UC Berkeley Fall 007 Economics 01A - Section 6 1 What we learnt this week Marina Halac Basics: trigger strategy Classes of games: infinitely repeated games, static Bayesian games Solution concepts: subgame perfect Nash euilibrium, Bayesian Nash euilibrium Applications: auction model Problems Problem 1: Infinitely repeated asymmetric prisoners dilemma Consider an infinitely repeated game with discount factor δ and the following stage-game C D C 3, -1, 3 D 5, -1 0, 0 (i) What are the payoffs in the worst possible SPNE? (ii) What is the lowest discount factor δ for which the players achieve cooperation in a SPNE? (iii) Is it possible to achieve payoffs better than (0, 0) in a SPNE for discount factors lower than δ? Hint: Think about a SPNE in which the players alternate between (C, D) and (D, C) on the euilibrium path. Problem : Euilibria with payoffs worse than Nash Consider an infinitely repeated game with discount factor δ and the following stage-game D E D 0,0-1,-1 E -1,-1 -,- Assume δ 1/. Can (E,E) be played in a SPNE? Note that (E,E) gives payoffs (-,-) which are lower than the payoffs achieved in the stage-game Nash euilibrium, (0,0). If your answer is yes, propose strategies that sustain (E,E) in some period in a SPNE. If your answer is no, prove it. Problem 3: Cournot oligopoly Let Y k be player k s payoff in his worst Nash euilibrium of the stage game. Let D k (a 1,..., a N ) max a Ak U k (a 1,..., a k 1, a, a k+1,..., a N ). In problem 11 of Problem Set Nagano, you are asked to prove the following theorem: Consider a stage-game G and an action profile of G, (a 1,..., a N ), with D payoffs (X 1, X,..., X N ) such that X k > Y k. Then for all δ > max k X k k D k Y k, there exists a SPNE of G(T=,δ) where players play (a 1,..., a N ) each period. 1
Now consider n firms in a Cournot oligopoly. Inverse demand is given by P (Q) = a Q, where Q = 1 +... + n. Total costs for firm i are c i ( i ) = c i. Consider the infinitely repeated game based on this stage game. Using the theorem above, find the lowest value of δ such that the firms can sustain the monopoly output level in a SPNE. How does you answer vary with n? Problem 4: Bayesian game C D A x,y 4,5 B,y x,3 Let x {3, 5} and y {4, 6}, independently drawn. Both values of x and both values of y are eually likely. Suppose that each player knows his own payoff: player 1 observes x and player observes y. Player 1 does not observe y and player does not observe x. Find the BNE. Problem 5: Cournot with asymmetric information (Gibbons 3.) Consider a Cournot duopoly operating in a market with inverse demand P (Q) = a Q, where Q = 1 + is the aggregate uantity on the market. Both firms have total costs c i ( i ) = c i, but demand is uncertain: it is high (a = a H ) with probability θ and low (a = a L ) with probability 1 θ. Furthermore, information is asymmetric: firm 1 knows whether demand is high or low, but firm does not. All of this is common knowledge. The two firms simultaneously choose uantities. What are the strategy spaces for the two firms? Make assumptions concerning a H, a L, θ, and c such that all euilibrium uantities are positive. What is the BNE of this game? Problem 6: Second-price auction (MWG 8B3) Consider the following action (known as second-price or Vickrey auction). An object is auctioned off to I bidders. Bidder i s valuation of the object is v i. Each bidder observes only his own valuation. The auction rules are that each player submit a bid (a nonnegative number) in a sealed envelope. The envelopes are then opened, and the bidder who has submitted the highest bid gets the object but pays the auctioneer the amount of the second highest bid. If more than one bidder submits the highest bid, each gets the object with eual probability. Show that submitting a bid of v i with certainty is a weakly dominant strategy for bidder i. Also argue that this is bidder i s uniue weakly dominant strategy. Problem 7: First-price auction Consider a first-price sealed-bid auction of an object with two bidders. Each bidder i s valuation of the object is v i. Each bidder observes only his own valuation. The valuation is distributed uniformly and independently on [0, 1] for each bidder. The auction rules are that each player submit a bid in a sealed envelope. The envelopes are then opened, and the bidder who has submitted the highest bid gets the object and pays the auctioneer the amount of his bid. If the bidders submit the same bid, each gets the object with probability 1/. Derive the BNE of this auction. Hint: Look for an euilibrium in which bidder i s strategy is b i (v i ) = c i v i for some constant c i.
3 Answers Problem 1 (i) The payoffs in the worst SPNE are (0, 0). They are achieved by the repetition of the stagegame Nash euilibrium (D,D). It is not possible to achieve payoffs worse than 0 since either player can always guarantee himself a payoff of at least 0 by playing D in every period. (ii) According to part (a) the worst available punishment has a payoff of 0 for both players. So for neither player to want to deviate from (C, C), both of the following conditions must hold 5 3/(1 δ) δ /5 (deviation of player 1) 3 /(1 δ) δ 1/3 (deviation of player ) We find that δ = /5. (iii) Yes. Define the following regimes: regime 1: play (C, D); regime : play (D, C). Then consider the following strategies (for each player): Start in regime 1. If the outcome in regime 1 is (C,D), go to regime ; otherwise, play (D,D) forever. If the outcome in regime is (D,C), go to regime 1; otherwise, play (D,D) forever. We now prove that there exists δ < δ for which these strategies constitute a SPNE with payoffs better than (0,0). In this SPNE, the players alternate between (C,D) and (D,C) as long as neither player deviates, and deviations are punished by the stage-game Nash euilibrium forever. For no player to want to deviate in any regime, the following conditions must hold 3 5 δ + 5δ δ 3... = (5 δ)/(1 δ ) always holds (player 1 in regime (D,C)) 0 1 + 3δ δ + 3δ 3... = ( 1 + 3δ)/(1 δ ) holds if δ 1/3 (player in regime (D,C)) 0 1 + 5δ δ + 5δ 3... = ( 1 + 5δ)/(1 δ ) holds if δ 1/5 (player 1 in regime (C,D)) 3 δ + 3δ δ 3... = (3 δ)/(1 δ ) always holds (player in regime (C,D)) Note also that no player deviates from (D,D) (in the subgame following deviation) as (D,D) is a Nash euilibrium of the stage game. Hence, a SPNE in which the players alternate between (C, D) and (D, C) exists if δ 1/3 (note that 1/3 < δ ). If players start in regime (C, D), they get strictly positive payoffs of ( 1 + 5δ)/(1 δ ) and (3 δ)/(1 δ ). Problem The answer is yes. For δ 1/, a SPNE that implements (E,E) in some period exists. Define the following regimes: regime 1: play (E,E); regime : play (D,D). Consider the following strategies (for each player): Start in regime 1. If outcome is (E,E) in regime 1, go to regime. Otherwise, stay in regime 1. Regime is absorbing. We check that neither player wants to deviate in regimes 1 and. Regime is a repetition of a stage-game Nash euilibrium, so no player has incentives to deviate. In regime 1, if a player does not deviate, his payoff is. If a player deviates once, his payoff is 1 δ + 0δ +... = 1 δ Hence, a SPNE in which the players play (E,E) in the first period exists if δ 1/. (Note that I used the One-Stage Deviation Principle: In a repeated game, a combination of players strategies s is a SPNE iff no player can gain by deviating in a single stage, and confirming to s thereafter.) 3
Problem 3 The monopoly output is Q M = (a c)/. Thus, if the n firms share the monopoly output eually, each produces M = (a c)/(n), with profit π M = (a c) 4n A firm s optimal stage-game deviation, D, when the n 1 other firms choose M is D = arg max (a (n 1) a c n c) = n + 1 (a c) 4n This gives the deviator a profit level of π D = (n + 1) 16n (a c) The uniue stage-game NE is for each firm to choose i = arg max i (a i j c) = a j i j c i j i Solving the system of euations, the n-firm Cournot NE outcome involves each firm producing C = (a c)/(n + 1), with profit π C = (a c) (n + 1) Consider the proposed trigger-strategy for each player: Play M in period 1. In period t, play M if all firms have played M in all previous periods; play C otherwise. Using the theorem above, these trigger strategies constitute a SPNE if 1 16n 4n 1 16n δ πd π M (n+1) π D π C = (n+1) (n+1) δ Note that δ 1 as n. Hence, collusion is harder to sustain when there are more firms. Problem 4 Each player has two types: T 1 = {t 1L, t 1H }, where t 1L knows that x = 3 and t 1H knows that x = 5, and T = {t L, t H }, where t L knows that y = 4 and t H knows that y = 6. To find the BNE, note that for type t 1L, playing A strictly dominates playing B (regardless of his belief about player s action, A is t 1L s best response), and for type t H, playing C strictly dominates playing D. Hence, in any BNE, t 1L plays A and t H plays C. Consider next type t 1H. His expected payoff from playing A is 5p 1 (s = C t 1H ) + 4(1 p 1 (s = C t 1H )) = 4 + p 1 (s = C t 1H ) 4
and his expected payoff from playing B is p 1 (s = C t 1H ) + 5(1 p 1 (s = C t 1H )) = 5 3p 1 (s = C t 1H ) Thus, for t 1H, playing A is a best response iff the probability that player plays C is at least 1/4. Now note that in euilibrium, the probability that player plays C is always greater than 1/4, since for any [0, 1], p 1 (s = C t 1H ) = p 1 (t H t 1H ).1 + p 1 (t L t 1H ) = 1/ + / 1/ Hence, in any BNE, t 1H plays A. Finally, consider type t L. Regardless of what player 1 does, the payoff from playing C is always y = 4 for this type. His expected payoff from playing D is U (s 1L, D; t 1L, t L )p (t 1L t L ) + U (s 1H, D; t 1H, t L )p (t 1H t L ) = U (A, D; t 1L, t L )p (t 1L t L ) + U (A, D; t 1H, t L )p (t 1H t L ) = 5 > 4 Hence, in any BNE, t L plays D. We have found that there exists a uniue BNE with (s 1L, s 1H, s L, s H ) = (A, A, D, C). Problem 5 Firm 1 s strategy space is [0, ) [0, ), and firm s strategy space is [0, ). The strategies (1L, 1H, ) constitute a BNE iff 1L = arg max 1H = arg max = arg max (a L c), (a H c), and θ(a H 1H c) + (1 θ)(a L 1L c) The three first-order conditions are: 1L = a L c, 1H = a H c, and = θa H + (1 θ)a L c (θ1h + (1 θ) 1L ) Therefore, the uniue BNE is: ( (1L, 1H, ) θah + ( + θ)a L c =, (3 θ)a H (1 θ)a L c, θa ) H + (1 θ)a L c 6 6 3 Problem 6 We prove that bidding v i is at least as good as any other bid no matter what other bids are. Consider an alternative bid b i > v i. Case 1: There is another bid higher than b i. Then neither of the bids wins the auction and so both give a payoff of 0. Case : The second highest bid is less than v i. Then bidder i wins the auction and pays the same second highest bid in both cases. Both bids give the same payoff. Case 3: The highest bid of other players is between v i and b i. Then bid v i would lose the auction and give a payoff of 0. Bid b i would win the auction but the bidder would 5
have to pay more than his valuation, obtaining a negative payoff. Hence, bidding b i > v i is weakly dominated by bidding v i. Consider next an alternative bid b i < v i. Case 1: There is another bid higher than v i. Then neither of the bids wins the auction and so both give a payoff of 0. Case : The second highest bid is less than b i. Then bidder i wins the auction and pays the same second highest bid in both cases. Both bids give the same payoff. Case 3: The highest bid of other players is between b i and v i. Then bid b i would lose the auction and give a payoff of 0. Bid v i would win the auction and the bidder would have to pay less than his valuation, obtaining a positive payoff. Hence, bidding b i < v i is weakly dominated by bidding v i. This argument implies that bidding v i is the uniue weakly dominant strategy for player i. Problem 7 Following the hint, suppose player j s strategy is b j (v j ) = c j v j. For a given value of v i, player i s best response solves max b i (v i b i )Pr(b i > b j ) = (v i b i )Pr(b i > c j v j ) (Note that we can ignore a tie since Pr(b i = b j ) = 0 because b j (v j ) = c j v j and v j is uniformly distributed, so b j is uniformly distributed.) Now note that Pr(b i > c j v j ) = Pr(v j < b i /c j ) = b i /c j where the last euality uses b i /c j [0, 1], which follows from the fact that player i would not bid below 0 nor above c j. Hence, player i s best response is b i (v i ) = v i / The same analysis for player j yields b j (v j ) = v j / So each player submits a bid eual to half his valuation. This reflects the trade-off that a player faces in an auction: the higher the bid, the more likely to win; the lower the bid, the larger the gain if the player wins. 6