Solution to Assignment 3

Transcription

1 Solution to Assignment 3 0/03 Semester I MA6 Game Theory Tutor: Xiang Sun October 5, 0. Question 5, in Tutorial set 5;. Question, in Tutorial set 6; 3. Question, in Tutorial set 7. Solution for Question Calculate firm i s price and profit in the collusion, Bertrand competition, and deviation from punishment cases, respectively: Cooperative price and profit: In the collusion, the price is p c i a+c, and profit is πi c a c 8 ; Non-cooperative price and profit: In the Bertrand competition, price is p m i profit is πi m 0; Deviation price and profit: Firm j s price is p c j a+c profit by choosing a price p d i < a+c almost equal to monopoly profit π d i a c. c, and, Firm i j can increases its, and profit is, but as close as possible to a+c For each i, consider the following trigger strategy T i for Firm i: In the first stage, choose price p c i. In the t-th stage, choose p c i if Firm j chooses price pc j stages; otherwise, choose price p m i. in each of the t previous For any i, assume that Firm j i chooses the trigger strategy T j. We want to find the condition which guarantees the trigger strategy T i to be Firm i s best response. If Firm i does not choose the trigger strategy, then we consider the following two cases: If Firm i always chooses the cooperative production p c i in every stage game it is a strategy for Firm i, but not the trigger strategy, then the payoff is as same as the payoff when it chooses trigger strategy. xiangsun@nus.edu.sg. Suggestion and comments are always welcome.

2 MA6 Game Theory /5 Assignment If Firm i deviates in some stage and the profit maxizer is p d i. Without loss of generality, we assume that the t-th stage is the first stage when Firm i deviates, then it can get at most πi d at this stage. From the t + -th stage on, Firm j i will choose non-cooperative price p m j. Thus Firm i will receive at most πm i 0 in each of the subsequent stages, and the t-th stage s present value of its payoff from the t-th stage onwards is at most πi d. It is easy to understand when looking at the following table, where means we do not know exactly the action of Firm i at that stage. Stage t t t + t + t + 3 Firm j i p c j p c j p c j p m j p m j p m j Firm i p c i p c i p d i Firm i s payoff πi c πi c πi d πi m πi m πi m If Firm i chooses the trigger strategy T i, then it will receive πi c in each stage, and the present value of its payoff from t-th stage onwards is π c i + δπ c i + δ π c i + πc i δ. In order for firm i to play trigger strategy T i, we should have that is δ. π c i δ πd i, Solution for Question There are players: Player I and Player II; Type spaces: T },, 3}}, and T, 3}, }}; Action spaces: A T, B}, and A L, R}; Strategy spaces: S T T, T B, BT, BB}, and S LL, LR, RL, RR}. Now we will find the best-response correspondence for each player and each associated type: let a and a be Player I s actions in Game, and Games and 3, respectively, b and b Player II s actions in Games and 3, and Game, respectively. If Game is drawn, then Player I s best-response correspondence is a T, if b L; b T, if b R. If Game is not drawn, then by considering the expected payoff, Player I s bestresponse correspondence is T, if b b LL; a T, if b b LR; b b B, if b b RL; B, if b b RR.

3 MA6 Game Theory 3/5 Assignment If Game is drawn, then Player II s best-response correspondence is b L, if a T ; a R, if a B. If Game is not drawn, then by considering the expected payoff, Player II s bestresponse correspondence is R, if a a T T ; b L, if a a T B; a a R, if a a BT ; L, if a a BB. Therefore, by definition, there is no Bayesian Nash equilibrium. The reason is as follows: If Player I chooses T T, then Player II should choose RL; on the other hand, T T is not a best response for RL. So there is no Bayesian Nash equilibrium when Player I chooses T T. If Player I chooses T B, then Player II should choose LR; on the other hand, T B is not a best response for LR. So there is no Bayesian Nash equilibrium when Player I chooses T B. If Player I chooses BT, then Player II should choose RL; on the other hand, BT is not a best response for RL. So there is no Bayesian Nash equilibrium when Player I chooses BT. If Player I chooses BB, then Player II should choose LR; on the other hand, BB is not a best response for LR. So there is no Bayesian Nash equilibrium when Player I chooses BB. Solution for Question 3 There are two players: seller s and buyer b; Type spaces: T s [α s, β s and T b [α b, β b ; Action spaces: A s A b [0, ; Strategy spaces: S b function from T b to A b }, and S s function from T s to A s }; Payoff: u s p s, p b ; v s, v b u b p s, p b ; v s, v b ps+pb v s, p b p s, 0, p b < p s v b ps+p b, p b p s. 0, p b < p s Suppose p s, p b is a linear Bayesian Nash equilibrium, where p sv s a s + v s, p b v b a b + v b. Note that a s,, a b, are to be determined. Here we should assume, > 0.

4 MA6 Game Theory /5 Assignment For seller, when v s is drawn, given buyer s strategy p b, p sv s will maximize his expected payoff E[u s p s, p b ; v s, v b β b α b p s p b v b p b β b βb p s + p b v b p s + a b + v b v β b α ps a s dv b b b c [ b ps + a b v s β b p s a b β b α b [ ps + a b v s β b p s a b β b α b β b p [ s a b ps + a b β b α b β b α b β b p s + a b Therefore, by the first order condition, v s dv b + 0 dv b β b α b p b α b p b v b<p s + + v s + βb ps a b [ v s + 3 p s + a b + β b p sv s 3 v s + 3 a b + 3 β b, v b dv b β b p s a b β b + p s a b β b + p s a b and hence 3, a s 3 a b + β b. For buyer, when v b is drawn, given seller s strategy p s, p b v b will maximize his expected payoff E[u b p s, p b ; v s, v b p s αs p s vs p b p b as cs v b p sv s + p b v b a s + v s + p b dv s α s [ v b a s + p b pb a s [ v b a s + p b pb a s [ pb a s v b a s + p b p b a s α s Therefore, by the first order condition, dv s + 0 dv s p b <p s vs p s βs [ v b 3 p b a s + α s p b as cs α s p b v b 3 v b + 3 a s + 3 α s, v s dv s pb a s pb a s + α s pb a s + α s and hence 3, a b 3 a s + α s.

5 MA6 Game Theory 5/5 Assignment Solving Equations and, we will have a s α s + β b, a b β b + α s.