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.
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