Wyatt Earp and the Gun Slinger

A Bayesian gunslinger game

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

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

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

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.

• 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.