Part 1 Nocooperative Equilibria in Normal Form Games

Part 1 NocooperativeEquilibria 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（strategyunpredictability ），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

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.

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

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

Part 1 Nocooperative Equilibria in Normal Form Games