for Kenny's Alliance

2010-04-25T11:59:00.000-07:00

In the last episode of The Challenge: Fresh Meat II, Kenny's alliance lost the elimination challenge. It also became clear that one of the teams in Kenny's alliance was a mole. That puts Wes's coalition at 7 teams and Kenny's alliance at 3 teams. Kenny's alliance is pretty much doomed. They would have to win four reward challenges and three elimination challenges in a row to get back to equal footing. A loss in any of those challenges would lead to one of the teams in Kenny's alliance going home. If every elimination challenge is a 50-50 proposition and all teams are equally likely to win a reward challenge, the probability of Kenny's alliance winning three reward challenges and two elimination challenges in a row is 0.2% (30%*50%*33%*50%*38%*50%*43%= 00.2%). There is basically no chance of Kenny's alliance seizing the majority unless Kenny's alliance gets a team from Wes's coalition to defect.

What are some possible ways to get a defection? Remember that the winner of the reward challenge gets to vote one of the teams into the elimination challenge. One possible way is to use the threat of voting someone into the elimination challenge as a threat to get someone into your alliance. Kenny could have used that threat in the last episode. "Hey join my alliance and vote with me or I'll throw you into the elimination round." and perhaps gotten one of the teams to defect. My preferred style would to be to gather the two teams in Wes's coalition who are the least loyal and use that threat. Whoever defects first is in your alliance, the other person is voted into the elimination challenge. The high pressure of the situation and the threat of going into the elimination challenge could get someone to not think strategically and defect. The problem is there isn't much of a way for Kenny to enforce the threat. Someone could promise to join your alliance and then when Kenny chooses to put the other team into the elimination round, the team could simply keep voting with Wes.

Only four teams from Wes's coalition will make it to the final round. At some point the teams on the outside of Wes's coalition will realize that Wes doesn't intend for them to be one of the four teams in the final round. Kenny's alliance only has three teams in it so joining Kenny's alliance means a 100% chance of making it to the final round (that is if Kenny's alliance lasts that long). If their odds of making the final round are better by joining Kenny's alliance than by staying in Wes's alliance it makes sense to defect. If Kenny's alliance wins a few challenges and gets the balance to 3 teams in Kenny's alliance and 6 teams in Wes's coalition, then there is incentive for one of the teams on the outside of Wes's coalition to jump to Kenny's alliance. Kenny's alliance should continue to speak to the teams on the outside of Wes's coalition to try to get them to defect. CJ and Sandra and Katelynn and Brandon in particular appear to be on the outs in Wes's alliance. I also suspect Wes's coalition to turn on Landon's team at some point since he is very good at the challenges.

Are there any other strategies for Kenny's alliance that make sense? So far the presumption has been that the 3 teams in Kenny's alliance are best served by sticking together. Defecting to Wes's coalition isn't really an option as Wes's coalition already has enough teams in it and will have to start turning on itself soon. Even if you wanted to join Wes's coalition they don't want you. Perhaps if one of the teams in Kenny's alliance wins the reward challenge they can bargain with Wes's coalition and get a promise from Wes that they will not be thrown into a subsequent elimination challenge, but Wes's promise would be unenforceable and unlikely to be upheld. Plus Wes has already made an broken such a promise to Kenny. Perhaps if Kenny is eliminated, the teams remaining in Kenny's alliance and the teams on the outs in Wes's coalition could form a new alliance, but that could only happen if Wes completely mismanages his coalition.

Despite the chances of Kenny's alliance seizing control of the game being exceedingly small, sticking together is really the only option for the teams in Kenny's alliance. A couple of wins will prolong the amount of time until they are eliminated and a series of wins and a defection could put allow them to seize control of the game. It's a slim chance but it's still a chance. In my next post, I'll consider what the Wes's coalition should do going forward.

Fresh Meat II

2010-04-24T21:07:00.000-07:00

A couple of strategic decisions came up in the MTV show The Challenge: Fresh Meat II. For those who don't watch reality TV here's a quick run down. Thirteen teams of two compete to win prizes in a reward challenge. The winner gets a prize and is safe from the elimination round. At the end of the show, two teams are picked to go to an elimination round where the losing team goes home. One of the two teams that goes to the elimination round is picked by the winner of the reward challenge, the other is picked by majority vote. After nine teams have been eliminated, the final four teams get to compete in the final challenge with the top three teams winning money.

As you would expect in a voting situation, a coalition of teams joined together to form a majority so that they would control the voting round. Their optimal strategy is simple. Win the reward challenge and select the strongest team not in your coalition to go into the elimination challenge. Then vote the second strongest team not in your coalition into the elimination challenge. That way regardless of who wins the elimination challenge, one of your opponents goes home and your coalition stays strong. As an added bonus, a strong team is eliminated from the game and you have a better chance of winning the reward challenges and cash in the final challenge. This is what happened in the first two episodes when Wes's coalition formed a majority. They won the first two reward challenges and voted in their strongest competition in the first two elimination challenges. What happens if you lose the reward challenge and the winner selects someone in your coalition to go into the challenge? Vote in the weakest team in your opponent's coalition so your team has the best chance of winning the elimination challenge.

The teams not in Wes's coalition banded together to form their own alliance, lead by Kenny. So what is their optimal strategy? Obviously you want to win the reward challenge and send one of your opponent's teams into the elimination challenge. In the third episode, Kenny's alliance won the reward challenge. At that time, Wes's coalition had a majority with 6 teams and Kenny's alliance only had 5 teams. There was a lot of confusion in Wes's coalition about who Kenny would choose. Wes thought he would be picked because Kenny disliked him. The team Kenny ended up choosing--Eve's team--thought they were chosen because they were good at reward challenges. Neither of those decisions would be good strategically. Kenny knew that Wes's coalition would vote one of his teams into the elimination challenge. The best strategy for Kenny was to vote in the weakest team in Wes's coalition so that his alliance had the best chance of winning the elimination challenge.

Kenny knew that one of his alliance members would be voted into the elimination challenge. Kenny knew that if his alliance member won the elimination challenge against the team in Wes's coalition, then Wes would no longer have a majority in the voting (it would then be tied five teams to five teams). Obviously his best choice was to vote in the team in Wes's group that he thought was the weakest and that's what he did. Kenny chose Eve's team--part of Wes's coalition--to go into the elimination round. For some reason Eve thought that she was chosen because her team was good at the reward challenges. As I will explain, that makes no sense.

The next reward challenge will have 10 teams competing in it. The best chance Kenny's alliance has at winning the reward challenge is to have as many of his own teams in it as possible. The difference between Eve's team winning it versus any other of Wes's team is minimal. A simple example: let's say each team has a 10% chance of winning the next reward challenge. If Kenny's alliance has 5 teams in it, they have a 50% chance of winning it and Wes's coalition of 5 teams has a 50% chance of winning it. However if Kenny's alliance loses the elimination challenge, then he only has 4 teams remaining and a 40% chance of winning the reward challenge. So of course Kenny wanted to keep as many of his alliance teams in the game and voted for Eve's team because he thought they were the weakest team.

I'm not really sure what Eve was thinking. The only way for Kenny's team to seize control of the game is to win the elimination challenge and the subsequent reward challenge. Losing either of them puts Kenny's alliance at a huge disadvantage. Eve's team would have to have an extraordinary high chance of winning the reward challenge for Kenny to vote her in in order to eliminate her as a reward challenge threat. However if she was such a reward challenge threat, he would know that voting her in would mean that his alliance member would have a tough time winning the elimination challenge against her. The only way to rationalize voting Eve's team in to eliminate them as a reward challenge threat would be if they were also mediocre at elimination challenges.

How good would Eve's team have to be at reward challenges for Kenny to choose them for the elimination round? Hypothetically, let's say Eve's team only wins 60% of elimination challenges. Kenny chooses Eve's team for the elimination round. Kenny's alliance team wins this match 40% of the time and the subsequent reward challenge is 5 of Kenny's alliance teams versus 5 of Wes's coalition teams so Kenny's alliance has a 50% chance of winning. Kenny's alliance thus has a 20% chance of seizing control of the game (40%*50% = 20%). Instead let's say Kenny chooses the weakest member of Wes's alliance to go into the challenge. Kenny's alliance team now wins the elimination match 60% of the time. How good would Eve's team have to be at the reward challenge to make this a bad move by Kenny? Eve's team would have to be good enough at the reward challenge that Kenny's alliance would have a less than the 20% chance described above.

Let Wes's coalition chances of winning the subsequent reward challenge be x%. 60% * (1-x) < 20% . For that inequality to hold true, x must be 67% or greater. Now let's say that of the 10 teams in the reward challenge, the nine non-Eve teams have an equal chance of winning and Eve's team has a greater chance of winning. Let Eve's team chance of winning be 'e' and the the other 9 teams' chance of winning be y. The subsequent system of equation is:

4y + e = .67

5y = .33

Solving for e we get 40%. Eve's team would have to have a 40% chance of winning the next reward challenge for it to make sense for Kenny to put vote her into the elimination round instead of voting the weakest team into the elimination round. Considering there are 10 teams competing in the reward challenge, it is highly unlikely that one of them would have a 40% chance of winning the reward challenge. Kenny picked Eve's team to go into the elimination challenge because he wanted to maximize his alliances chances of winning the elimination challenge not because of the subsequent reward challenge.

Quick Response to Fangraphs War Article

2009-10-07T16:20:00.001-07:00

A response to this article on Fangraphs.

WAR has an R^2 of .83, but what does that really mean? Some context (all stats from ESPN.com):

ALwins ALera ALba ALops ALrbi

ALwins 1.000 -0.642 0.588 0.704 0.672

That is how American League wins correlates (little "r" not R^2) with ERA, Batting Average, OPS, and RBI. And the NL:

NLwins NLera NLba NLops NLrbi

NLwins 1.000 -0.751 0.369 0.503 0.572

I made three linear models for each league to find the R^2 of wins versus ERA and BA, ERA and RBI, and ERA and RBI. The R^2 values are .84, .86, and .87 respectively for the AL. For the NL .61, .82, and .81. Below are the print outs from R.

Call:

lm(formula = ALwins ~ ALera + ALba)

Residuals:

Min 1Q Median 3Q Max

-7.6862 -2.7943 -0.7403 1.9202 9.0998

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -49.532 46.485 -1.066 0.309461

ALera -24.762 4.224 -5.862 0.000109 ***

ALba 907.270 166.522 5.448 0.000201 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.265 on 11 degrees of freedom

Multiple R-squared: 0.8414, Adjusted R-squared: 0.8126

F-statistic: 29.18 on 2 and 11 DF, p-value: 3.995e-05

> summary(al2)

Call:

lm(formula = ALwins ~ ALera + ALrbi)

Residuals:

Min 1Q Median 3Q Max

-8.888 -2.575 1.766 3.284 5.192

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 96.20702 22.96652 4.189 0.001513 **

ALera -22.33402 3.96918 -5.627 0.000154 ***

ALrbi 0.11425 0.01941 5.885 0.000105 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.971 on 11 degrees of freedom

Multiple R-squared: 0.8586, Adjusted R-squared: 0.8329

F-statistic: 33.4 on 2 and 11 DF, p-value: 2.124e-05

> summary(al3)

Call:

lm(formula = ALwins ~ ALera + ALops)

Residuals:

Min 1Q Median 3Q Max

-8.8988 -2.1200 0.6058 2.9829 8.0973

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) -6.139 34.441 -0.178 0.861764

ALera -21.489 3.783 -5.681 0.000142 ***

ALops 240.741 38.383 6.272 6.07e-05 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 4.734 on 11 degrees of freedom

Multiple R-squared: 0.8718, Adjusted R-squared: 0.8485

F-statistic: 37.41 on 2 and 11 DF, p-value: 1.239e-05

summary(nl1)

Call:

lm(formula = NLwins ~ NLera + NLba)

Residuals:

Min 1Q Median 3Q Max

-9.467 -4.923 -1.258 3.634 12.506

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 69.111 70.851 0.975 0.34714

NLera -16.635 4.173 -3.986 0.00155 **

NLba 312.338 251.322 1.243 0.23590

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 7.43 on 13 degrees of freedom

Multiple R-squared: 0.6113, Adjusted R-squared: 0.5515

F-statistic: 10.22 on 2 and 13 DF, p-value: 0.002150

> summary(nl2)

Call:

lm(formula = NLwins ~ NLera + NLrbi)

Residuals:

Min 1Q Median 3Q Max

-7.950 -3.843 1.426 3.937 5.915

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 85.42980 19.82645 4.309 0.000849 ***

NLera -16.63429 2.77925 -5.985 4.55e-05 ***

NLrbi 0.09444 0.02190 4.311 0.000845 ***

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.042 on 13 degrees of freedom

Multiple R-squared: 0.821, Adjusted R-squared: 0.7935

F-statistic: 29.82 on 2 and 13 DF, p-value: 1.39e-05

> summary(nl3)

Call:

lm(formula = NLwins ~ NLera + NLops)

Residuals:

Min 1Q Median 3Q Max

-7.792 -3.753 1.456 4.205 5.731

Coefficients:

Estimate Std. Error t value Pr(>|t|)

(Intercept) 2.526 39.070 0.065 0.94944

NLera -17.588 2.853 -6.164 3.41e-05 ***

NLops 204.851 50.095 4.089 0.00128 **

---

Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 5.198 on 13 degrees of freedom

Multiple R-squared: 0.8098, Adjusted R-squared: 0.7805

F-statistic: 27.67 on 2 and 13 DF, p-value: 2.065e-05

Yet Another Way to Look at Free Throw Percentage

2009-04-26T06:28:00.000-07:00

John Branch wrote an article in the New York Times noting that in the past 50 years, Free Throw Percentage (FT%) has not improved in the NBA. The Freakonomics blog posted a follow up to this article that suggested that free throw percentage among the best Free Throw (FT) shooters has improved during that period. Unfortunately, I read this post and wasted the best weather of the year fooling around on R rather than going outside and getting sunburnt.

Ashley Smart, the person who supplied the data and analysis the blog post is based on, seems intelligent and probably knows more about stats than I do. The data she has for her analysis is deeply flawed. The biggest mistake she made was getting the data from N.B.A. Encyclopedia. The place for NBA data is Basketball Reference. You can download several spreadsheets of NBA data stretching back to 1946. As several commentators noted, Ms. Smart's other mistake is not controlling for the number of players.

That said there are two major flaws in Ms. Smart's analysis:

The NBA Encyclopedia is arbitrary and inconsisent. The data collectors changed their standards on who qualified as a top 20 FT shooter 14 times between 1950 and 2007. The biggest supposed gain in FT% in Ms. Smart's study occurs between the 1972 and 1973 seasons when qualification standards go from 350 Free Throw Attempts (FTA) to 160 FTA.
The study does not account for the number of players in the league. Of course the top 20 players in the league now are going to have a better FT% than those in 1950. In 1950 there were 135 players, in 2007 595 players. If I took a group of 100 random people and made them shoot a bunch of Free Throws and then took a group of 500 people and made them shoot a bunch of Free Throws which group do you think would have the best average among the top 20 Free Throw shooters?

I corrected the first problem by getting my data from Basketball Reference. Anyone who had 100 FTA qualified for my analysis. Let us take a look at the Top Twenty players in FT% versus the number of players in the league.

And a look at the number of players by year:

For what it's worth, there is an R^2 value of .82 for Number of Players vs. Top Twenty FT% and an R^2 value of .76 for Year vs. Top Twenty FT% for fitted linear models. The two models have too much serial correlation to be taken seriously. Year and Number of Players has a .93 correlation value.

A hopefully correct method for examining the best FT shooters is to look at the Top Tenth Percentile of FT% for each year:

There appears to be a slight increase in the FT% of the top tenth percentile but once again there is serial correlation. A Durbin Watson test confirms this; D-W Statistic of .978--anything below 1.38 for this sample size is suspicious--with a p-value of 0 and estimated rho value of .51. I did a basic transformation of the Year (X) and Top Tenth Percentile FT% (Y) variables:

Ytransformed = Yi+1 - rho*Yi

Xtransformed = Xi+1 - rho*Xi

I then fit a new model of Xtransformed verus Ytransformed. The D-W statistic was 1.89 with a p value of .58 and an estimated rho of .05, thus removing the serial correlation. A summary of this model:

With such a poor fitting model (R^2 of .1056) the data does not really explain anything. If I were going to make an inference I would point out the underlined value of .0004034 (b1). The regression coeffecient for X for the transformed model will remain the same when the model is tranformed back to Y = b1*X + b0. So over the 57 years of data, 57*b1 = 2.3% increase in the Top Tenth Percentile FT%. This is really stretching the model but there may be a slight increase (2.3% over 57 years) among the best FT shooters.

More Poker Odds

2008-09-20T22:34:00.000-07:00

I decided to finish up the poker post I had earlier. After I had calculated a few probabilities, I found a wikipedia site which covers the topic in a much more thorough manner. Last post I calculated the odds of having these starting hands: a pair (5.9%), suited connectors (3.9%), two cards both a 10 or higher (14.3%), and or of having any of these hands (20.6%).

How do these hands relate to the starting hands the other people at the table have? Using the binomial distribution and the hypergeometric distribution, I came up with this table:

The first column is the number of people at the table who have that hand, assuming that there are 10 people at the table. The Pair column says that the probability of exactly two people having a pair is 9.6%. The Ace column says that about 86.7% of the time at least one ace is dealt to the table and about 50% (34.8+13.5+1.8) of the time two or more aces are dealt to the table. So having an ace 2 for a starting hand means that it is about even odds that you do not have the best hand at the table with an ace in it.

The most useful part of the wikipedia page is the section detailing the approximation of hitting outs. Say that you have two spades and the flop gives you two more spades. That leaves 9 more spades in the deck or 9 outs to get a flush. Simply times the number of outs by 4 to get the odds of getting a flush by the river. 4*9 = 36 so 36% chance. This is an approximation with the actual probability being 34.96%. For 10 or more outs after the flop the formula is 3x+9 with x being the number of outs. The approximation on the turn is 2x (or the better approximation of 2x+(2x/10)). Pretty useful and gives you a handy way of calculating pot odds during a hand.

Before the flop what are the odds of improving the starting hands that I mentioned above? For a pair there is roughly a 11.5% chance of getting three of a kind on the flop, 15% by the turn, and 18.5% by the turn. Chance of four of a kind is .24%, .49%, and .82% respectively. If you have two cards of the same suit it is about a .8% chance of getting a flush on the flop, 2.8% of getting it by the turn, and 5.8% of getting it by the river. The odds of getting a straight or pairing one of your cards can be found using the post-flop approximations.

Some Numbers from the 2008 Olympics

2008-08-10T20:07:00.000-07:00

During the opening ceremonies, the US broadcast displayed a graphic with the size of the delegation and population of the country. I wondered what the correlation was between population and the number of athletes competing and what other variables might influence the delegation size. In addition to population GDP, climate, and some metric measuring civil rights for women might also explain delegation size. When I went to research this I had trouble finding the country and delegate data. What I did find was a list of all the athletes online. I turned the data into an excel file and played around a little with the data. The file can be downloaded here if anyone wants it. It is a .csv, which makes it easy to inport and analyze with R.

I created some pivot tables in Excel. It is possible to do similar things in R with the tapply() function but Excel makes pivot tables so easy to create and alter I did not bother. I uploaded them to a Google spreadsheet and embedded a few of them at the bottom of this post. The rest of the data I mainly found using R. There are 204 countries competing in this Olympics. The largest delegation belongs to the United States, with 618 athletes. 10 countries have one athlete competing: Arba, Belize, Burundi, Central Africa Republic, Dominica, Gabon, Niger, and Nauru (which was featured in a surreal This American Life episode). The mean delegation size is 49.24 athletes. Surprisingly, the median delegation size was only 9. Roughly half the countries send 9 or fewer athletes. Random fact: Only 27 of the 204 delegations have more women competing than men. Which two countries have the largest female-positive (ie more women than men) delegation?

Another column of data on the offical site listed the disciplines (sports) which athletes compete in. There are 38 discipline classifications. The most competed in discipline is Athletics (Track & Field I would guess) with 1943 competitors. Cycling BMX is the smallest event with only 24 competitors. The median and mean are 182.5 (between Baseball and Table Tennis) and 264 respectively. Random fact: there are five sports that are specific to only one gender. Which ones are they?

A little manipulation with R turned the Date of Birth data into Year of Birth and then into an 'age estimate' where I took 2008 and subtracted Year of Birth to get current age. The oldest athlete is Hoketsu Hiroshi, a 67 year-old man representing Japan in Equestrian. The youngest competitor is 12 year-old swimmer Antoinette Joyce Guedia Mouafo from Cameroon. The Median and Mean ages estimates are 26 and 26.37 years. The Random Fact was going to be the average oldest and youngest delegation, but Excel started acting up and I was a bit tired to do more R (date modification can be tricky). There is plenty to explore in this data. Let me know if you find anything neat in the above file and don't be afraid to add more columns of data.

Answers: Norway and Sweden; baseball, softball, boxing, synchronised swimming, rhythmic gymnastics

Hoops Analyst and R

2008-07-26T20:13:00.000-07:00

Rather than write another convuluted post on poker and probability, I decided to play around a bit with the statistics program R. A post by Harlan Schreiber at the much enjoyed site Hoops Analyst and a recently purchased book on R inspired the effort. I highly recommend R and this post will demonstrate some of the things one can do in R, even one inexperienced in R and statistics such as myself.

Mr. Schreiber explores the thinking that defense wins championships and offense is not as important. He presents a table of Offensive and Defensive Ranks of past champions that shows that they are equally important. The data leads to a simple analysis and conclusion: defense and offense have been equally important for past champions. Mr. Schreiber still manages to make the article interesting by adding depth and insight to a simple table. He can find history and anectdotes in the driest of data.

Statistical analysis can support Mr. Schreiber's conclusion. A paired t-test and sign test are two simple ways to compare data. I copied the data into Excel and then loaded it into R. I added two more columns to the data table. One labelled "Difference," which is Offensive Rank minus Defensive Rank. The second column I labelled "Sign" and it assigned a 1 for values of Difference that were positive and 0 for Differences that were not. To perform a paired t-test the data has to be normally distributed. A Schapiro test for normality (R command>schapiro.test(Difference))on Difference produced the following result:

Shapiro-Wilk normality test
data: Difference W = 0.9682, p-value = 0.5107

Typically we look for a p-value <>qqnorm(Difference) >qqline(Difference)) and histogram test (R command>hist(Difference)) confirm this.

The pictures show a roughly normal distribution. A t-test can be used. Simply type t.test(Offensive.Rank,Defensive.Rank,paired=T) and R does the rest of the work:

Paired t-test
data: Offensive.Rank and Defensive.Rank t = 0.1257, df = 28, p-value = 0.9009

alternative hypothesis: true difference in means is not equal to 0 95 percent

confidence interval: -2.637672 2.982499

sample estimates: mean of the differences 0.1724138

With such a high p-value, we fail to reject the null hypothesis. There is no proof that Offensive or Defensive Rank has been statistically more significant for prior champions.

Another possible test is the non-parametric sign test. This test may be preferable to the t-test in this instance since the data we are working with are ranks rather than continuous variables. Like many non-parametric tests, the sign test has fewer necessary conditions and does not require the data to be normally distributed. A simple binomial test can be used. A binomial test works much the same way that the binomial probability function does. A binomial probability function calculates the chances that there will be k successes in n trials. For example, the binomial probability function can tell you the chance of getting 2 heads in 10 trials. To use binomial, the outcome of the test must only have two outcomes. Heads or tails, success or failure, infected or not infected, ale or bad beer and so on.

So how can we apply the principals of binomial probability to our data? That is the purpose of the Sign column explained earlier. All Differences > 0 were labelled as successes and assigned a value of 1. Running the binomial test in R is simple > binom.test(14,27) where 14 is the number of successes and 27 is the total number of trials (29 trials - the 2 trials where the Differences = 0). I typed binom.test(sum(Sign),length(Sign)-2) which will make sense as you familiarize yourself with R. The result:

Exact binomial test
data: sum(Sign) and length(Sign) - 2

number of successes = 14, number of trials = 27, p-value = 1

alternative hypothesis: true probability of success is not equal to 0.5

95 percent confidence interval: 0.3194965 0.7133275

sample estimates:probability of success 0.5185185

So such an event as our data would occur roughly 51.8% of the time. There is no reason to reject the null hypothesis. Once again, the data does not support the claim that defense has been more important to past champions than offense. How many success would we need to get a p-value less than .05? 頑張って R-さん。 The command is > qbinom(.975,27,.5), which returns a value of 19. 19 successes would be needed for there to be statistically significant difference between Offensive and Defensive ranks. For the heck of it here is a box plot and summary of the data.

The Basics of Poker

2008-07-20T23:53:00.000-07:00

No post last week. The author was a bit worried that someone had give Martin another box of silver, but in fact it was the something quite different. Enough personal stuff, onto the math.

Say one was to go play a game of Texas Hold'em. What should the player know? Blackjack and craps have fairly simple strategies to follow to maximize expected play, or rather to minimize expected losses. For blackjack all someone has to do is remember a simple table. Poker is different. The basic requirement is to know the rules. What hands beat which hands, the order in which people bet, and generally how the game unfolds. Once the basics are learned, the two most important concepts are the Fundamental Theorem of Poker and bluffing. The fundamental theorem dictates that a player play his hand as if he could see everyone's cards and always applied the correct pot odds. Every time a player does this he increases his expected gain. Every time a player fails to do this, he is losing money.

Bluffing adds depth to poker. A player cannot follow the fundamental theorem perfectly. He can only guess what his opponents have. If a player plays his hand based on the fundamental theorem and never bluffs, his opponents will be able to accurately guess what cards he has and put him at a disadvantage. The skill in poker comes from balancing the concepts of the fundamental theorem and bluffing.

Those are the basics. Next it is important to get a sense for how often certain hands occur. Playing repeated hands of poker will give a player a good feel. A player gets two cards before he has to decide whether to play the hand or fold. What types of hands should a player ante up for and how often do those hands occur? Let us say a player likes to play the following types of hands: a pair, cards of the same suit that are adjacent (suited connectors), and two high cards (two cards that come from the set of 10, Jack, Queen, King, and Ace). Those are generally regarded as good hands to ante up on as they can lead to strong hands. Let the sets be defined as A for pairs, B for suited connectors, and C for high cards. The probability of getting one of those hands is the union of those three sets:

P(A U B U C) = P(A) + P(B) + P(C) -[P(AB) + P(AC) + P(BC)] + P(ABC)

Where 'U' indicates a union of sets and 'AB, ABC etc.' indicates an intersection of sets. For unions of sets, the basic rule is to add the sets of odd intersections and subtract the even intersections.

P(A) = probability of getting a pair. The first card can be anything. For a pair to occur the second card had to be one of the 3 remaining cards from the deck of 51. The first card in P(B) can be any card. For any card, there are two remaining cards that are suited connectors. If the first card was the Ace of Spades, the second card has to be the King or 2 of Spades. P(C) can only have a 10, Jack, Queen, King or Ace for its first card and second card. There are 20 such cards in the deck and 19 remaining after the first one has been dealt.
P(A) = (52/52)*(3/51)
P(B) = (52/52)*(2/51)
P(C) = (20/52)*(19/51)

The intersection P(ABC) and P(AB) cannot occur since two cards cannot be both a pair and suited connectors. P(AC) is the set of cards that are pairs of 10, Jack, Queen, King, or Ace. There are 13 different types of cards and P(AC) makes up 5 of those types (10, 10; Jack, Jack etc). P(AC) can be expressed as the portion of A that falls into C. Likewise with P(BC) with the caveat that only 4/13 types of suited connectors of B occur in C (since 10, 9 and Ace, 2 do not occur in C).
P(AC) = P(A)*(5/13)
P(BC) = P(B)*(4/13)

Plugging those into the formula:
P(A U B U C) = 20.6%
Thus a player that plays these types of hands will play roughly one of every five hands. To keep opponents from getting a clear read on what type of hands you prefer it may be advisable to play a junk hand occasionally.

I like playing suited connectors and pairs because if the next five cards improve your hand, a clear advantage can emerge. For example, if you play a suited connector and three of the next five cards are the same suit you get a flush. Unless there is a pair among those five cards, a flush will very likely be the best hand. But given that you played a pair or suited connector, what are the odds that the next five cards will improve your hand? I will attempt this question in the next post.

The Gauntlet Revisited

2008-07-06T19:29:00.000-07:00

An earlier post described how a player in the TV show The Gauntlet III could calculate his odds of winning a Gauntlet and who he should select as an opponent. This post will explore team strategy, in particular why the men of the Veteran team decided to purposely lose team challenges.

If a team loses a challenge, two of its members have a duel in the Gauntlet. The losing player has to leave the show. After an unknown number of Gauntlets, the show has one final challenge in which the two teams compete for $300,000. The winning team divides the $300,000 equally between the remaining members. Going into the team challenges, the participants know whether it is a "guys'" or "girls'" day. On guys' days two men from the losing team duel in the Gauntlet. On girls' days two women compete. On guys' days the women face no punishment (i.e. the Gauntlet) for losing, and the men face no punishment for losing on girls' days. To entice the opposite sex to compet on their respective days, the show offers a prize to the winning team. On girls' days, men of the winning team each get a prize of roughly $500. However, if a team loses then one of its members will leave the show, meaning a larger portion of the $300,000 grand prize will go to those who remain.

First let's take a look at how much each remaining member stands to gain when a teammate loses in the Gauntlet. The left column is the number of people remaining on the team, the middle column is how much each member will receive if the team wins the final challenge, and the right column is how much more a player gets when a teammate leaves the show.

Teams start with 16 people and every time someone leaves, the remaining members can potentially win more prize money. The 'Difference' column shows that the more people leave, the more the rest stand to gain. When the first person leaves, everyone stands to win $1,250 more. When the 10th person leaves, the team members can win over $7,000 more. For example, in the final challenge the Veteran team stood to win roughly $30,000 each and the smaller Rookie team $60,000.

Let's take a look at the normal form representation of a single stage of this game. Let us assume that it is a girls' day and we are looking at the representation from the men's view. Assume that the men think they have a 50% possibility of winning the final challenge, thus they stand to gain $625 if they lose the challenge and a teammate (1250*.5 = 625). The men from both teams have the same strategy set: Try or Shirk. A team that chooses to Try will win 100% of the time against a Shirking team. Or perhaps more accurately, the Shirking team can make sure they lose with 100% certainty. If both teams Try they each have a 50% chance of winning, if both teams Shirk they each have a 50% chance of winning.

Thus Shirk dominates Try, since the chance of $1250 ($625) is greater than the $500 prize for winning the challenge. In the next stage game, the team that lost a member has a chance to win even more than $625, since the figures in the Difference column grow as more people leave the show. Using the binomial theorem one can calculate how much present value one gains by having the chance to lose more teammates. In the first stage matrix, the $625 jumps to over $3,000 in present value. The question is not why the Veteran men decided to Shirk challenges, but why didn't they start Shirking earlier?

The payoff matrix shows that the $500 prize is no deterrent. Are there other deterrents to Shirking? The final challenge determines the grand prize. If teams with more members had an advantage in that challenge, then that would be a deterrent to Shirking. However the format of the final challenge, common knowledge to the players because of previous versions of the show, does not favor large teams. It favors teams that do not have weak links and ones with strong athletes. Many Veteran men purposely Shirked because they thought it would improve their chances of winning the final challenge. The women realized that their probability of winning the final challenge would decrease if they lost strong athletes and were less likely to Shirk. A third deterrent might be an emotional reason. Perhaps pride, a competitive nature or a connection to members of the opposite sex motivated players not to Shirk. That might explain some of the romantic relationships.

The only tool the women have to deter Shirking is threat of reciprocal punishment. "If you Shirk and send one of us to the Gauntlet, we will Shirk and send one of you to the Gauntlet tomorrow." This strategy is undermined since losing too many athletic members hurts everyone on the team and because the remaining women also benefit from losing weaker members. By Shirking and losing strong men, the women hurt themselves. By losing the remaining women, they stand to make more money, and have a better chance of winning the final challenge. The Veteran men chose to Shirk, and the women had little recourse.

If both teams' men had realized the dominance of Shirk, there would have been some interesting repercussions. Both teams' men would be trying to lose on girls' days. To try to deter this, both teams' women might play the 'Mad President' strategy (act irrational-not a stretch for this group) and try to Shirk, the final challenge be damned. The Nash Equilibrium would probably involve an agreement between the men and women on each team with both sides agreeing that a certain amount of Shirking, to lose the weaker male and female competitors, is ideal and to only Shirk strategically in order to maximize the probability of winning the final challenge.

Late Goals in Euro 2008

2008-06-29T19:35:00.000-07:00

While watching Spain defeat Russia in the semi-finals of the Euro 2008 soccer tournament, a stat flashed on the screen. 23 out of the 79 goals in the tournament were scored after the 75th minute. Disregarding overtime and stoppage time, the 75th to 90th minute is about 1/6 of the game. So one would presume that 1/6 (roughly 17%) of the goals would occur in that interval. But in fact 23/79, roughly 30%, of the goals were scored then. Is this statistically significant?

Let p^ = the sample statistic (23/79) and p(o) = the expected population statistic (1/6). Let the α =.05 be the threshold. If the discovered p-value is < .05 we reject the null hypothesis. The null hypothesis is that p^ = p(o). For this let's use a One-proportion z-test.

At this point in the tournament there had been 29 games, so let n = 29. p^ = p(hat) (I am unsure how to type hats or subscripts), p(o) = p and z-score = z. Plug the values into the formula to get the z-score... the z-score = 1.79, making the p-value < .05 so we reject the null hypothesis.

Does it make sense that such a high percentage of goals would be scored in the final 15 minutes? It brings to mind an analysis I read about the US presidential elections. The author asserts that the trailing candidate should pursue the "pull the goalie strategy" used in hockey. The trailing hockey team is so desperate to score, the coach pulls the goalie and puts a better goal score into the game. By pulling the goalie the team expects to have more chances to score but will certainly be easier to score upon. The team that is trailing wants to utilize a strategy that increases the overall variance at the expense of the optimal strategy. Over the long-term this high variance strategy is not as effective as the optimal strategy, but the trailing team is not playing for the long term. It is playing for the short term. The trailing team will use a high variance strategy, resulting in it and the opposition scoring more goals.

This was evident today in the Euro 2008 finals. Germany pursued the high variance strategy in the final 15 minutes at the expense of the optimal strategy. The German team had more chances to score, but at the same time allowed the Spanish side some great opportunities.

Which Half is the What Now?

2008-06-29T18:27:00.000-07:00

The New York Times posted an article on the questionable efficacy of cardiac CT scans. There is a suprising quote from the website of doctor who supports CT scans. "Half of Americans have died of heart attacks and strokes. Which one are you?" This statement is absurd. Rather than dissect it gramatically, I am unsure why the NYTimes would include it, it can be taken apart with logic and algebra. There are about 300 million Americans alive right now. Let x equal the amount of Americans who have died of heart attacks or strokes (I assume the doctor meant 'or' (the union of two sets) not 'and' (the intersection)) and let y equal the amount of Americans who have died of anything else. Making x + y = the total amount of Americans who have died. From that quote we know that:

x / (x+y+3.0*10^6) = 1/2
2x = x + y +3.0*10^6
x - 3.0*10^6 = y

This tells us that x > 3.0*10^6 and x > y, assuming that x + y > 0. The percentage of dead Americans who have died of heart attacks or strokes is equal to x/(x+y). Since x > y, x/(x+y) > 1/2. For example, suppose that 500 million Americans have died. Plugging that number into the formula:

x/(5.0*10^6+3.0*10^6) = 1/2
x = 4.0*10^6
x / (x+y) = % of dead Americans who have died of heart attacks or strokes
4.0*10^6 / 5.0*10^6 = .8 or 80% of dead Americans have died of heart attacks or strokes

I am unaware of what twisted game the doctor is playing. He is trying to emphasize the risk of heart attacks and strokes with the "1/2 comment", when he could quote the x / (x+y) = % of Americans who have died of heart attacks or strokes. The latter is larger than 1/2. A mystery of a man who is trying to shock the public and undersell the problem at the same time.

On a less mathematical note, there are worse things than heart attacks or strokes to have as the leading cause of death in a society. I am reading the book Chances Are... . I am surprised by how short the life expectancy for Londoners was in prior centuries and how resistant to statistical studies the medical community was. Unfortunately, as the NYTimes article details, there are still resistance to evidence based medicine.

When to Foul the Shooter

2008-06-20T19:48:00.001-07:00

The conventional basketball wisdom is to foul the shooter rather than give up an easy lay-up. If a shooter has an easy shot, say a shot he makes 95% of the time, it is better to foul him and let him attempt two free throws rather than take the easy shot. As long as he does not make over 95% of his free throws (which almost nobody does), the defensive team will allow less expected points. If the offensive player only makes 50% of his free throws, fouling him saves .9 expected points. 2*.95 - (1*.5+1*.5) = .9 . Fouling the league average shooter, who makes 75.2% of his free throws saves about .4 points. Fouling has another detriment that is not accounted for in that math. Once a team has commited five fouls, the other team goes to the free thow line for every subsequent defensive or loose ball foul, regardless of whether the player was in the act of shooting.

During the 2006-2007 NBA season, teams scored about 1.1 points per possession and made 75.5% of free throws. If the offensive team is in the bonus, a non-shooting foul results in the defense allowing .43 more expected points than they do on an average possession. 2*.752 - 1.1 = .43. It is better to let the possession elapse without commiting a non-shooting foul. The defensive team would rather not be in the bonus. Should you still foul the shooter on an easy shot?

Before estimating the additional penalty that committing a foul detracts, here are some statistics:

Free throw % and the other stats needed to calculate Points Per Possession and Mean Fouls were found here. Points Per Possession was found by taking the league average Points Per Game, and dividing by league average 'Pace' statistic. PPG/Pace = PPP. I calculated Mean Fouls by taking the league average minutes per season, divided by 5 to make the stat minutes per team, and divided it by the league average fouls per team to get team fouls per minute. I multiplied it by 12 (minutes in a quarter) to get fouls per quarter. (Minutes/5)/(Fouls per Team) * 12 = FPQ.

I will take an extreme case to see if there is an instance where it would be better to let the offensive player have any easy shot, rather than foul. Suppose that the offensive player has an easy shot at the start of the quarter. Should he be fouled? The Poisson Distrubution, can be used to show how often a team reaches a certain number of fouls.

In this case λ = 5.51 (average fouls per quarter) and k = fouls in a quarter. On the first line of the table below is the random variable k, ranging from 0 to 14. The percentage below it the Poisson probability of that exact amount of fouls happening in a quarter. For example, the most likely outcome is a team commiting 5 fouls in a quarter, which happens 17.1% of the time. Below that number is 'Points Lost.' Points Lost is the amount of points the defense loses by fouling a player and letting him shoot free throws. As shown above, by fouling the defense allows .43 more points than they would if it did not foul. Multiplying it by the Poisson probability gives the points lost. For example, by fouling 5 times in a quarter, on the 5th foul the team goes to the free throw line and gets .43 more points. This happens in 17.1% of quarters. The penalty increases as the team fouls more often. Fouling 6 times a quarter send the opposing team to the free throw line on two occassions, allowing the offensive team to get .86 more points per quarter. The formula is .43*Poisson%*(k-4) = Points Lost. 4 being used because every time a team commits k > 4 fouls the opposing team goes to the free throw line k-4 times. Total Points Lost is the sum of expected Points Lost for each value of k.

To return to the extreme case, fouling at the beginning of the quarter has the effect of the offensive team needing to draw only 4 more fouls (rather than 5) in order to shoot free throws. Compared to the previous example, this increases the Total Points Lost by the defense. Committing 4 fouls results in Points Lost and the penalty for committing more fouls increases. For this case, Points Lost = .43*Poisson%*(k-3).

If the difference between Total Points Lost (Early Foul) and Total Points Lost (normal case)increases by more than .4 points, the amount of points the defense saves by fouling the league average player on an easy shot, then the defense would be better off not fouling. The table below shows the comparison:

The difference in Total Points Lost is .343 which is < .4. Although the defense's Total Points Lost increases with the early foul it does not increase enough to justify letting a player have an easy shot. Thus the conventional wisdom is reaffirmed for the league average player; foul him rather than let him have an easy shot. If Total Points Lost had been more than .4, the next step would have been to calculate a more accurate Total Points Lost by accounting for offensive fouls (which do not result in free throws) and shooting fouls (which always result in free throws). However that is not needed and the reader will never learn that 9.8% of all fouls committed during the 2006-2007 NBA season were offensive fouls.

While all I did was reaffirm the conventional wisdom, I believe the data is suggests that there may be special cases when the defense should not commit the early foul. If the offensive player makes easy shots somewhat than 95% of the time and makes free throws somewhat more than 75% of the time, it may be worth not commiting an early foul on him. NBA teams with access to more exact stats would be advised to, especially during 7 game playoff series, calculate some exact figures for specific players to see who and when not to foul.

Bringing Math to the Gauntlet

2008-06-15T08:07:00.000-07:00

Real World/Road Rules Challenge: The Gauntlet III is a reality show that I embarrisingly happen to enjoy. Two teams compete in random challenges. The losing team has to participate in the Gauntlet, a one-on-one duel where the winner gets to stay on the show and the loser leaves the show. The winning team selects the first player to enter the duel and the losing team chooses his opponent. Often the losing team would let the first player pick who he would face in the duel. I hope to show how a player can choose an opponent that will give him the highest probability of winning the Gauntlet.

To determine which game the players will play in the Gauntlet, a wheel with six outcomes (five different games and a spin again) is spun. Since all five games have an equal chance of being selected, Player 1 can calculate his odds by estimating his subjective probability of winning in each individual game, summing the probabilities and dividing by five (the number of games). For example, if Player 1 believes that he is evenly match with his opponent in all five games, his chances of winning are 50%, shown by the formula Pv:

Pv (Probability of Victory) = (.5 + .5 + .5 +.5 + .5)/5 = .5 = 50%

The games are varied and certain games favor certain types of players. It is unlikely that a player would estimate himself as evenly matched with his opponent in all five games. Force Field-basically tug of war with pulleys-favors body strength and weight. Ankle Breakers-reverse tug of war with a rope tieing a player to his opponents ankle-and Ram it Home-a shoving match of sorts also favor strength and body weight. Sliders is a puzzle game that does not require athletic ability. Ball Brawl-a race to grab and carry a ball across a goal line-gives the advantage to the faster player. Let's do another example. This time Player 1 can select a weaker, smaller Player 2, who is faster and smarter (one would assume giving Player 2 an advantage in the puzzle game) than Player 1. Player 1 calculates the games in his favor of giving him a 70% chance of winning and only 30% for the games that favor Player 2.

Pv = (.7 + .7 + .7 +.3 + .3)/5 = .54 = 54%

Another wrinkle. Ball Brawl is a repeated stage game. The winner is the first who scores 4 points. Repeated stage games, compared to a single elimination games, favor the team with a higher probability of winning. Much like the 7 game series of the NBA playoffs favor the better team more than the NCAA college basketball tournament does. If Player 1 believes that his chances of scoring in each stage game of Ball Brawl is 30%, his overall odds of winning the game drop to roughly 18%. The logic being it is easier for Player 1 to convert one 30% chance than it is to convert multiple 30% chances. The actual math can be calculated using the binomial theorem:

The winning player needs to score 4 points. There are five stage games in which to score points. In the first three stages, grabbing a ball and returning it over the goal line is worth 1 point. In the last two stages, a successful score is worth two points. The winning player needs to score 2 or 3 points in the first three stages and 2 points in the final two stages or to score 4 points in the final two stages. Say the probability of scoring in a stage game is .3. The function b(x) can be created to calculate the chance of winning ball brawl. b(.3) =

= .181 = 18% =

Or in Excel:

=BINOMDIST(2,2,.3,FALSE)+BINOMDIST(1,2,.3,FALSE)*(BINOMDIST(3,3,.3,FALSE)+BINOMDIST(2,3,.3,FALSE))

BINOMDIST(s, n, p, false) where s = number of successes, n = number of trials, p = probability of success for a trial, false = not cumulative

Formally, Pv = (p1 + p2 + p3 + p4 + b(p5)) / 5

where p1 = estimated probability of winning game 1, p2 = game 2, ..., p5 = prob of winning a stage in ball brawl

So what should Player 1 look for in an opponent? Since three games emphasize strength and body weight, Player 1's first criteria is to choose a weaker opponent. After that, the repeated stages of Ball Brawl, and the advantage it gives to the faster player, dictate choosing a slower opponent. The last criteria to evaluate is your potential opponent's intelligence. Big players will pick on small players and small players will choose weaker, slower, and or less intelligent small players.

More formally, the dominant strategy is to pick a person i ∈ S (set of of all players) for Player 2 such that Pv(i) ≥ Pv(j) for all players j ∈ S.

While I would be surprised if anybody on the Gauntlet reasoned his or her opponent selection out to this degree, it does explain why a player like Eric survived to the end of the competition. Eric was not suited for the final competition, but his sizable body weight and strength advantage made him an opponent no one wanted to face in the Gauntlet.