Bringing Math to the Gauntlet

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.