A Dynamic Level-k Model in Games

Transcription

1 Dynamic Level-k Model in Games Teck Ho and Xuanming Su UC erkeley March, 2010 Teck Hua Ho 1

2 4-stage Centipede Game Outcome Round % 30.3% 35.9% 20.0% 7.6% % 41.2% 38.2% 10.3% 2.2% ackward Induction 100% 0% 0% 0% 0% March, 2010 Teck Hua Ho 2

3 6-Stage Centipede Game Outcome Round % 5.5% 17.2% 33.1% 33.1% 9.00% 2.10% % 7.4% 22.8% 44.1% 16.9% 6.60% 0.70% ackward Induction 100% 0% 0% 0% 0% 0% 0% March, 2010 Teck Hua Ho 3

4 Outline ackward did induction i and dits systematic violations il i Dynamic Level-k model and the main theoretical results The centipede game Resolving well-known paradoxes: o o Cooperation in finitely repeated prisoner ss dilemma Chain-store paradox n empirical application: The centipede game lternative explanations o o Reputation-based story Social preferences March, 2010 Teck Hua Ho 4

5 ackward Induction Principle ackward did induction is the most widely idl accepted tdprinciple il to generate prediction in dynamic games of complete information Extensive-form games (e.g., Centipede) Finitely repeated games (e.g., Repeated PD and chain-store paradox) Multi-person dynamic programming For the principle p to work, every yplayer must be willingness to bet on others rationality March, 2010 Teck Hua Ho 5

6 Violations of ackward Induction Well-known violations in economic experiments include: ( ): Passing in the centipede game Cooperation in the finitely repeated PD Chain-store paradox Likely to be a failure of mutual consistency condition (different people make initial different bets on others rationality) March, 2010 Teck Hua Ho 6

7 Standard ssumptions in Equilibrium nalysis ssumptions ackward DLk Solution Method Induction Model Strategic Thinking X X est Response X X Mutual Consistency X? Instant t Equilibration X? March, 2010 Teck Hua Ho 7

8 Notations S : Total number of subgames (indexed by s) I : Total number of players (indexed by i) N s : Total number of players who are active at subgame s S = 4, I = 2, N = N = N = N = = 1 March, 2010 Teck Hua Ho 8

9 Deviation from ackward Induction δ ( I 1 L 1,..., L, G ) = ) S s 1 N s i 1 S N s 1 i D s ( L, L = = D s (L i,l ) = 1, 0, a ( L i ) a ( L otherwise ) 0 δ (.) 1 March, 2010 Teck Hua Ho 9

10 Examples Examples } { } { }; { T T T T L T P L T P L E ] 0 1 [1 ),, ( },,, { },,,, { };,,, { = = = = = G L L T T T T L T P L T P L δ Ex1: 1 0 ] 0 0 [1 ) ( },,, { },,,, { };,,, { = = = G L L T T T T L T T L T P L δ Ex2: March, 2010 Teck Hua Ho ] 0 0 [1 ),, ( = = G L L δ

11 Systematic Violation 1: Limited Induction δ ( L, L, G ) < δ ( L, L, G ( 4 G6 ) March, 2010 Teck Hua Ho 11

12 Limited Induction in Centipede Game Figure 1: Deviation in 4-stage versus 6-stage game (1 st round) March, 2010 Teck Hua Ho 12

13 Systematic Violation 2: Time Unraveling δ ( L ( t ), L ( t ), G ) 0 as t March, 2010 Teck Hua Ho 13

14 Time Unraveling in Centipede Game Figure 2: Deviation in 1 st vs. 10 th round of the 4-stage game March, 2010 Teck Hua Ho 14

15 Outline ackward did induction i and dits systematic violations il i Level-k model and the main theoretical results The centipede game Resolving well-known paradoxes: o o Cooperation in finitely repeated prisoner ss dilemma Chain-store paradox n empirical application: The centipede game lternative explanations o o Reputation-based story Social preferences March, 2010 Teck Hua Ho 15

16 Research question To develop a good descriptive model to predict the probability of player i (i=1,,i) choosing strategy j at subgame s (s=1,.., S) in any dynamic game of complete information P ij (s) March, 2010 Teck Hua Ho 16

17 Criteria of a Good Model Nests backward induction as a special case ehavioral plausible Heterogeneous in their bets on others rationality Captures limited induction and time unraveling Fits data well Simple (with as few parameters as the data would allow) March, 2010 Teck Hua Ho 17

18 Standard ssumptions in Equilibrium nalysis ssumptions ackward Hierarchical Induction Strategizing Solution Method Strategic Thinking X X est Response X X Mutual Consistency X Heterogenous ets Instant Equilibration X Learning March, 2010 Teck Hua Ho 18

19 Dynamic Level-kk Model: Summary Players choose rule from a rule hierarchy h Players make differential initial bets on others chosen rules fter each game play, players observe others rules (e.g., strategy method) Players update their beliefs on rules chosen by others Players always choose a rule to maximize their subjective expected utility in each round March, 2010 Teck Hua Ho 19

20 Dynamic Level-k Model: Rule Hierarchy Players choose rule from a rule hierarchy h generated dby bestresponses Rule hierarchy: L, L1, L2,... 0 L L ( ) k = R Lk 1 Restrict t L 0 to follow behavior proposed in the existing literature L = I March, 2010 Teck Hua Ho 20

21 Dynamic Level-k Model: Poisson Initial elief Different people make different initial i i bets on others chosen rules Poisson distributed initial beliefs: f ( K) = e τ K λ K! λ : average belief of rules used by opponents f(k) fraction of players think that their opponents use L k-1 rule. March, 2010 Teck Hua Ho 21

22 Dynamic Level-k model: elief Updating at the End of Round t Initial belief strength: N k (0) = β Update after observing which rule opponent chose i i N ( t ) = Ν ( t 1 ) + I(k,t) 1 k k i k ( t ) = S N k ' = 0 i k N ( t) i k ( t) I(k, t) = 1 if opponent chose L k and 0 otherwise ayesian updating involving a multi-nomial distribution with a Dirichlet prior (Fudenberg and Levine, 1998; Camerer and Ho, 1999) March, 2010 Teck Hua Ho 22

23 Dynamic Level-k model: : Optimal lrule in Round dt+1 Optimal rule k * : k * = arg max k = 1,.., S S S s= 1 k ' = 1 i k ' ( t) π ( a ks, a k ' s ) Let the specified action of rule L k at subgame s be a ks March, 2010 Teck Hua Ho 23

24 The Centipede Game (Rule Hierarchy) Player Player (P,-,P-) (,, (-,P,-,P),,, (P,-,P-) (P,-,T,-) (P,-,T,-) (T,-,T,-) (-,P,-,T) (-,T,-,P) (-,T,-,T) (-,T,-,T) March, 2010 Teck Hua Ho 24

25 Player in 4-Stage Centipede Game N i k(t) β=0.5 Round (t) L 0 L 1 L 2 L 3 L 4 Rule Used by Opponent Optimal Rule (Player ) 0 β L 2 1 β 1 L 3 L 2 2 β 2 L 3 L 2 3 β 3 L 3 L 4 March, 2010 Teck Hua Ho 25

26 Dynamic Level-kk Model: Summary Players choose rule from a rule hierarchy h Players make differential initial bets on others chosen rules fter each game play, players observe others rules (e.g., strategy method) Players update their beliefs on rules chosen by others Players always choose a rule to maximize their subjective expected utility in each round 2-paramter extension of backward induction (λ and β) March, 2010 Teck Hua Ho 26

27 Main Theoretical Results: Limited Induction δ ( L, L, G ) < δ ( L, L, G ) ( 4 G6 March, 2010 Teck Hua Ho 27

28 Main Theoretical Results: Time Unraveling δ ( L ( t ), L ( t ), G ) 0 as t March, 2010 Teck Hua Ho 28

29 Iterated Prisoner ss Dilemma (Rule Hierarchy) 33 3,3 05 0,5 5,0 1,1 Level Strategy 0 TFT* 1 TFT,D 2 TFT,D,D 3 TFT,D,D,D K TFT,D 1,D k * Kreps et al (1982) March, 2010 Teck Hua Ho 29

30 Main Theoretical Results δ ( L, L, GT ) < δ ( L, L, GT '); T' > T March, 2010 Teck Hua Ho 30

31 Main Theoretical Results δ ( L ( t), L ( t), G) 0 as t March, 2010 Teck Hua Ho 31

32 Properties of Level-0 Rule Maximize group payoff: level-0 player always chooses a decision that if others do the same will lead to the largest total payoff for the group (e.g., TFT in RPD) Protect individual id payoff: While maximizing i i group payoff, a level-0 player also ensures that the chosen decision rule is robust against continued exploitation by others (e.g., TFT in RPD) March, 2010 Teck Hua Ho 32

33 Chain-Store Paradox (Rule Hierarchy) E OUT IN 5 1 CS FIGHT SHRE Level Chain Store (CS) Entrant 0 FIGHT(F) GTR: OUT unless CSi is observed to share (then F,F,F,..,F,F,S 1 ENTER(E) 2 F,F,F,..,F,S,S GRE, E 3 FF F,F,,..F,S,S,S FSSS GTR,E,E E K F,..,F,S 1,..,S k GTR,E 1,..,E k-1 March, 2010 Teck Hua Ho 33

34 Main Theoretical Results March, 2010 Teck Hua Ho 34

35 Outline ackward did induction i and dits systematic violations il i Level-k model and the main theoretical results The centipede game Resolving well-known paradoxes: o o Cooperation in finitely repeated prisoner ss dilemma Chain-store paradox n empirical application: The centipede game lternative explanations o o Reputation-based story Social preferences March, 2010 Teck Hua Ho 35

36 4-Stage versus 6-Stage Centipede Games March, 2010 Teck Hua Ho 36

37 Empirical Regularities Outcome Round % 6.2% 30.3% 3% 35.9% 20.0% 0% 76% 7.6% % 41.2% 38.2% 10.3% 2.2% Outcome Round % 5.5% 17.2% 33.1% 33.1% 9.00% 2.10% % 7.4% 22.8% 44.1% 16.9% 6.60% 0.70% March, 2010 Teck Hua Ho 37

38 Dynamic Level-k Model s Prediction in 4-stage game March, 2010 Teck Hua Ho 38

39 Dynamic Level-k Model s Prediction in 6-stage game March, 2010 Teck Hua Ho 39

40 MLE Model Estimates Special cases are rejected oth heterogeneity and learning are important March, 2010 Teck Hua Ho 40

41 Model Predictions March, 2010 Teck Hua Ho 41

42 lternative 1: Gang of Four s Story y( (Kreps, et al, 1982) large θ = proportion of altruistic players (level 0 players) March, 2010 Teck Hua Ho 42

43 Gang of Four s Predictions (LL=-955.7) March, 2010 Teck Hua Ho 43

44 lternative 2: Social Preferences March, 2010 Teck Hua Ho 44

45 Conclusions Dynamic level-k l model lis an empirical i alternative to I Captures limited induction and time unraveling Explains violations of I in centipede game Explains paradoxical behaviors in 2 well-known games (cooperation finitely fiitl repeated tdpd, chain-store hi paradox) Dynamic level-k model can be considered a tracing procedure for backward induction (since the former converges to the latter as time goes to infinity) March, 2010 Teck Hua Ho 45