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Game Theory

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O A B C $5 $1 $3 $2 $4 -$3 $0 $1 White plays first Then Black White’s move is not disclosed until After Black chooses x y z

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Prisoners’ Dilemma Two crooks questioned by police for a serious crime. Separate cells, no communication. If neither confesses then you a 2 year sentence for each is all you have on the lesser charge. Turn state witness, confess and you will be set free! * The other gets 4 years plus 1 more for obstructing justice. (But if both of you confess you both get 4 yrs).

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Keep Quiet (Q) or Squeal (S) Payoffs are in years lost in jail Payoffs (P1, P2); prisoners 1 and 2 Q -2, -2 -5, 0 S 0, -5 -4, -4 Q S Prisoner 2 Prisoner 1 J. Webb, “Game Theory”, Springer.

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Both prisoners, by following the optimum strategy (self interest) are actually worse off! Honor among thieves? Revenge! (that’s another game, with a different set of payoffs!) J. Webb, “Game Theory”, Springer.

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Two firms are merging into two divisions of a large firm, and have to choose the computer system to use. In the past the firms have used different systems, I and A; each prefers the system it has used in the past. They will both be better off if they use the same system then if they continue to use different systems. http://www.economics.utoronto.ca/osborne/2x3/t utorial/NEFEX.HTM

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(I,I) Neither player can increase her payoff by choosing an action different from her current one. Thus this action profile is a Nash equilibrium. (I,A) By choosing A rather than I, player 1 obtains a payoff of 1 rather than 0, given player 2's action. Thus this action profile is not a Nash equilibrium. [Also, player 2 can increase her payoff by choosing I rather than A.] (A,I) By choosing I rather than A, player 1 obtains a payoff of 2 rather than 0, given player 2's action. Thus this action profile is not a Nash equilibrium. [Also, player 2 can increase her payoff by choosing A rather than I.] (A,A) Neither player can increase her payoff by choosing an action different from her current one. Thus this action profile is a Nash equilibrium. We conclude that the game has two Nash equilibria, (I,I) and (A,A).

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Game Theory and Decisions How are societies formed and sustained? The Social Contract largely unoptimal equilibrium Collective of individual decisions optimizing payoff Taxes: – Most would want to do without it (wealthier without it) – If no one pays, then no services, everyone worse off – Equilibrium condition is still unsatisfying

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http://www.wired.com/wiredscience/2012/04/probability-and- game-theory-in-the-hunger-games/ In Puccini’s opera, the woman Tosca’s lover is condemned to death. Scarpia, the police chief offers to fake the execution in exchange for Tosca’s honor. The deal is made. However, Tosca stabs and kills Scarpia to keep her virute. Scarpia however, had reneged on the deal and actually had Tosca’s lover executed.

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Fishing Two Nations Combined sustainable fish catch proportional to your fleet size and your competitor’s fleet size Nash equilibrium

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From James Case, “Competition,” Hill and Wang

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Prisoner’s Dilemma An illustration of Nash Equilibrium Art’s Strategies Bob’s Strategies Confess Deny Confess Deny 10 yrs. 1 yr. 3 yrs. 1 yr. 10 yrs.

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