The Sheriff’s Dilemma is an example of a simultaneous move Bayesian game. In a standard game, the Nash equilibrium is formed by a player’s understanding of the other player. Whereas in Bayesian, the type (of the other) also matters. We will see that through an example. But before that, the rules.
Civilian vs criminal
The sheriff encounters an armed suspect, and they must decide whether to shoot at the other.
- The suspect could be a criminal with probability p and civilian with (1-p).
- The sheriff shoots if the suspect shoots
- The criminal always shoots
The payoffs are
| civilian (1-p) | sheriff | ||
| Shoot | Not | ||
| suspect | Shoot | -3,-1 | -1,-2 |
| Not | -2,-1 | 0,0 |
| criminal (p) | sheriff | ||
| Shoot | Not | ||
| suspect | Shoot | 0,0 | 2,-2 |
| Not | -2,-1 | -1,1 |
Before moving to the sheriff, let’s find out the strategy of the suspect. If the suspect is a civilian, his dominant strategy is not to shoot (-2 > -3 AND 0 > -1). For the criminal, the dominant strategy is to shoot (0 > -2 AND 2 > -1).
The sheriff’s payoff
The expected payoffs if the sheriff shoots is = -1 x (1-p) + 0 x p = p – 1
The expected payoffs if the sheriff doesn’t shoot is = 0 x (1-p) -2 x p = -2p
So, the payoffs match for p = 1/3. If p is greater than 1/3, the sheriff is better off if he shoots.
