more bayes-nash
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More Bayes-Nash. 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 - PowerPoint PPT PresentationTRANSCRIPT
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More Bayes-Nash
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Wyatt Earp and the Gun Slinger
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A Bayesian gunslinger game
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The gunfight game when the stranger is (a) a gunslinger or (b) a cowpoke
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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
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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
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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.
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• 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.