Repeated Games. Econ 400. University of Notre Dame. Econ 400 (ND) Repeated Games 1 / 48
|
|
- Harvey Osborne
- 5 years ago
- Views:
Transcription
1 Repeated Games Econ 400 University of Notre Dame Econ 400 (ND) Repeated Games 1 / 48
2 Relationships and Long-Lived Institutions Business (and personal) relationships: Being caught cheating leads to punishment or exclusion Econ 400 (ND) Repeated Games 2 / 48
3 Relationships and Long-Lived Institutions Business (and personal) relationships: Being caught cheating leads to punishment or exclusion Government: We are willing to give up resources now in the expectation that we will be paid back later (Fiat money, Social Security) Econ 400 (ND) Repeated Games 2 / 48
4 Relationships and Long-Lived Institutions Business (and personal) relationships: Being caught cheating leads to punishment or exclusion Government: We are willing to give up resources now in the expectation that we will be paid back later (Fiat money, Social Security) Social Norms: By fixing what society expects, we can achieve better coordination than acting alone Econ 400 (ND) Repeated Games 2 / 48
5 Relationships and Long-Lived Institutions Is cooperation always good? Econ 400 (ND) Repeated Games 3 / 48
6 Relationships and Long-Lived Institutions Is cooperation always good? Is anonymity always bad? Econ 400 (ND) Repeated Games 3 / 48
7 Relationships and Long-Lived Institutions Is cooperation always good? Is anonymity always bad? Do we value relationships because of the relationship per se, or for the stream of benefits it provides? Is that bad? Econ 400 (ND) Repeated Games 3 / 48
8 Relationships and Long-Lived Institutions Is cooperation always good? Is anonymity always bad? Do we value relationships because of the relationship per se, or for the stream of benefits it provides? Is that bad? Magellan s Victoria Econ 400 (ND) Repeated Games 3 / 48
9 Repeated Prisoners Dilemma Suppose two players are going to play prisoners dilemma t = 1,2,...,T times, where the payoffs are given by S C S 1,1-1,2 C 2,-1 0,0 Econ 400 (ND) Repeated Games 4 / 48
10 Repeated Prisoners Dilemma Suppose two players are going to play prisoners dilemma t = 1,2,...,T times, where the payoffs are given by S C S 1,1-1,2 C 2,-1 0,0 What are the subgame perfect Nash equilibria of the game? Econ 400 (ND) Repeated Games 4 / 48
11 Repeated Prisoners Dilemma But what if the other players was known to be a nice guy/gal, who plays S as long as you have played S in all previous periods, then chooses C for all future periods once you have chosen C i.e., they play nice until you cheat them. Maybe now there will be cooperation for some number of periods? Econ 400 (ND) Repeated Games 5 / 48
12 Repeated Prisoners Dilemma But what if the other players was known to be a nice guy/gal, who plays S as long as you have played S in all previous periods, then chooses C for all future periods once you have chosen C i.e., they play nice until you cheat them. Maybe now there will be cooperation for some number of periods? Consider the payoff from choosing S for the first T 1 periods, then choosing C in the final period to get the 2. Then the payoff from the strategy (S,S,...,S,C) is: 1+δ +δ δ T 1 +2δ T = 1 δt 1 δ +2δT Econ 400 (ND) Repeated Games 5 / 48
13 Repeated Prisoners Dilemma But what if the other players was known to be a nice guy/gal, who plays S as long as you have played S in all previous periods, then chooses C for all future periods once you have chosen C i.e., they play nice until you cheat them. Maybe now there will be cooperation for some number of periods? Consider the payoff from choosing S for the first T 1 periods, then choosing C in the final period to get the 2. Then the payoff from the strategy (S,S,...,S,C) is: 1+δ +δ δ T 1 +2δ T = 1 δt 1 δ +2δT What if we choose the C one period sooner? Would this improve our payoff? This strategy is (S,S,...,S,C,C): 1+δ +δ δ T 2 +2δ T 1 +0 = 1 δt 1 1 δ +2δ T 1 +0 Econ 400 (ND) Repeated Games 5 / 48
14 Repeated Prisoners Dilemma But then comparing confessing first at T 1 to confessing first at time T gives ( 1 δ T 1 ( 1 δ +2δ ) T 1 T ) ( ) 1+δ 1 δ 1 δ +2δT = δ T 1 1 δ +2 δ > 0 So it looks like the unravelling is going to occur, even if one player is willing to cooperate. Econ 400 (ND) Repeated Games 6 / 48
15 Repeated Prisoners Dilemma But let s look at the difference in payoffs between confessing for the first time in period T 1 and confessing for the first time in period T again: ( ) 1+δ δ T 1 1 δ +2 δ Econ 400 (ND) Repeated Games 7 / 48
16 Repeated Prisoners Dilemma But let s look at the difference in payoffs between confessing for the first time in period T 1 and confessing for the first time in period T again: ( ) 1+δ δ T 1 1 δ +2 δ As T gets large, δ T 1 becomes small, and cooperation is almost an equilibrium of the repeated game early on. But as the end of the game approaches, the unravelling motive will kick in. Econ 400 (ND) Repeated Games 7 / 48
17 Repeated Games Definition A repeated game is (i) a terminal date, T, giving the number of times the players interact, where T = 1,2,3,... The calendar date is given by t = 1,2,...,T. (ii) a discount factor, 0 δ 1, that represents both how patient the players are and how likely the game is to continue. (iii) a stage game of finite length: A specification of the players, actions, payoffs, and timing, which is usually independent of the calendar date. Econ 400 (ND) Repeated Games 8 / 48
18 Repeated Games Definition A repeated game is (i) a terminal date, T, giving the number of times the players interact, where T = 1,2,3,... The calendar date is given by t = 1,2,...,T. (ii) a discount factor, 0 δ 1, that represents both how patient the players are and how likely the game is to continue. (iii) a stage game of finite length: A specification of the players, actions, payoffs, and timing, which is usually independent of the calendar date. If s t = (s1 t,st 2,...,st N ) is the strategy profile that occurs in period t, the players discounted payoff is u i (s 1 )+δu i (s 2 )+δ 2 u i (s 3 )+...+δ T 1 u i (s T ) = T δ t 1 u i (s t ) t=1 Econ 400 (ND) Repeated Games 8 / 48
19 Histories Presumably, players will want to keep track of how their opponents have behaved in previous periods; this allows them to choose strategies that reward or punish other players for good or bad behavior. But how do we keep track of what has happened in repeated games? Econ 400 (ND) Repeated Games 9 / 48
20 Histories Presumably, players will want to keep track of how their opponents have behaved in previous periods; this allows them to choose strategies that reward or punish other players for good or bad behavior. But how do we keep track of what has happened in repeated games? In prisoners dilemma, all of the possible outcomes from one repetition of the game are Σ = {(S,S),(S,C),(C,S),(C,C)} Econ 400 (ND) Repeated Games 9 / 48
21 Histories Presumably, players will want to keep track of how their opponents have behaved in previous periods; this allows them to choose strategies that reward or punish other players for good or bad behavior. But how do we keep track of what has happened in repeated games? In prisoners dilemma, all of the possible outcomes from one repetition of the game are Σ = {(S,S),(S,C),(C,S),(C,C)} When the game goes two periods, however, it becomes Σ 2 = {[(S,S),(S,S)],[(S,C),(S,S)],[(C,S),(S,S)],[(C,C),(S,S)], [(S,S),(S,C)],[(S,C),(S,C)],[(C,S),(S,C)],[(C,C),(S,C)], [(S,S),(C,S)],[(S,C),(C,S)],[(C,S),(C,S)],[(C,C),(C,S)], [(S,S),(C,C)],[(S,C),(C,C)],[(C,S),(C,C)],[(C,C),(C,C)]} since we need to keep track of what happened in the first period, and the second period. Econ 400 (ND) Repeated Games 9 / 48
22 Histories Let Σ be the set of all the strategy profiles for the stage game. (For example, in prisoners dilemma, Σ = {(S,S),(S,C),(C,S),(C,C)}). Econ 400 (ND) Repeated Games 10 / 48
23 Histories Let Σ be the set of all the strategy profiles for the stage game. (For example, in prisoners dilemma, Σ = {(S,S),(S,C),(C,S),(C,C)}). If we want to keep track of the outcomes of a repeated game, we re interested in sequences of observations from Σ. For two periods, Σ Σ = Σ 2 is the set of all possible outcomes for two repetitions of the game. For three periods, Σ Σ Σ = Σ 3 is the set of all possible outcomes for three repetitions of the game, and so on. Econ 400 (ND) Repeated Games 10 / 48
24 Histories Definition Let the set of all strategy profiles for the stage game be Σ. Then the set Σ t = Σ Σ... Σ contains all lists of the possible outcomes in terms of what strategies the players have used in each of the t periods. Then any element h t of the set Σ t = H t is a history at time t. Econ 400 (ND) Repeated Games 11 / 48
25 Equilibria in Repeated Games Definition A set of strategies is a Subgame Perfect Nash Equilibrium of a repeated game if, for any t-period history h t, there is no subgame in which any player has a profitable deviation. Econ 400 (ND) Repeated Games 12 / 48
26 Equilibria in Repeated Games Definition A set of strategies is a Subgame Perfect Nash Equilibrium of a repeated game if, for any t-period history h t, there is no subgame in which any player has a profitable deviation. Note that no player can have a profitable deviation for any history, even though, given the strategies, only one history actually occurs. But it is precisely because the players know the consequences of their actions that the equilibrium history arises. Econ 400 (ND) Repeated Games 12 / 48
27 Equilibria in Repeated Games Suppose the stage game has an equilibrium s = (s1,...,s N ). Then the strategy for the repeated game where each player i plays si after every history is a Subgame Perfect Nash Equilibrium of the repeated game. Econ 400 (ND) Repeated Games 13 / 48
28 Equilibria in Repeated Games Suppose the stage game has an equilibrium s = (s1,...,s N ). Then the strategy for the repeated game where each player i plays si after every history is a Subgame Perfect Nash Equilibrium of the repeated game. But is this the only equilibrium of a repeated game? Econ 400 (ND) Repeated Games 13 / 48
29 Prisoners Dilemma Consider the following behavior strategy for an infinitely repeated prisoners dilemma: If the history at time t is {(S,S),(S,S),...,(S,S)}, play S. For any other history at time t, player C. Econ 400 (ND) Repeated Games 14 / 48
30 Prisoners Dilemma Consider the following behavior strategy for an infinitely repeated prisoners dilemma: If the history at time t is {(S,S),(S,S),...,(S,S)}, play S. For any other history at time t, player C. Is this a subgame perfect Nash equilibrium of the infinitely repeated game? Econ 400 (ND) Repeated Games 14 / 48
31 Prisoners Dilemma: Equilibrium Analysis Well, we have to check all the possible histories to see if there are any profitable deviations. There s really just two cases: and anything else. {(S,S),(S,S),...,(S,S)} Econ 400 (ND) Repeated Games 15 / 48
32 Prisoners Dilemma: Equilibrium Analysis Well, we have to check all the possible histories to see if there are any profitable deviations. There s really just two cases: {(S,S),(S,S),...,(S,S)} and anything else. Let s start with anything else : Suppose the history at time t is not {(S,S),(S,S),...,(S,S)}. Then today and in all future periods, all my opponents will choose C. Then I should choose C, since it maximizes my discounted payoff. So there are no profitable deviations from the proposed strategies for these histories. Econ 400 (ND) Repeated Games 15 / 48
33 Prisoners Dilemma: Equilibrium Analysis Suppose the history at time t is {(S,S),(S,S),...,(S,S)}. Is playing S an optimal strategy, given the behavior strategies of the players? If I choose S this period, the next period s history is {(S,S),(S,S),...,(S,S),(S,S)}, and given the strategies, the players will choose S from then on forever. The payoff from that is 1+δ +δ 2 +δ = 1 1 δ Econ 400 (ND) Repeated Games 16 / 48
34 Prisoners Dilemma: Equilibrium Analysis Suppose the history at time t is {(S,S),(S,S),...,(S,S)}. Is playing S an optimal strategy, given the behavior strategies of the players? If I choose S this period, the next period s history is {(S,S),(S,S),...,(S,S),(S,S)}, and given the strategies, the players will choose S from then on forever. The payoff from that is 1+δ +δ 2 +δ = 1 1 δ If I deviate this period, next period s history is {(S,S),(S,S),...,(S,S),(C,S)} {(S,S),(S,S),...,(S,S),(S,S)} so all players will confess, generating a sequence of histories where players confess forever. Econ 400 (ND) Repeated Games 16 / 48
35 Prisoners Dilemma: Equilibrium Analysis Suppose the history at time t is {(S,S),(S,S),...,(S,S)}. Is playing S an optimal strategy, given the behavior strategies of the players? If I choose S this period, the next period s history is {(S,S),(S,S),...,(S,S),(S,S)}, and given the strategies, the players will choose S from then on forever. The payoff from that is 1+δ +δ 2 +δ = 1 1 δ If I deviate this period, next period s history is {(S,S),(S,S),...,(S,S),(C,S)} {(S,S),(S,S),...,(S,S),(S,S)} so all players will confess, generating a sequence of histories where players confess forever. The payoff from that is 2 δ0 δ = 2 Econ 400 (ND) Repeated Games 16 / 48
36 Prisoners Dilemma: Equilibrium Analysis So it all comes down to whether it s better to cooperate than cheat in any period, or 1 1 δ 2 δ 1 2 Econ 400 (ND) Repeated Games 17 / 48
37 Prisoners Dilemma: Equilibrium If δ > 1 2, both players using the strategy If the history at time t is {(S,S),(S,S),...,(S,S)}, play S. For any other history at time t, play C. is a Subgame Perfect Nash Equilibrium of the infinitely repeated prisoners dilemma. Econ 400 (ND) Repeated Games 18 / 48
38 Collusion and Bertrand In the Bertrand pricing game, there are two firms 1 and 2 who have the same marginal costs c and compete in prices, choosing p 1,p 2 = {0,1,...,c,...,8,...,10}. Econ 400 (ND) Repeated Games 19 / 48
39 Collusion and Bertrand In the Bertrand pricing game, there are two firms 1 and 2 who have the same marginal costs c and compete in prices, choosing p 1,p 2 = {0,1,...,c,...,8,...,10}. Fix demand at 1 for all prices less than or equal to 10. Econ 400 (ND) Repeated Games 19 / 48
40 Collusion and Bertrand In the Bertrand pricing game, there are two firms 1 and 2 who have the same marginal costs c and compete in prices, choosing p 1,p 2 = {0,1,...,c,...,8,...,10}. Fix demand at 1 for all prices less than or equal to 10. If p 1 < p 2, all the consumers go to firm 1 and firm 2 gets no business. If p 1 = p 2, the firms split the market equally. If p 2 > p 1, all consumers go to firm 2 and firm 1 gets no business. Econ 400 (ND) Repeated Games 19 / 48
41 Collusion and Bertrand In the Bertrand pricing game, there are two firms 1 and 2 who have the same marginal costs c and compete in prices, choosing p 1,p 2 = {0,1,...,c,...,8,...,10}. Fix demand at 1 for all prices less than or equal to 10. If p 1 < p 2, all the consumers go to firm 1 and firm 2 gets no business. If p 1 = p 2, the firms split the market equally. If p 2 > p 1, all consumers go to firm 2 and firm 1 gets no business. The profits for firm 1 are: (1 (1)(p ) 1 c),p 1 < p 2 π 1 (p 1,p 2 ) = 2 (p1 c),p 1 = p 2 0,p 1 > p 2 and similarly for firm 2. Econ 400 (ND) Repeated Games 19 / 48
42 Collusion and Bertrand Consider the repeated game: T = Discount factor: 0 < δ < 1 Stage game: Bertrand Competition Econ 400 (ND) Repeated Games 20 / 48
43 Collusion and Bertrand Consider the repeated game: T = Discount factor: 0 < δ < 1 Stage game: Bertrand Competition Notice that if there are 11 price increments, there are (22) t possible outcomes we might observe by time t. If t = 5, there are 5,153,632 histories. Even on a very good computer, computing the extensive form and payoffs would take a lot of time. Econ 400 (ND) Repeated Games 20 / 48
44 Bertrand: Stage Game Equilibrium and Trigger Strategies We know from the first part of the class that p1 = p 2 = c is a Nash equilibrium of the stage game. Let s use this as the punishment for a breakdown in cooperation, but otherwise have the players use p 1 = p 2 = 10. Econ 400 (ND) Repeated Games 21 / 48
45 Bertrand: Stage Game Equilibrium and Trigger Strategies We know from the first part of the class that p1 = p 2 = c is a Nash equilibrium of the stage game. Let s use this as the punishment for a breakdown in cooperation, but otherwise have the players use p 1 = p 2 = 10. Consider the strategies: If the history is {(10,10),(10,10),(10,10),...,(10,10)}, play 10 this period. For any other history, play c this period. Econ 400 (ND) Repeated Games 21 / 48
46 Bertrand: The Optimal Deviation If your opponent adopts these strategies and plays nice up to some date t = 0,1,..., what is the best way to stab him in the back? Econ 400 (ND) Repeated Games 22 / 48
47 Bertrand: The Optimal Deviation If your opponent adopts these strategies and plays nice up to some date t = 0,1,..., what is the best way to stab him in the back? (If you re going to ruin a relationship, at least do it optimally) Econ 400 (ND) Repeated Games 22 / 48
48 Bertrand: The Optimal Deviation If your opponent adopts these strategies and plays nice up to some date t = 0,1,..., what is the best way to stab him in the back? (If you re going to ruin a relationship, at least do it optimally) By charging the price just below the cutoff point, 9 1, the deviator captures the whole market and only losses a dollar on each unit: π d = (1)(9 1 c) Econ 400 (ND) Repeated Games 22 / 48
49 Bertrand: Equilibrium Analysis Once a deviation has occurred, the players use p1 = p 2 = c in all future periods. In any of these scenarios, there are no profitable deviations, because if your opponent is playing c in these periods, your best response is c. The payoff to cooperating is 10 c 2 +δ 10 c 2 +δ 210 c 2 The payoff to optimally deviating is +... = 10 c δ (9 c)+δ0+δ = 9 c Econ 400 (ND) Repeated Games 23 / 48
50 Bertrand: Equilibrium Analysis Then cooperating is better than deviating if or 10 c δ 9 c δ 8 c 19 2c Econ 400 (ND) Repeated Games 24 / 48
51 Bertrand: Equilibrium Analysis Then cooperating is better than deviating if or 10 c δ 9 c δ 8 c 19 2c For example, if c = 2, we have δ 6/15. But if c = 6, we have δ = 2/7. Econ 400 (ND) Repeated Games 24 / 48
52 Bertrand: Equilibrium Then as long as the strategies δ 8 c 19 2c If the history is {(10,10),(10,10),(10,10),...,(10,10)}, play 10 this period. For any other history, play c this period. are a Subgame Perfect Nash Equilibrium of the infinitely repeated Bertrand game. Econ 400 (ND) Repeated Games 25 / 48
53 Bertrand: Equilibrium Then as long as the strategies δ 8 c 19 2c If the history is {(10,10),(10,10),(10,10),...,(10,10)}, play 10 this period. For any other history, play c this period. are a Subgame Perfect Nash Equilibrium of the infinitely repeated Bertrand game. So collusive is possible in the infinite-horizon version of the repeated game. (What about the finite version of the repeated game?) Econ 400 (ND) Repeated Games 25 / 48
54 The Pattern Solve for all of the equilibria of the stage game. (Competitive Play) Econ 400 (ND) Repeated Games 26 / 48
55 The Pattern Solve for all of the equilibria of the stage game. (Competitive Play) Find a strategy profile that gives all the players a higher payoff. (Cooperative Play) Econ 400 (ND) Repeated Games 26 / 48
56 The Pattern Solve for all of the equilibria of the stage game. (Competitive Play) Find a strategy profile that gives all the players a higher payoff. (Cooperative Play) Enforce cooperation through trigger strategies: If all players have previously cooperated, continue cooperating. If any player has previously defected, play competitively. Econ 400 (ND) Repeated Games 26 / 48
57 The Pattern Solve for all of the equilibria of the stage game. (Competitive Play) Find a strategy profile that gives all the players a higher payoff. (Cooperative Play) Enforce cooperation through trigger strategies: If all players have previously cooperated, continue cooperating. If any player has previously defected, play competitively. For sufficiently high values of the discount factor δ, this will be an equilibrium of the repeated game. Econ 400 (ND) Repeated Games 26 / 48
58 Examples of Equilibria in Repeated Games For any game, playing the equilibrium of the stage game forever is a Subgame Perfect Nash Equilibrium. In prisoners dilemma, as long as the players didn t discount their payoffs too much (δ >.5), one Subgame Perfect Nash Equilibrium of the game was to be silent in all periods unless your opponent had previously confessed at some point, and then to confess forever. In Cournot, as long as the players didn t discount their payoffs too much (δ >.377), one Subgame Perfect Nash Equilibrium of the game was to collude in all periods unless your opponent had previously played competitively, and then to play the Cournot quantity forever Econ 400 (ND) Repeated Games 27 / 48
59 Examples of Equilibria in Repeated Games For any game, playing the equilibrium of the stage game forever is a Subgame Perfect Nash Equilibrium. In prisoners dilemma, as long as the players didn t discount their payoffs too much (δ >.5), one Subgame Perfect Nash Equilibrium of the game was to be silent in all periods unless your opponent had previously confessed at some point, and then to confess forever. In Cournot, as long as the players didn t discount their payoffs too much (δ >.377), one Subgame Perfect Nash Equilibrium of the game was to collude in all periods unless your opponent had previously played competitively, and then to play the Cournot quantity forever Today, we want to make the argument that As long as players are patient, they can cooperate in infinitely repeated games in ways that aren t possible in finitely repeated games. Econ 400 (ND) Repeated Games 27 / 48
60 The Nash Threats Folk Theorem Theorem Consider any N-player infinitely repeated game with a stage game equilibrium s = (s 1,s 2,...,s N ) yielding payoffs u = (u 1,u 2,...,u N ). Suppose there is another strategy profile ŝ = (ŝ 1,ŝ 2,...,ŝ N ) yielding payoffs û = (û 1,û 2,...,û N ), where, for every player i, û i u i If the players are sufficiently patient, then there is a Subgame Perfect Nash Equilibrium in which the players use ŝ in every period of the infinitely repeated game. Econ 400 (ND) Repeated Games 28 / 48
61 The Folk Theorem Let s fill in the blanks for the prisoners dilemma game: Theorem Consider prisoners dilemma with a stage game equilibrium s = (C,C) yielding payoffs u = (0,0). Suppose there is another strategy profile ŝ = (S,S) yielding payoffs û = (1,1), where, for every player i, 1 0 If the players are sufficiently patient, then there is a Subgame Perfect Nash Equilibrium in which the players use (S,S) in every period of the infinitely repeated game. Econ 400 (ND) Repeated Games 29 / 48
62 The Folk Theorem Let s fill in the blanks for the prisoners dilemma game: Theorem Consider prisoners dilemma with a stage game equilibrium s = (C,C) yielding payoffs u = (0,0). Suppose there is another strategy profile ŝ = (S,S) yielding payoffs û = (1,1), where, for every player i, 1 0 If the players are sufficiently patient, then there is a Subgame Perfect Nash Equilibrium in which the players use (S,S) in every period of the infinitely repeated game. What do we need to think about in proving the Folk Theorem? Econ 400 (ND) Repeated Games 29 / 48
63 The Folk Theorem: Trigger Strategies Consider the following trigger strategy for player i: If the history at t is h t = (ŝ,ŝ,...,ŝ), play ŝ i in period t. For any other history at time t, play s i in period t. Econ 400 (ND) Repeated Games 30 / 48
64 The Folk Theorem: Trigger Strategies Consider the following trigger strategy for player i: If the history at t is h t = (ŝ,ŝ,...,ŝ), play ŝ i in period t. For any other history at time t, play s i in period t. This is called a trigger strategy because it starts in cooperative mode, but after any defection by any player, it switches to punishment or competitive mode, and they play the stage game strategies forever. Econ 400 (ND) Repeated Games 30 / 48
65 The Folk Theorem: Optimal Deviations Since û is presumably not a Nash equilibrium of the stage game, there are at least some players for whom u d j > û j u j i.e., while they prefer cooperating to the equilibrium of the stage game, they prefer defection to cooperation. Econ 400 (ND) Repeated Games 31 / 48
66 The Folk Theorem: Optimal Deviations Since û is presumably not a Nash equilibrium of the stage game, there are at least some players for whom u d j > û j u j i.e., while they prefer cooperating to the equilibrium of the stage game, they prefer defection to cooperation. Notice that if a player is tempted to deviate, the above inequality implies that u d j û j u d j û j Econ 400 (ND) Repeated Games 31 / 48
67 The Folk Theorem: Cooperating and Deviating The payoff to cooperating to player j is û j +δû j +δ 2 û j +... = 1 1 δûj Econ 400 (ND) Repeated Games 32 / 48
68 The Folk Theorem: Cooperating and Deviating The payoff to cooperating to player j is û j +δû j +δ 2 û j +... = 1 The payoff to deviating to player j is 1 δûj û d j +δu j +δu j +... = û d j +u j δ(1+δ +δ2 +...) = û d j +u j δ 1 δ Econ 400 (ND) Repeated Games 32 / 48
69 The Folk Theorem: Cooperating and Deviating The payoff to cooperating to player j is û j +δû j +δ 2 û j +... = 1 The payoff to deviating to player j is 1 δûj û d j +δu j +δu j +... = û d j +u j δ(1+δ +δ2 +...) = û d j +u j Then cooperating is better than deviating for player j if or 1 1 δ j û j û d j +u j δ j ud j û j u d j u j But 1 δ j, by the work on the previous slide. δ j 1 δ j δ 1 δ Econ 400 (ND) Repeated Games 32 / 48
70 The Folk Theorem: Equilibrium: Let δ = {δ 1,δ 2,...,δ N } so we have selected the highest discount factor for which cooperating is better than deviating, for all the players. Econ 400 (ND) Repeated Games 33 / 48
71 The Folk Theorem: Equilibrium: Let δ = {δ 1,δ 2,...,δ N } so we have selected the highest discount factor for which cooperating is better than deviating, for all the players. Then if all players are sufficiently patient, in the sense that each of their discount factors are greater than δ, then the trigger strategies are a subgame perfect Nash equilibrium of the infinitely repeated game, and they will play ŝ in every period. Econ 400 (ND) Repeated Games 33 / 48
72 The Folk Theorem: Geometry Econ 400 (ND) Repeated Games 34 / 48
73 Solving for Equilibria in Repeated Games 1. Solve for all equilibria of the stage game. (Competition) Econ 400 (ND) Repeated Games 35 / 48
74 Solving for Equilibria in Repeated Games 1. Solve for all equilibria of the stage game. (Competition) 2. Find a strategy profile (equilibrium or not) where all the players do at least as well as in the stage game. (Cooperation) Econ 400 (ND) Repeated Games 35 / 48
75 Solving for Equilibria in Repeated Games 1. Solve for all equilibria of the stage game. (Competition) 2. Find a strategy profile (equilibrium or not) where all the players do at least as well as in the stage game. (Cooperation) 3. Design trigger strategies that support cooperation and punish with competition. Econ 400 (ND) Repeated Games 35 / 48
76 Solving for Equilibria in Repeated Games 1. Solve for all equilibria of the stage game. (Competition) 2. Find a strategy profile (equilibrium or not) where all the players do at least as well as in the stage game. (Cooperation) 3. Design trigger strategies that support cooperation and punish with competition. 4. Find the optimal deviation for any player. Compute the minimum discount factor for which cooperating is an equilibrium. Econ 400 (ND) Repeated Games 35 / 48
77 Solving for Equilibria in Repeated Games 1. Solve for all equilibria of the stage game. (Competition) 2. Find a strategy profile (equilibrium or not) where all the players do at least as well as in the stage game. (Cooperation) 3. Design trigger strategies that support cooperation and punish with competition. 4. Find the optimal deviation for any player. Compute the minimum discount factor for which cooperating is an equilibrium. 5. Conclude that the trigger strategies are an equilibrium of the infinitely repeated game as long as all the players are sufficiently patient. Econ 400 (ND) Repeated Games 35 / 48
78 Solving for Equilibria in Repeated Games 1. Solve for all equilibria of the stage game. (Competition) 2. Find a strategy profile (equilibrium or not) where all the players do at least as well as in the stage game. (Cooperation) 3. Design trigger strategies that support cooperation and punish with competition. 4. Find the optimal deviation for any player. Compute the minimum discount factor for which cooperating is an equilibrium. 5. Conclude that the trigger strategies are an equilibrium of the infinitely repeated game as long as all the players are sufficiently patient. Note that things are a little more complicated than this: We don t prove directly that this is a Subgame Perfect Nash Equilibrium (which is just a little bit more work), we appeal to the Nash Threats Folk Theorem and that takes care of the boilerplate details. Econ 400 (ND) Repeated Games 35 / 48
79 Trade and Bonuses/Tips Why do people tip for services or firms pay bonuses to workers? Econ 400 (ND) Repeated Games 36 / 48
80 Equilibria with Forgiveness The grim trigger strategies of the Nash Threats Folk Theorem are pretty harsh: Mess up once, and cooperation is cut off forever. What if punish by playing the stage game equilibrium K rounds and then return to cooperative mode, instead? Econ 400 (ND) Repeated Games 37 / 48
81 Equilibria with Forgiveness The grim trigger strategies of the Nash Threats Folk Theorem are pretty harsh: Mess up once, and cooperation is cut off forever. What if punish by playing the stage game equilibrium K rounds and then return to cooperative mode, instead? Then for Prisoners Dilemma, cooperating is better than deviating if 1+δ +δ δ0+δ δ K 0+δ K+1 +δ K Econ 400 (ND) Repeated Games 37 / 48
82 Equilibria with Forgiveness The grim trigger strategies of the Nash Threats Folk Theorem are pretty harsh: Mess up once, and cooperation is cut off forever. What if punish by playing the stage game equilibrium K rounds and then return to cooperative mode, instead? Then for Prisoners Dilemma, cooperating is better than deviating if 1+δ +δ δ0+δ δ K 0+δ K+1 +δ K δ 1 2+δK+1 1 δ By making K sufficiently large and taking δ sufficiently close to 1, the equality will hold. Econ 400 (ND) Repeated Games 37 / 48
83 Equilibria with Forgiveness Then cooperating is better than deviating if 2δ 1+δ K+1 or K log(2δ 1) log(δ) 1 Econ 400 (ND) Repeated Games 38 / 48
84 Equilibria with Forgiveness Then cooperating is better than deviating if or K 2δ 1+δ K+1 log(2δ 1) log(δ) If you compute the limit as δ 1 (use L Hopital s rule twice), you get that the minimal punishment period is 0. So players that are sufficiently patient will never cheat on each other. 1 Econ 400 (ND) Repeated Games 38 / 48
85 Collusion and Cournot Recall the Cournot game: Two firms simultaneously choose quantities q 1,q 2 > 0, where the market price is p = A q 1 q 2 and the firms have no costs. (Step 1) The Nash equilibrium of the stage game is q 1 = q 2 = A 3 giving profits ) 2 π 1 = π 2 = ( A 3 Econ 400 (ND) Repeated Games 39 / 48
86 Collusion and Cournot Recall the Cournot game: Two firms simultaneously choose quantities q 1,q 2 > 0, where the market price is p = A q 1 q 2 and the firms have no costs. (Step 1) The Nash equilibrium of the stage game is q 1 = q 2 = A 3 giving profits ) 2 π 1 = π 2 = ( A 3 What if the firms made an agreement to work together to improve their profits? What could they achieve? Econ 400 (ND) Repeated Games 39 / 48
87 Collusion and Cournot Summing the firms profits, we get π 1 +π 2 = (A q 1 q 2 )q 1 +(A q 1 q 2 )q 2 = (A q 1 q 2 )(q 1 +q 2 ) or Π = (A Q)Q Econ 400 (ND) Repeated Games 40 / 48
88 Collusion and Cournot Summing the firms profits, we get π 1 +π 2 = (A q 1 q 2 )q 1 +(A q 1 q 2 )q 2 = (A q 1 q 2 )(q 1 +q 2 ) or maximizing gives Π = (A Q)Q Q = A 2, Π = ( ) A 2 2 Since Π = A2 4 > 2A2 9 = π 1 +π 2, collusion is potentially profitable. Econ 400 (ND) Repeated Games 40 / 48
89 Collusion and Cournot (Step 2) Suppose the two firms are playing the Cournot game an infinite number of times, and they share a discount factor δ. Let ˆq = A 4, ˆπ = ( A 4 ) 2 This is half the monopoly quantity, A/2, and strictly less than the Cournot equilibrium quantity A/3. Econ 400 (ND) Repeated Games 41 / 48
90 Collusion and Cournot (Step 2) Suppose the two firms are playing the Cournot game an infinite number of times, and they share a discount factor δ. Let ˆq = A 4, ˆπ = ( A 4 ) 2 This is half the monopoly quantity, A/2, and strictly less than the Cournot equilibrium quantity A/3. (Step 3) Consider the strategy: If the two firms have both used ˆq in all previous periods, use ˆq = A/4 this period. If either firm ever did anything besides ˆq, play the stage Cournot quantity q = A/3. Is this a subgame perfect Nash equilibrium of the infinitely repeated game? Econ 400 (ND) Repeated Games 41 / 48
91 Equilibrium Analysis: The Optimal Deviation To decide if this is an equilibrium, we need to know the payoff from optimally stabbing your partner in the back (if we re going to break up a relationship, we might as well do it optimally, right?). (Step 4) Then we need to solve max(a ˆq q )q = max (A A4 ) q q q q Maximizing gives q = 3A 8, π = ( ) 3A 2 8 Econ 400 (ND) Repeated Games 42 / 48
92 Equilibrium Analysis: Cooperation Cooperating in some period t after a history in which all the players previously cooperated keeps the industry in collusive mode (remember, given the proposed strategies), giving a payoff of ˆπ +δˆπ +δ 2ˆπ +... = ˆπ 1 1 δ which equals ( ) A δ Econ 400 (ND) Repeated Games 43 / 48
93 Equilibrium Analysis: Deviation Deviating optimally in some period t after a history in which all the previously players cooperated switches the game from collusion to competition (remember, given the proposed strategies), giving a payoff of π +δπ +δ 2 π +... = ( ) 3A 2 +δ 8 ( ) A 2 ( 1+δ +δ ) 3 Which is ( ) 3A 2 +δ 8 ( ) A δ Econ 400 (ND) Repeated Games 44 / 48
94 Equilibrium Analysis Then cooperating is better than deviating if or or ( ) A δ ( ) 3A 2 +δ δ δ 91 δ δ ( ) A δ So if δ 17 45, there are no profitable deviations as long as the players have previous cooperated. Econ 400 (ND) Repeated Games 45 / 48
95 Cournot and Collusion (Step 5) So if the players are sufficiently patient, or then the strategies δ If the two firms have both used ˆq in all previous periods, use ˆq = A/4 this period. If either firm ever did anything besides ˆq, play the stage Cournot quantity q = A/3. are a subgame perfect Nash equilibrium of the infinitely repeated Cournot game. Econ 400 (ND) Repeated Games 46 / 48
96 Money as a long-lived institution Suppose each calendar date t, a generation of new people are born, and a previous generation die. Each generation lives for two periods. In the first period, they receive a wage of 4 during their working years, but receive no wage in period two of their lives. Their utility function over consumption pairs is u(c 1,c 2 ) = c 1 + c 2, where c 1 is the amount consumed in period 1 and c 2 is the amount consumed in period 2. Econ 400 (ND) Repeated Games 47 / 48
97 Money as a long-lived institution Suppose that there is a dollar that each generation hands to the next one. The dollar obligates the young to give the old y units of their wealth so that the old have consumption in old age. Econ 400 (ND) Repeated Games 48 / 48
98 Money as a long-lived institution Suppose that there is a dollar that each generation hands to the next one. The dollar obligates the young to give the old y units of their wealth so that the old have consumption in old age. Suppose the players adopt the following strategies: If all previous generations have accepted the dollar and paid y to the old, do so this period. If any previous generation refused to pay, do not pay. Econ 400 (ND) Repeated Games 48 / 48
99 Money as a long-lived institution Suppose that there is a dollar that each generation hands to the next one. The dollar obligates the young to give the old y units of their wealth so that the old have consumption in old age. Suppose the players adopt the following strategies: If all previous generations have accepted the dollar and paid y to the old, do so this period. If any previous generation refused to pay, do not pay. Suppose the economy only lasts for T periods. For what values of y is it an equilibrium for the young to honor the dollar? Econ 400 (ND) Repeated Games 48 / 48
100 Money as a long-lived institution Suppose that there is a dollar that each generation hands to the next one. The dollar obligates the young to give the old y units of their wealth so that the old have consumption in old age. Suppose the players adopt the following strategies: If all previous generations have accepted the dollar and paid y to the old, do so this period. If any previous generation refused to pay, do not pay. Suppose the economy only lasts for T periods. For what values of y is it an equilibrium for the young to honor the dollar? Suppose the economy goes on infinitely. For what values of y is it an equilibrium for the young to honor the dollar? Econ 400 (ND) Repeated Games 48 / 48
The Nash equilibrium of the stage game is (D, R), giving payoffs (0, 0). Consider the trigger strategies:
Problem Set 4 1. (a). Consider the infinitely repeated game with discount rate δ, where the strategic fm below is the stage game: B L R U 1, 1 2, 5 A D 2, 0 0, 0 Sketch a graph of the players payoffs.
More informationIntroduction to Game Theory Lecture Note 5: Repeated Games
Introduction to Game Theory Lecture Note 5: Repeated Games Haifeng Huang University of California, Merced Repeated games Repeated games: given a simultaneous-move game G, a repeated game of G is an extensive
More informationWarm Up Finitely Repeated Games Infinitely Repeated Games Bayesian Games. Repeated Games
Repeated Games Warm up: bargaining Suppose you and your Qatz.com partner have a falling-out. You agree set up two meetings to negotiate a way to split the value of your assets, which amount to $1 million
More informationDuopoly models Multistage games with observed actions Subgame perfect equilibrium Extensive form of a game Two-stage prisoner s dilemma
Recap Last class (September 20, 2016) Duopoly models Multistage games with observed actions Subgame perfect equilibrium Extensive form of a game Two-stage prisoner s dilemma Today (October 13, 2016) Finitely
More informationG5212: Game Theory. Mark Dean. Spring 2017
G5212: Game Theory Mark Dean Spring 2017 Bargaining We will now apply the concept of SPNE to bargaining A bit of background Bargaining is hugely interesting but complicated to model It turns out that the
More informationIn reality; some cases of prisoner s dilemma end in cooperation. Game Theory Dr. F. Fatemi Page 219
Repeated Games Basic lesson of prisoner s dilemma: In one-shot interaction, individual s have incentive to behave opportunistically Leads to socially inefficient outcomes In reality; some cases of prisoner
More informationGame Theory. Wolfgang Frimmel. Repeated Games
Game Theory Wolfgang Frimmel Repeated Games 1 / 41 Recap: SPNE The solution concept for dynamic games with complete information is the subgame perfect Nash Equilibrium (SPNE) Selten (1965): A strategy
More informationCHAPTER 14: REPEATED PRISONER S DILEMMA
CHAPTER 4: REPEATED PRISONER S DILEMMA In this chapter, we consider infinitely repeated play of the Prisoner s Dilemma game. We denote the possible actions for P i by C i for cooperating with the other
More informationEconomics 431 Infinitely repeated games
Economics 431 Infinitely repeated games Letuscomparetheprofit incentives to defect from the cartel in the short run (when the firm is the only defector) versus the long run (when the game is repeated)
More informationEconomics 171: Final Exam
Question 1: Basic Concepts (20 points) Economics 171: Final Exam 1. Is it true that every strategy is either strictly dominated or is a dominant strategy? Explain. (5) No, some strategies are neither dominated
More informationAnswer Key: Problem Set 4
Answer Key: Problem Set 4 Econ 409 018 Fall A reminder: An equilibrium is characterized by a set of strategies. As emphasized in the class, a strategy is a complete contingency plan (for every hypothetical
More informationNot 0,4 2,1. i. Show there is a perfect Bayesian equilibrium where player A chooses to play, player A chooses L, and player B chooses L.
Econ 400, Final Exam Name: There are three questions taken from the material covered so far in the course. ll questions are equally weighted. If you have a question, please raise your hand and I will come
More informationRepeated Games. September 3, Definitions: Discounting, Individual Rationality. Finitely Repeated Games. Infinitely Repeated Games
Repeated Games Frédéric KOESSLER September 3, 2007 1/ Definitions: Discounting, Individual Rationality Finitely Repeated Games Infinitely Repeated Games Automaton Representation of Strategies The One-Shot
More informationREPEATED GAMES. MICROECONOMICS Principles and Analysis Frank Cowell. Frank Cowell: Repeated Games. Almost essential Game Theory: Dynamic.
Prerequisites Almost essential Game Theory: Dynamic REPEATED GAMES MICROECONOMICS Principles and Analysis Frank Cowell April 2018 1 Overview Repeated Games Basic structure Embedding the game in context
More informationFDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.
FDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.) Hints for Problem Set 3 1. Consider the following strategic
More informationECE 586BH: Problem Set 5: Problems and Solutions Multistage games, including repeated games, with observed moves
University of Illinois Spring 01 ECE 586BH: Problem Set 5: Problems and Solutions Multistage games, including repeated games, with observed moves Due: Reading: Thursday, April 11 at beginning of class
More informationInfinitely Repeated Games
February 10 Infinitely Repeated Games Recall the following theorem Theorem 72 If a game has a unique Nash equilibrium, then its finite repetition has a unique SPNE. Our intuition, however, is that long-term
More informationEarly PD experiments
REPEATED GAMES 1 Early PD experiments In 1950, Merrill Flood and Melvin Dresher (at RAND) devised an experiment to test Nash s theory about defection in a two-person prisoners dilemma. Experimental Design
More informationIn the Name of God. Sharif University of Technology. Graduate School of Management and Economics
In the Name of God Sharif University of Technology Graduate School of Management and Economics Microeconomics (for MBA students) 44111 (1393-94 1 st term) - Group 2 Dr. S. Farshad Fatemi Game Theory Game:
More informationIntroduction to Industrial Organization Professor: Caixia Shen Fall 2014 Lecture Note 5 Games and Strategy (Ch. 4)
Introduction to Industrial Organization Professor: Caixia Shen Fall 2014 Lecture Note 5 Games and Strategy (Ch. 4) Outline: Modeling by means of games Normal form games Dominant strategies; dominated strategies,
More informationProblem 3 Solutions. l 3 r, 1
. Economic Applications of Game Theory Fall 00 TA: Youngjin Hwang Problem 3 Solutions. (a) There are three subgames: [A] the subgame starting from Player s decision node after Player s choice of P; [B]
More informationFebruary 23, An Application in Industrial Organization
An Application in Industrial Organization February 23, 2015 One form of collusive behavior among firms is to restrict output in order to keep the price of the product high. This is a goal of the OPEC oil
More informationEconomics and Computation
Economics and Computation ECON 425/563 and CPSC 455/555 Professor Dirk Bergemann and Professor Joan Feigenbaum Reputation Systems In case of any questions and/or remarks on these lecture notes, please
More informationRepeated games. Felix Munoz-Garcia. Strategy and Game Theory - Washington State University
Repeated games Felix Munoz-Garcia Strategy and Game Theory - Washington State University Repeated games are very usual in real life: 1 Treasury bill auctions (some of them are organized monthly, but some
More informationCUR 412: Game Theory and its Applications Final Exam Ronaldo Carpio Jan. 13, 2015
CUR 41: Game Theory and its Applications Final Exam Ronaldo Carpio Jan. 13, 015 Instructions: Please write your name in English. This exam is closed-book. Total time: 10 minutes. There are 4 questions,
More informationEconS 424 Strategy and Game Theory. Homework #5 Answer Key
EconS 44 Strategy and Game Theory Homework #5 Answer Key Exercise #1 Collusion among N doctors Consider an infinitely repeated game, in which there are nn 3 doctors, who have created a partnership. In
More informationStochastic Games and Bayesian Games
Stochastic Games and Bayesian Games CPSC 532L Lecture 10 Stochastic Games and Bayesian Games CPSC 532L Lecture 10, Slide 1 Lecture Overview 1 Recap 2 Stochastic Games 3 Bayesian Games Stochastic Games
More informationEconomics 209A Theory and Application of Non-Cooperative Games (Fall 2013) Repeated games OR 8 and 9, and FT 5
Economics 209A Theory and Application of Non-Cooperative Games (Fall 2013) Repeated games OR 8 and 9, and FT 5 The basic idea prisoner s dilemma The prisoner s dilemma game with one-shot payoffs 2 2 0
More informationEcon 101A Final exam Mo 18 May, 2009.
Econ 101A Final exam Mo 18 May, 2009. Do not turn the page until instructed to. Do not forget to write Problems 1 and 2 in the first Blue Book and Problems 3 and 4 in the second Blue Book. 1 Econ 101A
More informationis the best response of firm 1 to the quantity chosen by firm 2. Firm 2 s problem: Max Π 2 = q 2 (a b(q 1 + q 2 )) cq 2
Econ 37 Solution: Problem Set # Fall 00 Page Oligopoly Market demand is p a bq Q q + q.. Cournot General description of this game: Players: firm and firm. Firm and firm are identical. Firm s strategies:
More informationMA300.2 Game Theory 2005, LSE
MA300.2 Game Theory 2005, LSE Answers to Problem Set 2 [1] (a) This is standard (we have even done it in class). The one-shot Cournot outputs can be computed to be A/3, while the payoff to each firm can
More informationChapter 8. Repeated Games. Strategies and payoffs for games played twice
Chapter 8 epeated Games 1 Strategies and payoffs for games played twice Finitely repeated games Discounted utility and normalized utility Complete plans of play for 2 2 games played twice Trigger strategies
More information13.1 Infinitely Repeated Cournot Oligopoly
Chapter 13 Application: Implicit Cartels This chapter discusses many important subgame-perfect equilibrium strategies in optimal cartel, using the linear Cournot oligopoly as the stage game. For game theory
More informationLecture 5 Leadership and Reputation
Lecture 5 Leadership and Reputation Reputations arise in situations where there is an element of repetition, and also where coordination between players is possible. One definition of leadership is that
More informationECONS 424 STRATEGY AND GAME THEORY MIDTERM EXAM #2 ANSWER KEY
ECONS 44 STRATEGY AND GAE THEORY IDTER EXA # ANSWER KEY Exercise #1. Hawk-Dove game. Consider the following payoff matrix representing the Hawk-Dove game. Intuitively, Players 1 and compete for a resource,
More informationSI Game Theory, Fall 2008
University of Michigan Deep Blue deepblue.lib.umich.edu 2008-09 SI 563 - Game Theory, Fall 2008 Chen, Yan Chen, Y. (2008, November 12). Game Theory. Retrieved from Open.Michigan - Educational Resources
More informationName. Answers Discussion Final Exam, Econ 171, March, 2012
Name Answers Discussion Final Exam, Econ 171, March, 2012 1) Consider the following strategic form game in which Player 1 chooses the row and Player 2 chooses the column. Both players know that this is
More informationRepeated Games. EC202 Lectures IX & X. Francesco Nava. January London School of Economics. Nava (LSE) EC202 Lectures IX & X Jan / 16
Repeated Games EC202 Lectures IX & X Francesco Nava London School of Economics January 2011 Nava (LSE) EC202 Lectures IX & X Jan 2011 1 / 16 Summary Repeated Games: Definitions: Feasible Payoffs Minmax
More informationStochastic Games and Bayesian Games
Stochastic Games and Bayesian Games CPSC 532l Lecture 10 Stochastic Games and Bayesian Games CPSC 532l Lecture 10, Slide 1 Lecture Overview 1 Recap 2 Stochastic Games 3 Bayesian Games 4 Analyzing Bayesian
More informationS 2,2-1, x c C x r, 1 0,0
Problem Set 5 1. There are two players facing each other in the following random prisoners dilemma: S C S, -1, x c C x r, 1 0,0 With probability p, x c = y, and with probability 1 p, x c = 0. With probability
More informationPrisoner s dilemma with T = 1
REPEATED GAMES Overview Context: players (e.g., firms) interact with each other on an ongoing basis Concepts: repeated games, grim strategies Economic principle: repetition helps enforcing otherwise unenforceable
More informationRepeated Games with Perfect Monitoring
Repeated Games with Perfect Monitoring Mihai Manea MIT Repeated Games normal-form stage game G = (N, A, u) players simultaneously play game G at time t = 0, 1,... at each date t, players observe all past
More informationGame Theory Fall 2003
Game Theory Fall 2003 Problem Set 5 [1] Consider an infinitely repeated game with a finite number of actions for each player and a common discount factor δ. Prove that if δ is close enough to zero then
More informationEconS 424 Strategy and Game Theory. Homework #5 Answer Key
EconS 44 Strategy and Game Theory Homework #5 Answer Key Exercise #1 Collusion among N doctors Consider an infinitely repeated game, in which there are nn 3 doctors, who have created a partnership. In
More informationSimon Fraser University Spring 2014
Simon Fraser University Spring 2014 Econ 302 D200 Final Exam Solution This brief solution guide does not have the explanations necessary for full marks. NE = Nash equilibrium, SPE = subgame perfect equilibrium,
More informationOutline for Dynamic Games of Complete Information
Outline for Dynamic Games of Complete Information I. Examples of dynamic games of complete info: A. equential version of attle of the exes. equential version of Matching Pennies II. Definition of subgame-perfect
More informationOptimal selling rules for repeated transactions.
Optimal selling rules for repeated transactions. Ilan Kremer and Andrzej Skrzypacz March 21, 2002 1 Introduction In many papers considering the sale of many objects in a sequence of auctions the seller
More information1 Solutions to Homework 3
1 Solutions to Homework 3 1.1 163.1 (Nash equilibria of extensive games) 1. 164. (Subgames) Karl R E B H B H B H B H B H B H There are 6 proper subgames, beginning at every node where or chooses an action.
More informationNoncooperative Oligopoly
Noncooperative Oligopoly Oligopoly: interaction among small number of firms Conflict of interest: Each firm maximizes its own profits, but... Firm j s actions affect firm i s profits Example: price war
More informationPRISONER S DILEMMA. Example from P-R p. 455; also 476-7, Price-setting (Bertrand) duopoly Demand functions
ECO 300 Fall 2005 November 22 OLIGOPOLY PART 2 PRISONER S DILEMMA Example from P-R p. 455; also 476-7, 481-2 Price-setting (Bertrand) duopoly Demand functions X = 12 2 P + P, X = 12 2 P + P 1 1 2 2 2 1
More informationSolution Problem Set 2
ECON 282, Intro Game Theory, (Fall 2008) Christoph Luelfesmann, SFU Solution Problem Set 2 Due at the beginning of class on Tuesday, Oct. 7. Please let me know if you have problems to understand one of
More informationEcon 711 Homework 1 Solutions
Econ 711 Homework 1 s January 4, 014 1. 1 Symmetric, not complete, not transitive. Not a game tree. Asymmetric, not complete, transitive. Game tree. 1 Asymmetric, not complete, transitive. Not a game tree.
More informationIterated Dominance and Nash Equilibrium
Chapter 11 Iterated Dominance and Nash Equilibrium In the previous chapter we examined simultaneous move games in which each player had a dominant strategy; the Prisoner s Dilemma game was one example.
More informationEcon 101A Final exam Th 15 December. Do not turn the page until instructed to.
Econ 101A Final exam Th 15 December. Do not turn the page until instructed to. 1 Econ 101A Final Exam Th 15 December. Please solve Problem 1, 2, and 3 in the first blue book and Problems 4 and 5 in the
More informationEco AS , J. Sandford, spring 2019 March 9, Midterm answers
Midterm answers Instructions: You may use a calculator and scratch paper, but no other resources. In particular, you may not discuss the exam with anyone other than the instructor, and you may not access
More informationECON/MGEC 333 Game Theory And Strategy Problem Set 9 Solutions. Levent Koçkesen January 6, 2011
Koç University Department of Economics ECON/MGEC 333 Game Theory And Strategy Problem Set Solutions Levent Koçkesen January 6, 2011 1. (a) Tit-For-Tat: The behavior of a player who adopts this strategy
More informationSequential-move games with Nature s moves.
Econ 221 Fall, 2018 Li, Hao UBC CHAPTER 3. GAMES WITH SEQUENTIAL MOVES Game trees. Sequential-move games with finite number of decision notes. Sequential-move games with Nature s moves. 1 Strategies in
More informationMicroeconomic Theory II Preliminary Examination Solutions Exam date: June 5, 2017
Microeconomic Theory II Preliminary Examination Solutions Exam date: June 5, 07. (40 points) Consider a Cournot duopoly. The market price is given by q q, where q and q are the quantities of output produced
More informationLecture 9: Basic Oligopoly Models
Lecture 9: Basic Oligopoly Models Managerial Economics November 16, 2012 Prof. Dr. Sebastian Rausch Centre for Energy Policy and Economics Department of Management, Technology and Economics ETH Zürich
More informationECON106P: Pricing and Strategy
ECON106P: Pricing and Strategy Yangbo Song Economics Department, UCLA June 30, 2014 Yangbo Song UCLA June 30, 2014 1 / 31 Game theory Game theory is a methodology used to analyze strategic situations in
More informationMicroeconomics of Banking: Lecture 5
Microeconomics of Banking: Lecture 5 Prof. Ronaldo CARPIO Oct. 23, 2015 Administrative Stuff Homework 2 is due next week. Due to the change in material covered, I have decided to change the grading system
More informationSpring 2017 Final Exam
Spring 07 Final Exam ECONS : Strategy and Game Theory Tuesday May, :0 PM - 5:0 PM irections : Complete 5 of the 6 questions on the exam. You will have a minimum of hours to complete this final exam. No
More informationEC 202. Lecture notes 14 Oligopoly I. George Symeonidis
EC 202 Lecture notes 14 Oligopoly I George Symeonidis Oligopoly When only a small number of firms compete in the same market, each firm has some market power. Moreover, their interactions cannot be ignored.
More informationFDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.
FDPE Microeconomics 3 Spring 2017 Pauli Murto TA: Tsz-Ning Wong (These solution hints are based on Julia Salmi s solution hints for Spring 2015.) Hints for Problem Set 2 1. Consider a zero-sum game, where
More informationGame Theory: Additional Exercises
Game Theory: Additional Exercises Problem 1. Consider the following scenario. Players 1 and 2 compete in an auction for a valuable object, for example a painting. Each player writes a bid in a sealed envelope,
More informationTopics in Contract Theory Lecture 1
Leonardo Felli 7 January, 2002 Topics in Contract Theory Lecture 1 Contract Theory has become only recently a subfield of Economics. As the name suggest the main object of the analysis is a contract. Therefore
More informationStatic Games and Cournot. Competition
Static Games and Cournot Introduction In the majority of markets firms interact with few competitors oligopoly market Each firm has to consider rival s actions strategic interaction in prices, outputs,
More informationNotes for Section: Week 4
Economics 160 Professor Steven Tadelis Stanford University Spring Quarter, 2004 Notes for Section: Week 4 Notes prepared by Paul Riskind (pnr@stanford.edu). spot errors or have questions about these notes.
More informationEconomics 51: Game Theory
Economics 51: Game Theory Liran Einav April 21, 2003 So far we considered only decision problems where the decision maker took the environment in which the decision is being taken as exogenously given:
More informationExtensive-Form Games with Imperfect Information
May 6, 2015 Example 2, 2 A 3, 3 C Player 1 Player 1 Up B Player 2 D 0, 0 1 0, 0 Down C Player 1 D 3, 3 Extensive-Form Games With Imperfect Information Finite No simultaneous moves: each node belongs to
More informationCMSC 474, Introduction to Game Theory 16. Behavioral vs. Mixed Strategies
CMSC 474, Introduction to Game Theory 16. Behavioral vs. Mixed Strategies Mohammad T. Hajiaghayi University of Maryland Behavioral Strategies In imperfect-information extensive-form games, we can define
More informationECON402: Practice Final Exam Solutions
CO42: Practice Final xam Solutions Summer 22 Instructions There is a total of four problems. You must answer any three of them. You get % for writing your name and 3% for each of the three best problems
More informationElements of Economic Analysis II Lecture X: Introduction to Game Theory
Elements of Economic Analysis II Lecture X: Introduction to Game Theory Kai Hao Yang 11/14/2017 1 Introduction and Basic Definition of Game So far we have been studying environments where the economic
More informationIntroduction to Game Theory
Introduction to Game Theory 3a. More on Normal-Form Games Dana Nau University of Maryland Nau: Game Theory 1 More Solution Concepts Last time, we talked about several solution concepts Pareto optimality
More informationDiscounted Stochastic Games with Voluntary Transfers
Discounted Stochastic Games with Voluntary Transfers Sebastian Kranz University of Cologne Slides Discounted Stochastic Games Natural generalization of infinitely repeated games n players infinitely many
More informationDynamic Games. Econ 400. University of Notre Dame. Econ 400 (ND) Dynamic Games 1 / 18
Dynamic Games Econ 400 University of Notre Dame Econ 400 (ND) Dynamic Games 1 / 18 Dynamic Games A dynamic game of complete information is: A set of players, i = 1,2,...,N A payoff function for each player
More information1 Solutions to Homework 4
1 Solutions to Homework 4 1.1 Q1 Let A be the event that the contestant chooses the door holding the car, and B be the event that the host opens a door holding a goat. A is the event that the contestant
More informationMicroeconomic Theory II Preliminary Examination Solutions
Microeconomic Theory II Preliminary Examination Solutions 1. (45 points) Consider the following normal form game played by Bruce and Sheila: L Sheila R T 1, 0 3, 3 Bruce M 1, x 0, 0 B 0, 0 4, 1 (a) Suppose
More informationFinitely repeated simultaneous move game.
Finitely repeated simultaneous move game. Consider a normal form game (simultaneous move game) Γ N which is played repeatedly for a finite (T )number of times. The normal form game which is played repeatedly
More informationCMPSCI 240: Reasoning about Uncertainty
CMPSCI 240: Reasoning about Uncertainty Lecture 23: More Game Theory Andrew McGregor University of Massachusetts Last Compiled: April 20, 2017 Outline 1 Game Theory 2 Non Zero-Sum Games and Nash Equilibrium
More informationAnswers to Problem Set 4
Answers to Problem Set 4 Economics 703 Spring 016 1. a) The monopolist facing no threat of entry will pick the first cost function. To see this, calculate profits with each one. With the first cost function,
More informationGame Theory and Economics Prof. Dr. Debarshi Das Department of Humanities and Social Sciences Indian Institute of Technology, Guwahati
Game Theory and Economics Prof. Dr. Debarshi Das Department of Humanities and Social Sciences Indian Institute of Technology, Guwahati Module No. # 03 Illustrations of Nash Equilibrium Lecture No. # 04
More informationProblem Set 2 Answers
Problem Set 2 Answers BPH8- February, 27. Note that the unique Nash Equilibrium of the simultaneous Bertrand duopoly model with a continuous price space has each rm playing a wealy dominated strategy.
More informationIntroduction to Multi-Agent Programming
Introduction to Multi-Agent Programming 10. Game Theory Strategic Reasoning and Acting Alexander Kleiner and Bernhard Nebel Strategic Game A strategic game G consists of a finite set N (the set of players)
More informationPlayer 2 L R M H a,a 7,1 5,0 T 0,5 5,3 6,6
Question 1 : Backward Induction L R M H a,a 7,1 5,0 T 0,5 5,3 6,6 a R a) Give a definition of the notion of a Nash-Equilibrium! Give all Nash-Equilibria of the game (as a function of a)! (6 points) b)
More informationJanuary 26,
January 26, 2015 Exercise 9 7.c.1, 7.d.1, 7.d.2, 8.b.1, 8.b.2, 8.b.3, 8.b.4,8.b.5, 8.d.1, 8.d.2 Example 10 There are two divisions of a firm (1 and 2) that would benefit from a research project conducted
More informationCUR 412: Game Theory and its Applications, Lecture 9
CUR 412: Game Theory and its Applications, Lecture 9 Prof. Ronaldo CARPIO May 22, 2015 Announcements HW #3 is due next week. Ch. 6.1: Ultimatum Game This is a simple game that can model a very simplified
More information6.6 Secret price cuts
Joe Chen 75 6.6 Secret price cuts As stated earlier, afirm weights two opposite incentives when it ponders price cutting: future losses and current gains. The highest level of collusion (monopoly price)
More informationIntroduction to Game Theory
Introduction to Game Theory Part 2. Dynamic games of complete information Chapter 1. Dynamic games of complete and perfect information Ciclo Profissional 2 o Semestre / 2011 Graduação em Ciências Econômicas
More informationw E(Q w) w/100 E(Q w) w/
14.03 Fall 2000 Problem Set 7 Solutions Theory: 1. If used cars sell for $1,000 and non-defective cars have a value of $6,000, then all cars in the used market must be defective. Hence the value of a defective
More informationIntroductory Microeconomics
Prof. Wolfram Elsner Faculty of Business Studies and Economics iino Institute of Institutional and Innovation Economics Introductory Microeconomics More Formal Concepts of Game Theory and Evolutionary
More informationIMPERFECT COMPETITION AND TRADE POLICY
IMPERFECT COMPETITION AND TRADE POLICY Once there is imperfect competition in trade models, what happens if trade policies are introduced? A literature has grown up around this, often described as strategic
More informationAnswers to Microeconomics Prelim of August 24, In practice, firms often price their products by marking up a fixed percentage over (average)
Answers to Microeconomics Prelim of August 24, 2016 1. In practice, firms often price their products by marking up a fixed percentage over (average) cost. To investigate the consequences of markup pricing,
More informationRationalizable Strategies
Rationalizable Strategies Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu Jun 1st, 2015 C. Hurtado (UIUC - Economics) Game Theory On the Agenda 1
More informationIPR Protection in the High-Tech Industries: A Model of Piracy. Thierry Rayna University of Bristol
IPR Protection in the High-Tech Industries: A Model of Piracy Thierry Rayna University of Bristol thierry.rayna@bris.ac.uk Digital Goods Are Public, Aren t They? For digital goods to be non-rival, copy
More informationMixed-Strategy Subgame-Perfect Equilibria in Repeated Games
Mixed-Strategy Subgame-Perfect Equilibria in Repeated Games Kimmo Berg Department of Mathematics and Systems Analysis Aalto University, Finland (joint with Gijs Schoenmakers) July 8, 2014 Outline of the
More informationSUPPLEMENT TO WHEN DOES PREDATION DOMINATE COLLUSION? (Econometrica, Vol. 85, No. 2, March 2017, )
Econometrica Supplementary Material SUPPLEMENT TO WHEN DOES PREDATION DOMINATE COLLUSION? (Econometrica, Vol. 85, No., March 017, 555 584) BY THOMAS WISEMAN S1. PROOF FROM SECTION 4.4 PROOF OF CLAIM 1:
More informationCS 331: Artificial Intelligence Game Theory I. Prisoner s Dilemma
CS 331: Artificial Intelligence Game Theory I 1 Prisoner s Dilemma You and your partner have both been caught red handed near the scene of a burglary. Both of you have been brought to the police station,
More informationChapter 7 Review questions
Chapter 7 Review questions 71 What is the Nash equilibrium in a dictator game? What about the trust game and ultimatum game? Be careful to distinguish sub game perfect Nash equilibria from other Nash equilibria
More information14.12 Game Theory Midterm II 11/15/ Compute all the subgame perfect equilibria in pure strategies for the following game:
4. Game Theory Midterm II /5/7 Prof. Muhamet Yildiz Instructions. This is an open book exam; you can use any written material. You have one hour and minutes. Each question is 5 points. Good luck!. Compute
More information