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Chapter 17: Making Complex Decisions April 1, 2004

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17.6 Decisions With Multiple Agents: Game Theory Assume that agents make simultaneous moves Assume that the game is a single move game.

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Uses Agent Design (2 finger Morra) Mechanism Design

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Game Components Players Actions Payoff Matrix e.g. rock-paper-scissors

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Terminology Pure Strategy – deterministic policy Mixed Strategy – randomized policy, [p: a; (1-p): b] Outcome – result of game Solution: player adopts a strategy profile that is a rational strategy

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Prisoners Dilemna B testifiesB refuses A testifiesA = -5 B = -5 A = 0 B = -10 A refusesA = -10 B = 0 A = -1 B = -1

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Terminology (testify, testify) is a dominant strategy s strongly dominates s – s is better than s for all other player strategies s weakly dominates s – s is better than s for one other strategy and is at least as good as all the rest

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Terminology An outcome is Pareto optimal if there is no other outcome that all players would prefer An equilibrium is a strategy profile where no player benefits by switching strategies given that no other player may switch strategies Nash showed that every game has an equilibrium Prisoners Dilemna!

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Example: Two Nash Equilibria no dominant strategy! B: dvdB: cd A: dvdA = 9 B = 9 A = -4 B = -1 A: cdA = -1 B = -4 A = 5 B = 5

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Von Neumanns Maximin zero sum game E maximizer (2 finger Morra) O minimizer (2 finger Morra) U(E = 1, O = 1) = 2 U(E = 1, O = 2) = -3 U(E = 2, O = 1) = -3 U(E = 2, O = 2) = 4

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Maximin E reveals strategy, moves first [p: one; 1-p: two] O chooses based on p one: 2p -3(1-p) two: -3p + 4(1-p) p = 7/12 U E,O = -1/12

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Maximin O reveals strategy, moves first [q: one; 1-q: two] E chooses based on q one: 2q -3(1-q) two: -3q + 4(1-q) q = 7/12 U O,E = -1/12

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Maximin [7/12: one, 5/12: two] is the Maximin equilibrium or Nash equilibrium Always exists for mixed strategies! The value is a maximin for both players.

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Repeated Move Games Application: packet collision in an Ethernet network Prisoners Dilemna – fixed number of rounds – no change! Prisoners Dilemna – variable number of rounds (e.g. 99% chance of meeting again) –perpetual punishment –tit for tat

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Repeated Move Games Partial Information Games – games that occur in a partially observable environment such as blackjack

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17.7 Mechanism Design Given rational agents, what game should we design Tragedy of the Commons

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Auctions Single Item Bidder i has a utility v i for the item v i is only known to Bidder i English Auction Sealed Bid Auction Sealed Bid Second Price or Vickrey auction (no communication, no knowledge of others)

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