习题课关于博弈论的note第一个例子最后一个NE应是aab)而不是abb)

Game theory 1. 纳什均衡（NE）: Given others strategy, nobody has incentive to deviate. Example 1. Majority Voting Model set-up: Three voters 1,2 and 3 want to decide whether to vote project a or b. The Voting process is majority voting. If project a is passed (two or more than two people agree a), 1,2 and 3 all have pay-off 1 If project b is passed, 1,2 and 3 all have pay-off 0, in other words, everyone prefer a to b. What is NEs in this game? Intuitive explain: (a, a, a) is obvious, but what about (b, b, b)? Its a NE too. Formal solution: Three players normal form 2 a b 1 a 1, 1,1 1, 1, 1 1, 1, 1 b 1,1,1 1,1,1 0, 0, 0 2 a b 1 a 1,1,1 0,0,0 b 0,0,0 0,0,0 0,0,0 Figure1 All NEs in figure 1: (a,a,a), (a,b,a), (b,a,a), (b,b,b), (a,b,b). Comment: Multi-equilibriums exist in example1. NE is not a desirable concept of equilibrium in this case. Sometimes we need to refine the equilibriums: a stronger concept of equilibrium. 2. strategy vs. action In static game like PD (prisoners dilemma): strategy=action. Voter3 b a But in dynamic games: strategy and action are two concepts. Example 2 1s action: L, S. 1s strategy: L, S 2s action: L, S. 2s strategy: if 1 choose L, he chooses L; if 1 choose S, he chooses L LL if 1 choose L, he chooses L; if 1 choose S, he chooses F LF if 1 choose L, he chooses F; if 1 choose S, he chooses L FL if 1 choose L, he chooses F; if 1 choose S, he chooses F FF Normal form: 2s strategies (L, L) (L, S) (S, L) (S, S) 1 L 2, 2* 2, 2* -1, -1 -1, -1 S -1, -1 1, 1 -1, -1 1, 1* One SPNE: (L, (L,S) ( using backward induction) Three NEs Comment: SPNE is a stronger concept of equilibrium than NE. 3. Applied NE Example 3 沙滩卖冰： 在充满泳客的海滩上(0, 1uniform distribution)，有两家冰店进驻，你若是冰店经理， 应选在何处设店？(The price of drink is fixed in p, in other words, the sellers cant compete by price) 1 L S 2 L L S S ( 2, 2 ) ( -1, -1 ) ( 1, 1 ) ( -1, -1 ) one SPNE: (L, (L,S) three NEs 2 0 1/4 1/2 3/4 1 NE: (1/2, 1/2) What about three sellers? No NE What about four sellers? NE: two sellers in 1/4, two sellers in 3/4. (Using the concept of NE to check) 4. Mixed strategy Example 4. Battle of Sexes given (q, 1q)，Boys payoff: 2q or (1q) If 2q(1q) q1/3, payoff is same between two choices. to boy given (p, 1p)，girls payoff: p or 2(1p) if p2(1p) p2/3，payoff is same between two choices. to girl Draw the best response function in Figure1.1 below: We find three NEs Two are pure strategy ones The other one is mixed strategy. Comment: pure strategy is a special mixed strategy (the probability is 1) boy F R p 1p girl F q 1, 2 0, 0 R 1q 0, 0 2, 1 If q 1/3，2q(1q)