The Badminton World Federation disqualified four female badminton teams today from the London Olympics for unsportsmanlike behavior. These teams purposefully tried to lose their first match in a round robin event in order to be paired against easier teams in subsequent rounds (video recap here). These teams were obviously cheating as purposely losing in this manner is against the badminton rules. So why would so many teams blatantly cheat in a very public and very important sporting event? Because they thought they could get away with it. Prior to this mass banning of teams, several Chinese teams had a history of purposefully trying to lose in badminton matches and they were never punished for their behavior.
Being able to cheat and get away with it gave the Chinese teams a huge advantage in the Olympics. Hence, other teams followed the example of the Chinese teams and tried to lose, so they would also have an advantage in later rounds of the competition. All the cheating teams believed there existed a low probability of being caught and/or punished for cheating given past examples.
I created a game to demonstrate why it was in the best interest of all the badminton teams to try to lose their first badminton matches, assuming there was a very low probability that they would be caught.
The Players
There are two players in this game, Team 1 and Team 2. Team 1 is the better team. They have a 60% chance of winning the match if both teams play fairly. Team 2 is the weaker team. They only have a 40% chance of winning the match if both teams play fairly.
Both teams want to lose the match (while the A team is the better team in this particular round, they would rather have easier competition in subsequent rounds of tournament play). If a team loses, they receive 10 points. Alternatively, if a team wins, they receive 0 points.
Nature also plays a role in this game. Nature represents both the probability that the team will be found cheating and punished for their behavior. If Nature decides that a team played unfairly, Nature will punish the team, but only 5% of the time. Given that punishment has been rare in the past, this probability is reasonable, but can be altered to change the risk/reward ratio for cheating. If a team is caught cheating, they receive -10 points.
The Moves (See Figure 1)
Team 1 goes first. They can “Cooperate” and try to win the game, or they can “Defect”, and purposefully try to lose.
Team 2 moves next. They observed Team 1’s move. Team 2 can also “Cooperate” and try to win the game, or “Defect”, and purposefully try to lose.
The Outcomes
There are four outcomes.
1. Both teams Cooperate
For this outcome, both teams played fairly and tried to win the game. Team 1 wins 60% of the time when this occurs, and Team 2 wins 40% of the time. A team receives 0 points for winning and 10 points for losing. Hence, the expected utility for each team if this outcome occurs is:
([(0.6*0)+(0.4*10)] , [(0.4*0)+(0.6*10)]) = (4, 6)
2. Team 1 Cooperates, Team 2 Defects
For this outcome, Team 1 tried to win and Team 2 tried to lose. Team 1 wins with a 100% probability and receives 0 points for winning.
Since Team 2 cheated, there is a probability that Nature will punish them. Nature punishes Team 2 5% of the time. If Team 2 is punished, they receive -10 points. The other 95% of the time, Team 2 gets away with it and receives 10 points for losing.
The expected utility for each team if this outcome occurs is:
(0, [(0.95*10)+(0.05*-10)]) = (0, 9)
3. Team 1 Defects, Team 2 Cooperates
For this outcome, Team 1 tried to lose and Team 2 tried to win. Team 2 wins with a 100% probability and receives 0 points for winning.
Since Team 1 cheated, there is a probability that Nature will punish them. Nature punishes Team 1 5% of the time. If Team 1 is punished, they receive -10 points. The other 95% of the time, Team 1 gets away with it and receives 10 points for losing.
The expected utility for each team if this outcome occurs is:
([(0.95*10)+(0.05*-10)], 0) = (9, 0)
4. Team 1 Defects, Team 2 Defects
For this outcome, both teams tried to lose. Since both teams tried to lose, there is a 50% chance that any team could win.
If a team wins, and Nature does not catch them cheating (which happens 95% of the time), the team receives 0 points.
If a team loses, and Nature does not catch them cheating (which happens 95% of the time), the team receives 10 points.
If Nature catches a team cheating (which happens 5% of the time), regardless of whether the team won or lost, the team receives -10 points.
The expected utility for each team if this outcome occurs is:
([(0.5*(0.95*0))+(0.5*(0.95*10))+(0.05*-10)], [(0.5*(0.95*0))+(0.5*(0.95*10))+(0.05*-10)]) = (4.25, 4.25)
The Nash Equilibria (See Figures 2 and 3)
There is one Nash Equilibrium derived from backwards induction, {Defect; Defect, Defect}.
There are two Nash Equilibria derived from the strategic form game, {Defect; Cooperate, Defect} and {Defect; Defect, Defect}.
This game suggests that as long as the Badminton World Federation rarely punishes players for trying to lose in the first match of a round robin even, it is in a team’s best interest to try to cheat early on in order to improve their chances of winning in subsequent rounds. To change this outcome, the Badminton World Federation should either detect and punish cheating more often, or change the rules of the tournament where losing in an early round is not so advantageous.
(For another interesting sports match– the 1994 Caribbean Cup provides a classic example of how rules fundamentally change the expected behavior of players in an environment where the object of the game appears simple. By the end of the game, it was in the best interest of both teams to try to score points for the other team).
For one player I agree 100% but what I find interesting about this in a game theoretic fashion (in addition to your great article) is the impact of the choices of the other players on the state of nature. If we insert a state of nature that represents the updated quality of the opponent, would not the advantage of “losing” be conditioned by who else also played the “dominate” strategy. If the best teams all throw the game, then by playing the “dominated” strategy (last mover advantage?) the team assures that they will face an easier opponent. Maybe an evolutionary game?
Interesting, but one (parenthetical) point and two questions: Point: my calculations have that the payoff in the DD cell should be 4.25, 4.25. Questions: 1. Why the dual strategy for player 2 in the normal form game? 2. How did you decide on -10 as the payoff for getting caught cheating? Do you think a payoff for getting booted from the tournament, and probably any future competition, should have the same magnitude as remaining in the competition and potentially losing to a higher seed in the next round? It seems to me that forfeiting one’s career for the sake of getting slightly ahead in one tournament would warrant a negative payoff approaching infinity (or maybe I’m thinking about nuclear deterrence games too much). But you needn’t even take the payoff for getting caught cheating that far. All you need to do is find the point at which player 1’s payoff for defecting when player 2 cooperates is equal to his payoff for cooperating when 2 cooperates. In other words, at what point does the penalty for getting caught make make player 1 not risk defecting? It’s a bit complicated, since both the CD and DD cells rely on the penalty, but 1’s payoffs for CC and DC are equal at 4 when the penalty equals -110 (maintaining your 0.05 probability of getting caught). At this point, C is strictly dominant for 2. So really, if we agree that losing one’s badminton career is at least 11 times worse than losing in the second round, cheating is never beneficial, and I would feel pretty safe, given the consequences of getting caught cheating, that this is a fair statement.
Thanks for the math correction- I fixed it.
Question 1: Not sure if I am answering this correctly but… there are no information sets in the extensive form game. So Player 2 gets to observe Player 1 actions, and base its entire strategy on what Player 2 is doing. For example, Player 2 gets to decide, {Cooperate, Cooperate}, or in words, “If Player 1 Cooperates, then I Cooperate, and if Player 2 Defects, I Cooperate”.
IF there were an information set in the game, Player 2 could only play {Cooperate} or {Defect}. {Cooperate} would be “cooperate always”, and {Defect} would be “defect always”, no matter what Player 1 is doing.
Question 2: I just made up -10. I assumed that (Losing > Winning > Punishment). Since winning was 0, I chose a reasonable number less than 0.
As with Bruce’s suggestions above, you could make this game more complicated, taking into account several more factors, like updating, observing the actions of multiple teams, or varying the cost of punishment.
Also, as stated in the post, you could also vary the game to see how often the Badminton World Federation would have punish teams for cheating to make it beneficial to not cheat.