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A Dynamic Level- k Model in Games

A Dynamic Level- k Model in Games. Teck Ho and Xuanming Su UC Berkeley. Dual Pillars of Economic Analysis. Utility Specification Only final allocation matters Self-interests Exponential discounting Solution Method Nash and subgame perfect equilibrium (instant equilibration).

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A Dynamic Level- k Model in Games

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  1. A Dynamic Level-k Model in Games Teck Ho and Xuanming Su UC Berkeley Teck Hua Ho

  2. Dual Pillars of Economic Analysis • Utility Specification • Only final allocation matters • Self-interests • Exponential discounting • Solution Method • Nash and subgame perfect equilibrium (instant equilibration)

  3. Challenges: Utility Specification • Reference point matters: People care both about the final allocation as well as the changes with respect to a target level • Fairness: People care about others’ payoffs. We are nice to others who have been kind to us. We also get upset when others treat our peers better than us. • Hyperbolic discounting: People are impatient and prefer instant gratification

  4. Challenges: Solution Method • Nash and subgame perfect equilibrium: standard theories in marketing for predicting behaviors in competitive settings. • Subjects do not play Nash or subgame perfect equilibrium in experimental games. • Behaviors often converge to equilibrium with repeated interactions (especially when subjects are motivated by substantial financial incentives). • Multiplicity problem (e.g., coordination and infinitely repeated games). • Modeling subject heterogeneity really matters in games.

  5. Bounded Rationality in Markets: Revised Utility Function Ho, Lim, and Camerer (JMR, 2006)

  6. Bounded Rationality in Markets: Alternative Solution Methods

  7. Outline • Motivation • Backward induction and its systematic violations • Dynamic Level-k model and the main theoretical results • Empirical estimation • Alternative explanations: Reputation-based model and social preferences • Conclusions Teck Hua Ho

  8. A 4-stage Centipede Game P P P P 64 16 A B A B T T T T 4 1 2 8 16 4 8 32 Teck Hua Ho

  9. A 4-stage Centipede Game 64 16 A B A B 0 4 1 2 8 16 4 8 32 1 2 4 3 Teck Hua Ho

  10. A 6-Stage Centipede Game 256 64 A B A B A B 0 4 1 2 8 16 4 8 32 64 16 32 128 3 4 1 6 5 2 Teck Hua Ho

  11. Backward Induction Principle Nobel Prize, 1994 • Backward induction is the most widely accepted principle 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) • Dynamics in competitive interactions (e.g., repeated price competition) • Multi-person dynamic programming • For the principle to work, every player must be willingness to bet on others’ rationality Teck Hua Ho

  12. Violations of Backward Induction • Well-known violations in economic experiments include: (http://en.wikipedia.org/wiki/Backward_induction): • Passing in the centipede game • Cooperation in the finitely repeated PD • Chain-store paradox • Market settings? • Likely to be a failure of mutual consistency condition (different people make initial different bets on others’ rationality) Teck Hua Ho

  13. Standard Assumptions in Equilibrium Analysis Teck Hua Ho

  14. Notations 64 16 A B A B 4 1 2 8 16 4 8 32 Teck Hua Ho

  15. Deviation from Backward Induction Teck Hua Ho

  16. Examples 64 16 A B A B 4 1 8 32 2 8 16 4 Ex1: Ex2: Teck Hua Ho

  17. Systematic Violation 1: Limited Induction 64 16 A B A B 4 1 2 8 16 4 8 32 256 64 A B A B A B 4 1 2 8 16 4 8 32 64 16 32 128 Teck Hua Ho

  18. Limited Induction in Centipede Game Figure 1: Deviation in 4-stage versus 6-stage game Teck Hua Ho

  19. Systematic Violation 2: Time Unraveling 64 16 A B A B 4 1 2 8 16 4 8 32 Teck Hua Ho

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

  21. Outline • Motivation • Backward induction and its systematic violations • Dynamic Level-k model and the main theoretical results • Empirical estimation • Alternative explanations: Reputation-based model and social preferences • Conclusions Teck Hua Ho

  22. To develop a good descriptive model to predict the probability of player i (i=1,…,I) choosing strategy j at subgames(s=1,.., S)in any dynamic game of complete information Research question Teck Hua Ho

  23. Criteria of a “Good” Model • Nests backward induction as a special case • Behavioral 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) Teck Hua Ho

  24. Standard Assumptions in Equilibrium Analysis Teck Hua Ho

  25. Dynamic Level-k Model: Summary • Players choose rule from a rule hierarchy • Players make differential initial bets on others’ chosen rules • After each game play, players observe others’ rules • Players update their beliefs on rules chosen by others • Players always choose a rule to maximize their subjective expected utility in each round Teck Hua Ho

  26. Dynamic Level-k Model: Rule Hierarchy • Players choose rule from a rule hierarchy generated by best-responses • Rule hierarchy: • Restrict L0 to follow behavior proposed in the existing literature (i.e., pass in every stage) Teck Hua Ho

  27. Dynamic Level-k Model: Poisson Initial Belief • Different people make different initial bets on others’ chosen rules • Poisson distributed initial beliefs: • f(k) fraction of players think that their opponents use Lkrule. l : average belief of rules used by opponents Teck Hua Ho

  28. Dynamic Level-k model:Belief Updating at the End of Round t • Level k’s initial belief strength b entirely on k-1 • Update after observing which rule opponent chose • I(k, t) = 1 if opponent chose Lk and 0 otherwise • Bayesian updating involving a multi-nomial distribution with a Dirichlet prior (Fudenberg and Levine, 1998; Camerer and Ho, 1999) Teck Hua Ho

  29. Dynamic Level-k model: :Optimal Rule in Round t+1 • Optimal rule k*: • Let the specified action of rule Lkat subgames be aks Teck Hua Ho

  30. The Centipede Game (Rule Hierarchy) 0 1 2 3 4 Teck Hua Ho

  31. A 4-stage Centipede Game 64 16 A B A B 0 4 1 2 8 16 4 8 32 1 2 4 3 Teck Hua Ho

  32. Player A in 4-Stage Centipede Game Teck Hua Ho

  33. Dynamic Level-k Model: Summary • Players choose rule from a rule hierarchy • Players make differential initial bets on others’ chosen rules • After each game play, players observe others’ rules • Players update their beliefs on rules chosen by others • Players always choose a rule to maximize their subjective expected utility in each round • A 2-paramter extension of backward induction (l and b) Teck Hua Ho

  34. Main Theoretical Results: Limited Induction Theorem 1: The dynamic level-k model implies that the limited induction property is satisfied. Specifically, we have: Teck Hua Ho

  35. Main Theoretical Results: Time Unraveling Theorem 2: The dynamic level-k model implies that the time unraveling property is satisfied. Specifically, we have: Teck Hua Ho

  36. Outline • Motivation • Backward induction and its systematic violations • Dynamic Level-k model and the main theoretical results • Empirical estimation • Alternative explanations: Reputation-based model and social preferences • Conclusions Teck Hua Ho

  37. 4-Stage versus 6-Stage Centipede Games 64 16 A B A B 4 1 2 8 16 4 8 32 256 64 A B A B A B 4 1 2 8 16 4 8 32 64 16 32 128 Teck Hua Ho

  38. Caltech versus PCC Subjects Teck Hua Ho

  39. Caltech Subjects Teck Hua Ho

  40. Caltech Subjects: 6-Stage Centipede Game Teck Hua Ho

  41. Model Predictions; Caltech Subjects Teck Hua Ho

  42. Model Predictions: PCC subjects Teck Hua Ho

  43. Alternative 1:Reputation-based Model (Kreps, et al, 1982) large q = proportion of altruistic players (level 0 players) Teck Hua Ho

  44. Alternative 1: Reputation-based Model Teck Hua Ho

  45. Alternative 2: Social Preferences Teck Hua Ho

  46. Alternative 2: Empirical Estimation Teck Hua Ho

  47. Conclusions • Dynamic level-k model is an empirical alternative to BI • Captures limited induction and time unraveling • Explains violations of BI in centipede game • Dynamic level-k model can be considered a tracing procedure for BI (since the former converges to the latter as time goes to infinity) Teck Hua Ho

  48. p-Beauty Contests • n=7 players (randomly chosen) • Every player simultaneously chooses a number from 0 to 100 • Compute the group average • Define Target Number to be p=0.7 times the group average • The winner is the player whose number is the closet to the Target Number • The prize to the winner is US$20 (Ho & H0)

  49. Empirical Regularity 1: Groups with Smaller p Converge Faster Teck Hua Ho

  50. Empirical Regularity 2: Larger Groups Converge Faster Teck Hua Ho

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