1. Wyatt Earp and the Gun Slinger

2. A Bayesian gunslinger game

3. The gunfight game when the stranger is (a) a gunslinger or (b) a cowpoke

4. What are the strategies?
Earp
Draw
Wait
Stranger
Draw if Gunslinger, Draw if Cowpoke
Draw if Gunslinger, Wait if Cowpoke
Wait if Gunslinger, Draw if Cowpoke
Wait if Gunslinger, Wait if Cowpoke

5. One Bayes Nash equilibrium
Suppose that Earp waits and the other guy draws if he is a gunslinger, waits if he is a cowpoke.
Stranger in either case is doing a best response.
If stranger follows this rule, is waiting best for Earp?
Earp’s Payoff from waiting is 3/4x1+1/4x8=2.75
Earp’s Payoff from drawing, given these strategies for the other guys is (¾)2+(1/4) 4=2.5
So this is a Bayes Nash equilibrium

6. There is another equilibrium
Lets see if there is an equilibrium where everybody draws.
If Earp always draws, both cowpoke and gunslinger are better off drawing.
Let p be probability stranger is gunslinger.
If both types always draw, payoff to Earp from draw is 2p+5(1-p)=5-3p and payoff to Earp from wait is p+6(1-p)=6-5p
Now 5-3p>6-5p if p>1/2.

7. If Earp always draws, best response for stranger of either type is to draw.
If stranger always draws, best response for
Earp is to always draw, if he thinks stranger is a gunslinger with p>1/2.
Note that this is so, even though if he knew stranger was a cowpoke, it would be dominant strategy to wait.