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.

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).

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