ECO421: Reputation. Marcin P ski. March 29, 2018

Transcription

1 ECO421: Reputation Marcin P ski March 29, 2018

2 Plan Chain store game Model of reputation Reputations in innite games Applications Model No pure strategy equilibria Mixed strategy equilibrium Basic model Cost of keeping promises Basic model Reputation in economics Conclusions

3 Chain store game Class experiment Entrant ( ) 1 2 stay Out Enter Chain store Fight Accommodate ( ) 0 0 ( ) 2 1 One shot game: Entrant and Incumbent unique SPE: Enter, followed by Accommodate

4 Chain store game Class experiment Finitely repeated entry game. T regions, each one with a single entrant chain store plays the game in each region, consecultively the outcomes of games j < i are observed before game i. Chain store's payos are sum of payos from each region.

5 Chain store game Subgame perfect equilibrium Let h be a history at the beginning of period t = T. The game ends at the end of this period. At this moment, the game loks like one-shot. The SPE behavior in subgame h is the same as in the one-shot game: Enter, followed by Accommodate This observation does not depend on h.

6 Chain store game Subgame perfect equilibrium Let h be a history at the beginning of period t=t 1. The game in period T does not depend on what happens today. At this moment, the game loks like one-shot. The SPE behavior in subgame h is the same as in the one-shot game: Enter, followed by Accommodate This observation does not depend on h. Etc.

7 Chain store game Subgame perfect equilibrium Unique SPE: in each period. Enter, followed by Accommodate But?

8 Plan Chain store game Model of reputation Reputations in innite games Applications Model No pure strategy equilibria Mixed strategy equilibrium Basic model Cost of keeping promises Basic model Reputation in economics Conclusions

9 Chain store game Entrant ( ) le wc stay Out Enter Chain store Fight Accommodate ( ) 0 0 ( ) we lc More general payos: w i > l i > 0

10 Chain store with reputation A small modication of the chain store. Two types of the Incumbent: normal, with probability 1 ε < 1 the same payos as in the original game, crazy, with probability ε > 0, payo 1 if Fights, 0 otherwise, always Fights. ε is very small, but strictly positive. If it is small, does it matter?

11 Chain store with reputation One-shot game One-shot case: Nature normal 1 ε ε crazy Entrant Entrant stay Out Enter stay Out Enter ( le w c ) Chain store ( ) le 1 Chain store Fight Accommodate Fight Accommodate ( ) 0 0 ( we l c ) ( ) 0 1 ( ) we 0

12 Chain store with reputation One-shot game We can replace the crazy Incumbent's decision. Nature normal 1 ε ε crazy Entrant Entrant stay Out Enter stay Out Enter ( le w c ) Chain store ( ) le 1 ( ) 0 1 Fight Accommodate ( ) 0 0 ( we l c )

13 Chain store with reputation One-shot game If ε > 0 is too large, then the Entrant will stay Out: Enter:payo we (1 ε) + ε 0 if the normal Incumbent Accommodates = payo we, otherwise, it will Fight = payo 0 we, crazy Incumbent always Fights. Out= payo le, If we (1 ε) + ε 0 l e, or ε > w e l e w e =: p then Out is strictly dominant.

14 Chain store with reputation One-shot game Lemma In one shot game, If ε < p, then the Entrant Enters in each wpbe. If ε > p, then the Entrant stays Out in each wpbe. So, in one-shot case, small ε > 0 does not change anything.

15 Chain store with reputation One-shot game Lemma There is T large enough such that in T repeated game If ε > 0, then, in period t = 1, the Entrant stays Out in each wpbe. If ε > 0, there is a dramatic dierence.

16 Chain store with reputation Multiple periods Multiple T > 1 periods. wpbe equilibrium (strategies + beliefs) Beliefs p (h) that the Incumbent is crazy initially p ( ) = ε, we assume that if p (h) = 0, then p (h, a) = 0 (once recgnized as normal, always normal) also, p (h, Out) = p (h), no belief change if Entrant stays Out, reasonable restrictions (would arise from sequential equilibrium).

17 Chain store with reputation Multiple periods History h in (the beginning of) period t. If p (h) = 0, Incumbent is known (for ever) to be normal, the continuation behavior as in the SPE of complete information Entrant always Enters, Incumbent always Accommodates, Incumbent gets li Incumbent continuation payos: (T t + 1) l i.

18 Chain store with reputation Multiple periods If p (h) > p, Entrant Out, Incumbent payos wi, next periods beliefs p (h, Out) = p (h) p are the same, Incumbent continuation payos What if p (h) (0, p ]? (T t + 1) w i.

19 Chain store with reputation Multiple periods Suppose that p (h) (0, p ]. We will look at pure strategy equililbria only (for now). Two strategies for the normal type of the Incumbent: Accommodate, Fight. Is it possible to have an equilibrium in which in the very rst period, the normal Incumbent Accommodates? how does the answer depend on ε and T?

20 Chain store with reputation Multiple periods Suppose p (h) (0, p ] and the normal Incumbent Accommodates after history h. Beliefs : p (h, Enter, Accommodate) = 0, only normal type Accommodates, p (h, Enter, Fight) = 1! only crazy type Fights both histories are on-path (so we use Bayes formula)

21 Chain store with reputation Multiple periods Payos and continuation payos afrer Entrant Enters: if normal Incumbent Accomodates today li in the next periods, (T t) li together (T t + 1) li If normal Incumbent deviates and Fights, today 0, in the next periods (T t) wi. If (T t + 1) l i < (T t) w i, then Accommodate is not best response!

22 Chain store with reputation Multiple periods True also for t = 1. If l i < T 1 T w i, then, in pure strategy WPBE the normal type is going to Fight. More generally, let t be the rst period t such that l i T t T t + 1 w i, (and suppose that the inequality is strict). normal type will Fight for each t < t normal type will Accommodate after.

23 Chain store with reputation Multiple periods Lemma Suppose that ε (0, p ). Unique wpbe (with restrictions) is For each t < t, Entrant stays Out and Incumbent Fights, For t t, Entrant Enters, and normal Incumbent Accommodates.

24 Chain store with reputation Multiple periods For t t -> reputation maintanance. If l i 1 2 w i, we can check that t = T 1. Dramatic eect of the ε-probability craziness for the behavior.

25 Chain store with reputation Multiple periods Value of reputation building. Assume that l i 1w 2 i. Then, t = T 1. Complete information payos for the Incumbent: Tl i, With reputation: (t 1) w i + (T t + 1) l i = (T 2) w i + 2l i. Value = dierence (T 2) (w i l i ). Dramatic eect of the ε-probability craziness for the payos.

26 Chain store with reputation Multiple periods (mixed strategy equilibria) So far, we assumed pure strategies. Can be dealt with.

27 Model of reputation So far, crazy type always Fights. What if crazy type always Accommodates? What if dierent crazy types?

28 Model of reputation Crazy type = always Fights. What if crazy type always Accommodates? What if two dierent crazy types? It doesn't matter. The crazy type always Accommodate does not make a qualittative dierence:

29 Model of reputation Crazy type = always Fights. What if crazy type always Accommodates? What if two dierent crazy types? It doesn't matter. The crazy type always Accommodate does not make a qualitative dierence:

30 Model of reputation The chain store game is nite. most striking contrast. In innitely repeated games, there are typically multiple equilibria. But, when reputation (ie., small probability crazy type) is added, all equilibria but one disappear.

31 Plan Chain store game Model of reputation Reputations in innite games Applications Model No pure strategy equilibria Mixed strategy equilibrium Basic model Cost of keeping promises Basic model Reputation in economics Conclusions

32 Applications Never negotiate with terrorists. Reputation for cooperative behavior (centipede game). Reputation for promise-keeping.

33 Never negotiate with terrorists Two players: authority, terrorist, Terroritst decides whether to attack or not, Authority decides whether to yield to the demands or not. Authority normal, with probability 1 ε, crazy always Fights with probability ε > 0.

34 Never negotiate with terrorists Nature normal 1 ε ε crazy Terrorist Terrorist stay Out Enter stay Out Attack ( ) le wc Authority ( ) le 1 Authority Fight Yield Fight Accommodate ( ) 0 0 ( ) we lc ( ) 0 1 ( ) we 0 Exaxctly the same game. The same analysis.

35 Never negotiate with terrorists Never negotiate with terrorists is exaxctly the same game as Chain store game. The same analysis.

36 Centipede game There are two players, Ann and Bob Playeres move in alternating periods t = 1, 2,..., T, where T is even. In its period of action, the player decides whether to continue or stop. A player who stops in period t gets t + 2 and the other player gets t 1. If Bob does not stop in period T, payos as if Ann stopped in period T + 1.

37 Centipede game Example: T = 6 Ann C Bob C Ann C Bob C Ann C Bob C 9, 6 S S S S S S 3, 0 1, 4 5, 2 3, 6 7, 4 5, 8

38 Centipede game Unique SPE: Always stop backward induction Ann C Bob C Ann C Bob C Ann C Bob C 4, 3 S S S S S S 1, 0 0, 2 2, 1 1, 3 3, 2 2, 4

39 Centipede game Unique SPE: Always stop this does not depend on T But, that's not how this game is played for larger T, players Continue for a long time, and randomly Stop somewhere before the end, tons of experimental evidence, common sense, The Treasure of Sierra Madre. Problem for game theory.

40 Centipede game Suppose that each player i has one of two types: normal, with probabil ity 1 ε, crazy, with prob. ε > 0, the crazy type always Continues. If T is really large, then there is an wpbe* such that both players Continue for a long time, and then Stop randomly.

41 Centipede game No pure strategy equilibria Easy to check that if ε is small such an equilibrium cannot be pure strategy. Let t be the rst period that a player Stops. such t must exists as the last normal player will Stop for sure, let it be player i, Before t beliefs do not change. But then, after t, player i who Continued must be crazy, Player i best response must be to wait till the last moment, before t, player i Continues, after t, player i should continue vs crazy type.

42 Centipede game No pure strategy equilibria So, if player i Continues in period t, player i waits till period T or T 1 (whichever is her action.) But then, player i must wait till last moment before i Stops. Hence t T 2.

43 Centipede game No pure strategy equilibria So, the normal type of player i stops for the rst time in period t T 2. Let's check player i incentives to stop in period t 1. player i gets payo t from Stop in period t 1, and from Continue, at most payo εt + (1 ε) t, if ε is small, better to Stop.

44 Centipede game Mixed strategy equilibria For a mixed strategy equilibrium, we need some notation: t-player: player who moves in period t, p t - probability that the t-player t is crazy, α t - probability that the normal type of the t-player stops, q t - probability that the t-player stops: q t = (1 p t ) α t. If α t (0, 1), then the player must be indierent between stopping and continuing.

45 Centipede game Mixed strategy equilibria If α t (0, 1), then the player must be indierent between stopping and continuing. payo from Stopping t + 2, payo from Continuing and then Stopping the next time is q t+1 ((t + 1) 1) + (1 q t+1 ) ((t + 2) + 2). If player is indierent, we have t + 2 = q t+1 ((t + 1) 1) + (1 q t+1 ) ((t + 2) + 2), or q t+1 = 1 2.

46 Centipede game Mixed strategy equilibria The normal type in period T must stop for sure, α T = 1. the payo from stoping is T + 2 and the payo from continuing is T = T. Hence, 1 2 = q T = 1 p T, and p 2T = 1 2. Belief updating in periods t such that t + 1-player is indierent: q t = 1 2 p t+2 = P (crazy, C t) P (C t ) = p t 1 q t = 2p t. beliefs always go up every two periods, they become twice as high.

47 Centipede game Mixed strategy equilibria Assume that T ε > 1 2, and Let t the rst period such that (T t ) ε 1 2. We construct equilibirum such that for each t t, t-playere is indierent, q t+1 = 1 2, 2pt+1 = p t+3, p T = 1 2.

48 Centipede game Mixed strategy equilibria Both normal types play Continue for each t < t : α t = 1, no learning before t, p t = ε, We must choose α t to make sure that p t +2 = p t 1 q t = ε, 1 (1 ε) α t notice that qt 1 2 as (t 1)-player must prefer to play Continue.

49 Centipede game Mixed strategy equilibria Let t the rst period such that (T t ) ε 1 2. Lemma Suppose that 2 T ε i > 1 for some i. Then, there exists an equilibrium such that for periods t < t, players always choose Continue, in period t t players randomize.

50 Promise-keeping Simple model of promise keeping. In each period, Ann wants to exchange favors Ann decides whether to or not a promise. If Ann makes a promise, Bob chooses whether to trust Ann or not. Then, if she is trusted, Ann decides whether to full promise or not.

51 Promise-keeping Ann ( ) 0 0 No promise Promise Bob Trust Not trust Ann ( ) 0 0 Keep promise Betray ( ) 1 1 ( ) 2 1

52 Promise-keeping In all SPE always Betray, never Trust, make or not a promise : doesn't matter.

53 Promise-keeping But, if there is a small probability that Ann always Keeps promises: unique reputation equilibrium.

54 Promise-keeping Cost of keeping promises A small variation of the basic model: Prior to decision whether to make a promise, Ann observes the cost of keeping promises x F (.). F is c.d.f. on R + = {x : x 0}. Her payo from Keeping the promise is 2 x prebvious model corresponds to x = 1.

55 Promise-keeping Cost of keeping promises WE can check that, without reputation, unique SPE is that ASnn always breaks her promises. With reputation, unique SPE: she makes promises only if x 2, she always keeps them, and Bob always trusts.

56 Cournot duopoly Two rms. In each period, each rm simultaneously chooses q i 0. Payos π i (q i, q i ) = q i (a c (q 1 + q 2 )). Unique static Nash equilibrium q C 1 = qc 2 = 1 (a c). 3 Unique SPE.

57 Cournot duopoly Suppose that rm 1 is normal with probability 1 ε, and crazy and always plays q with probability ε. The normal type prefer that rm 2 believs that 1 is crazy if ( ) ( ) π 1 q, q BR 2 (q) > π 1 q C 1, q C 2. Reputation

58 Cournot duopoly What is the best crazy type to imitate?

59 Cournot duopoly What is the best crazy type to imitate? Stackelberg quantity.

60 Applications Examples of crazy strategies Kissinger and Nixon, never negotiate with terrorist

61 Applications Resources: Movie tropes: Never live it down, Respected by the Respected,

62 Plan Chain store game Model of reputation Reputations in innite games Applications Model No pure strategy equilibria Mixed strategy equilibrium Basic model Cost of keeping promises Basic model Reputation in economics Conclusions

63 Reputation in colloquial language Mirriam-Webster 1. overall quality or character as seen or judged by people in general; 2. recognition by other people of some characteristic or ability; has the reputation of being clever 3. a place in public esteem or regard : good name: trying to protect his reputation.

64 Reputation in colloquial language Wikipedia 1. Reputation or image of a social entity (a person, a social group, an organization) is an opinion about that entity, typically a result of social evaluation on a set of criteria. 2. It is important in business, education, online communities, and many other elds. 3. Highly ecient mechanism of social control in natural societies.

65 Reputation in economics Features of our model of reputation One builds reputation for behavior crazy type is not strategic, behaves in a particular way, no matter what. The reputation is valuable to the player.

66 Reputation in economics Features of our model of reputation Two players: Ann and Bob. Static game with unique equilibrium ( σ eq A, ) σeq B. Unique SPE in nitely repeated game. Instead, Ann could try to build a reputation for being crazy and always playing σ. If she is successful, Bob will best respond with σ B BR Bob (σ). Reputation is worthwile for Ann, if ( u A (σ, σ B ) > u A σ eq A, ) σeq B. The best strategy is called Stackelberg action.

67 Plan Chain store game Model of reputation Reputations in innite games Applications Model No pure strategy equilibria Mixed strategy equilibrium Basic model Cost of keeping promises Basic model Reputation in economics Conclusions

68 Conclusions Modelling reputations. wpbe with mixed strategies.

69 Conclusions Conditions for successful reputation building (payos, lentgh of the game, beliefs) Finding reputation equilibria Computing value of reputation