The Biggest Loserbarnes, Platner, Ridgway, Wantland

The Biggest LoserBarnes, Platner, Ridgway, Wantland

MBA 211 – Mid-Semester ProjectMarch 30, 2011

Background

The Biggest Loser is a reality television show that features obese people individually competing to lose the highest percentage of body weight. The American version of the show airs Tuesday evenings on NBC and features a $250,000 prize to the winning contestant that loses the highest percentage of weight among those still remaining after the final elimination round. An additional $100,000 prize is awarded to the “at-home winner,” i.e., the contestant that has lost the net highest percentage of weight among those who were eliminated. For the purposes of the games presented in this analysis, we assume contestants are only vying for the on-site $250,000 prize.

The current season began with 11 pairs of contestants (“couples”) comprised of siblings, spouses, parent/child, or friends. An initial weigh-in determined the baseline starting weight for each competitor. Two main groups were then formed on the basis of those who would reside on the show’s landmark ranch (the “Ranch” team) and those who were to reside offsite with a pair of mysterious new trainers (the “Unknowns” team). Couples were placed on the same team and could not be divided at this point in the game.

Each week of the show, contestants undertake a variety of workout and training sessions with the show’s personal trainers to expedite their weight loss. These workouts are interspersed with segments in which the trainers verbally draw out emotional issues carried by the contestants. Tears typically accompany these vignettes, either on the part of the contestant or from the home-viewing audience. At the conclusion of each week, contestants are individually weighed. The team with the lowest percentage of cumulative weight loss is required to eliminate one player via an open-ballot vote. The contestant on the losing team that has the highest percentage weight loss for the week has immunity and cannot be eliminated in that week’s vote.

Elimination Voting Equilibriums

As explained, every week one contestant is voted off of the on-site competition. During the portion of the game when contestants compete in teams, the team that lost the least percentage of their cumulative body weight must vote off one member of their team. Voting is public, so all members of the team know who votes for whom. There are two basic strategies to choose from:

• Dove: The cooperative strategy is to vote off the weakest performer on the team — someone who hasn’t been putting in as much effort as everyone else to lose weight.
• Hawk: Looking forward, you realize that each elimination round is an opportunity to vote off the person who is most likely to beat you in the individual competition.

To model this game, we can view the game as having two players: you and the other members of your team. You are in a simultaneous, repeated game and there are four distinct outcomes of each vote. The team could vote off the strongest or the weakest player of the team, and you could choose to vote with the team or against them.

The preferred strategy depends on the timing within the game. At the beginning of the game, players will prefer a dove strategy, but later in the game, a hawk strategy. To understand this change in preferences, we can look at players’ expected outcomes between the dove and hawk strategies in the first round of elimination voting, and in the final round just before the game turns into an individual competition.

Round 1:

Let’s assume the game starts with 20 people, 10 on each team. Each person would have 1/20th (5%) change of winning, but there is someone who is favored to win such that in any matchup between the favored player and any other player, the favorite player is twice as likely to win as anyone else. We will call this person the rival. Because the rival’s probability of winning is twice as high as any other player’s, we can calculate the expected valueof each player’s probability of winning as 1/(n+1) and the rival’s probability as twice this, 2*[1/(n+1)]. Accordingly, when there are 20 players in the game, each player’s chance of winning The Biggest Loser prize is 1/(20+1) = 4.8%, while the rival has a 2*4.8 = 9.6% chance of winning the $250,000 prize. Your initial expected value of playing the game is (1/21) * $250k = $11,904, and the rival’s expected value of winning is (2/21) * $250k = $23,810.

Dove: The objective of voting off the weakest player is making the team stronger overall. Let’s assume that if a team facing elimination votes off the weakest member, they are guaranteed not to have to eliminate someone in the next round. This means that two people (one from your team this round and one from the other team next round) will be eliminated, so your probability of winning increases as n drops from 20 to 18. The rival is still in the game, so your chances are (1/19) * $250k = $13,158. Thus, the value of the Dove strategy to you is the difference in the expected values before and after the vote, $13,158-$11,904 = $1,252.

Hawk: The advantage of voting off the rival is that each remaining player now has an equal share of winning. However, without the rival your team is weaker and more likely to have to vote off members of the team in subsequent rounds. To simplify, let’s assume that if you take the hawk strategy your team will absolutely have to vote off one of its players in the next round. Next round, there will be only 9 members left on the team, and you might the person who gets voted off. If you are as equally likely as any of the other 9 remaining members of your team to get voted off there is a (1/9) = 11% chance you get voted off, earning $0, and an 89% chance you make it two rounds and your rival is no longer a threat. If you do make it through the elimination round, your expected value of winning is (1/n) = 1/18 * $250,000 = $13,889. The weighted average of these two possibilities is 11% ($0) + 89% (13,889) = $12,361. Thus, the value of the hawk strategy is the change in your expected winning, or $12,361- $11,904 = $456.

Comparing these two options, the Dove strategy is more advantageous. This is primarily because in the first round of competition, with so many other people in the game, your rival is not your primary concern. Instead, you are most concerned by getting through the elimination rounds. As such, you should take the Dove strategy and vote off the weakest members of the team. The other members of your team would have the same preferences:

The Team
You / Dove / Hawk
Dove / +, + / + , -
Hawk / + , - / - , -

Thus, we would expect to see teams vote to eliminate the weakest member of the team, and the rivals to remain in the game. This, in fact, is exactly what we saw in the first rounds of voting this season (with the exception of Week 9, discussed in the next section of this paper). This also benefits the show’s producers, who aim to have the strongest player with the most stunning weight loss win.

Last Round of Team Play:

The situation changes later in the game, however. Say that we have gone through about half of the game, and only five members remain on each team. On your team are you and three other competitors that are equally as competitive as you are. There is also your rival, who has twice the probability of beating the other four members. There are 10 players total in the game; therefore n = 10. At the beginning of this round, the probability of you or any of the other three members of your team wins is (1/11) = 9% and the expected value is 9% * $250,000 = $22,727. Your rival has a 2/11 chance of winning, or 18%.

Dove: At this point, voting off the weakest member of your team does not help you as much. It would only mean that at the end of the “team” portion of the game, there is one less person that you compete with in the “individual” portion of the game. Thus, after taking the dove strategy, your expected winnings would be 1/(n+1) = 1/10 = 10% *$250,000 = $25,000. The increase in your expected value from taking this strategy is $25,000 - $22,727 = $2,273

Hawk: Voting of your rival is very attractive, because from this point on your rival is your primary threat. By voting off your rival, there will only be nine players left at the end of the round, all of whom have an equal probability of winning. Therefore, your probability of winning is 1/n = 1/9 = 11% *$250,000 = $27,778. Thus, the gain from this move is$27,778 - $22,727 = $5,051. At this point, we see that the Hawk strategy is far more attractive for you and the other team. This is primarily because the strength of your team is no longer important to you after this round.

The Team
You / Dove / Hawk
Dove / - , - / - , +
Hawk / +, - / + , +

Your dominant strategy for each player is now hawk, which means the new equilibrium would involve the rival getting voted off of the show. This analysis predicts that at some point, when players begin to suspect that the team-play portion of the game is coming to a close, they will start to vote off stronger rather than weaker players. When this happens will depend on each individual’s ability to look forward and reason back. While this is a predictable outcome, it is a problem for the producers of the show, because the solidarity within teams is part of the “feel good” element of the game. Moreover, the producers would like the person with the highest weight loss to win the game because of the shock value of their weight loss. (Recent winners have lost approximately half of their body weight.)

To avoid this outcome, the show’s producers keep secret the information about when the game will transition from being a team-based competition to an individual one. This technique has been successful, as past seasons reflect that contestants are consistently caught off-guard when the transition actually occurs. We suspect this stems in large part from the long length of the show, which allows producers to make the transition happen at various points from season-to-season, which in turn makes the transition difficult for contestants to predict.

Coalition Formation and Maintenance

In this season of The Biggest Loser, coalitions within teams are born primarily out of historic loyalties. During Week 6 of the show, the two groups are brought together to the Ranch and are formally divided into two teams based on the initial groupings from Week 1; the Ranch team officially becomes the Black team, while the Unknowns team officially becomes the Red team. Between Weeks 6 and 9, some players are shuffled around as the result of several challenges, and coalitions within each team begin to emerge as illustrated in the diagram below:

These coalitions become particularly important during elimination rounds. Because everyone’s votes are revealed at the same time, these elimination rounds are, in essence, a series of repeated, simultaneous mini-games within the larger competition. As discussed in the previous section, early on in the game, it is in each player’s best interest to retain the strongest players (those likely to lose the most weight in any given week) and vote off the weakest players to increase the probability of winning subsequent weigh-ins — i.e., a dove strategy prevails. Although payoffs change as the gametransitions from a team-based competition to an individual one, in order to illustrate the role of coalitions, wefocus here on a specific elimination round that occurred early enough in the game to implythe presence of a dove-dove equilibrium.

At the end of Week 9, the Black team won the weigh-in, leading the Red team to enter an elimination round. A player named Arthur, who had previously been on the Black team and is consequently outside of the Red team’s coalition, emerged as the second-strongest player on the team with the second-highest percentage weight loss. (Had he been the top player with the highest percentage weight loss, he would have been ineligible for elimination.) Per the dove-dove equilibrium, the dominant strategy for each team member would have been to vote off any member eligible for elimination other than Arthur. The game’s equilibrium did not emerge, however, as Arthur was ultimately voted off.

It was clear that the coalition had been the driving force behind the team members’ decision to eliminate Arthur. In order to determine exactly how the coalition motivated players to vote in the context of game theory, we first examined the role of threats, i.e., whether the threat of retaliation for voting against the coalition served to keep the coalition intact. Looking at the criteria of a “good” threat, however, we find that this threat was in fact a weak one:

• Detection: This is the one criterion that the threat satisfies well. Votes are not anonymous, and so defection from the coalition would be easy for other coalition members to detect.
• Clarity: Though the threat of retaliation for going against the coalition is implied, it is never explicitly stated.
• Repetition: The number of elimination rounds a given team member is likely to face in the future is relatively uncertain, given the continued swapping of players, the possibility that a team will win subsequent weigh-ins, and the constantly changing rules of the competition.
• Credibility: Players have reason to doubt the credibility of this threat, as non-coalition-based eliminations have historically been based on merit or need (from a health perspective).

Recognizing that the threat of retaliation was not the key factor keeping the coalition intact for Arthur’s elimination, we then examined the role of fairness value, or “F.” This seems a much more likely motivator for members voting with the coalition. The importance of personal relationships in the competition is reflected nearly every week, as players repeatedly refer to their respective teams as “family.” This emotional, familial tie runs deepest among those who started out the competition in the same group together, and trumps rationality during early elimination rounds, leading players to deviate from the game’s equilibrium.

In addition, looking at our assumptions in the payoff table more critically, we find that the incentive to defect from the coalition is not as strong as it may seem at the surface. Although a player does, in theory, experience a higher payoff for voting off the weakest player and retaining the strongest player, the competition’s history reflects that player performance in practice is relatively inconsistent. Arthur, for example, emerged as one of the strongest players in Week 9, but had previously been among the weakest players. Thus, “F” becomes even more influential, as the upside to “cheating” is diminished.

Avoiding the Coordination Challenge

In The Biggest Loser, players face costs to unsuccessful coordination in an elimination decision. In particular, players face the risk that without successful coordination a team vote will result in an outcome that does not match the player’s payoff maximizing outcome. In the face of this risk, there may be circumstances when a player prefers to avoid the coordination challenge and secure a certain payoff. The game structure of The Biggest Loser largely prevents certain outcomes in an elimination decision; however, there is one instance which occurs in nearly every season. This is the case when a player believes that his or her team is certain to face elimination and that a weak player on the team is likely to be eliminated. In this instance, if a player’s fairness value of keeping that weak player in the competition is greater than his or her payoff from staying in the competition, then the player can throw the round to force his or her own elimination over the weaker player. This relationship can be expressed as follows:

When F is relatively small, the player faces a positive payoff by remaining in the competition. However, as F grows sufficiently large the payoff can turn negative at which point the player would prefer to leave the competition than realize that negative payoff. To avoid the coordination challenge in this instance, a player can intentionally lose the round and change the payoffs his or her team faces in an elimination decision. When the team enters an elimination round, it must chose a member to leave the contest. In the absence of a member openly throwing the round, the vote leaves the team with the same payoff regardless of which player is eliminated. This is illustrated in the game tree below on the branch where the player does not throw the round; however, if a player throws the round, the team’s payoff changes. By throwing the round, the player becomes a liability to the team in the following round and makes it possible for the team to coordinate its elimination decision.