Part 1 Nocooperative Equilibria in Normal Form Games

Part1-4 Uncertainty, Risk, and Mixed Strategy

NE

prolog• Incomplete information （不完全訊息） complete but imperfect information （不完美訊息）• 虛擬參賽者：自然 以「類型」帶機率的方式出現，不在意報酬。• 不確定性可能來自「自然」，或來自於人類「有意的」選擇。 著眼information asymmetry （strategy unpredictability ），提高議價籌碼• 最適反應：既定對手機率分配下，最大期望報酬之機率分配 不存在純策略，也一定存在混合策略

自然

Mixed-strategy Nash equilibrium

A mixed-strategy Nash equilibrium is mixed- strategy profile having the property that no player could increase his payoff by switching to any other strategy, given the other player’s strategy.

• Note 1. is compared against each pure strategy rather than against all mixed strategies. 2. rationality of mixed strategies My opponent knows that he cannot out-think me. The pure strategy is not his best strategy. He will randomize in order to prevent me from out-think him by choosing probabilities that make the expected payoffs of my strategies equal. So I may as well randomize anyway.

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• We look for a mixed strategy for one player that makes the other player indifferent between his pure strategies.

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Graphics for Mixed Strategy 棒球中打者的期望值報酬

pure strategy and mixed strategy NE

• Result： every finite game （ having a finite number of players and a finite strategy space ） has at least one Nash equilibrium（ in pure or mixed strategies ）

Equilibriua with Mixed and Pure Strategies

• Pure strategy NE ：（前進 , 等候）、（等候，前進）• mixed strategy NE ：（ 1/2, 1/2 ），（ 1/2, 1/2 ）

A 之預期報酬

• In the coordination game, mixed strategies NE ( （ 1/2, 1/2 ），（ 1/2, 1/2 ） ) is unstable