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.
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