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 , whenever 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.