A Dynamic Level-k Model in Games Teck Ho and Xuanming Su UC Berkeley April 2011 Teck Hua Ho 1

Teck H. 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) April 2011

Teck H. Ho 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 April 2011

Teck H. Ho 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. April 2011

Teck H. Ho 5 Bounded Rationality in Markets: Revised Utility Function Ho, Lim, and Camerer (JMR, 2006) April 2011

Teck H. Ho 6 Bounded Rationality in Markets: Alternative Solution Methods April 2011

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 April 2011 Teck Hua Ho 7

A 4-stage Centipede Game April 2011 Teck Hua Ho 8 A A B B P P PP T T T T

A 4-stage Centipede Game April 2011 Teck Hua Ho 9 A A B B

A 6-Stage Centipede Game April 2011 Teck Hua Ho A ABB AB Outcome Round %5.5%17.2%33.1% 9.00%2.10% %7.4%22.8%44.1%16.9%6.60%0.70% Backward Induction100%0%

Backward Induction Principle  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 April 2011 Teck Hua Ho 11 Nobel Prize, 1994

Violations of Backward Induction  Well-known violations in economic experiments include: ( ) : 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) April 2011 Teck Hua Ho 12

April 2011 Standard Assumptions in Equilibrium Analysis 13 Teck Hua Ho

Notations April 2011 Teck Hua Ho 14 A A B B

Deviation from Backward Induction April 2011 Teck Hua Ho 15

Examples April 2011 Teck Hua Ho 16 A A B B Ex1: Ex2:

Systematic Violation 1: Limited Induction April 2011 Teck Hua Ho 17 A A B B A ABB AB

Limited Induction in Centipede Game April 2011 Teck Hua Ho 18 Figure 1: Deviation in 4-stage versus 6-stage game

Systematic Violation 2: Time Unraveling April 2011 Teck Hua Ho 19 A A B B

Time Unraveling in Centipede Game April 2011 Teck Hua Ho 20 Figure 2: Deviation in 1 st vs. 10 th round of the 4-stage game

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 April 2011 Teck Hua Ho 21

April 2011 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 22 Teck Hua Ho

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) April 2011 Teck Hua Ho 23

April 2011 Standard Assumptions in Equilibrium Analysis 24 Teck Hua Ho

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 April 2011 Teck Hua Ho 25

Dynamic Level-k Model: Rule Hierarchy  Players choose rule from a rule hierarchy generated by best- responses  Rule hierarchy:   Restrict L 0 to follow behavior proposed in the existing literature (i.e., pass in every stage)  April 2011 Teck Hua Ho 26

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 L k rule. April 2011 Teck Hua Ho 27 : average belief of rules used by opponents

April 2011 Dynamic Level-k model: Belief Updating at the End of Round t  Level k’s initial belief strength  entirely on k-1  Update after observing which rule opponent chose  I(k, t) = 1 if opponent chose L k and 0 otherwise  Bayesian updating involving a multi-nomial distribution with a Dirichlet prior (Fudenberg and Levine, 1998; Camerer and Ho, 1999) 28 Teck Hua Ho

April 2011 Dynamic Level-k model: : Optimal Rule in Round t+1  Optimal rule k * :  Let the specified action of rule L k at subgame s be a ks 29 Teck Hua Ho

April 2011 The Centipede Game (Rule Hierarchy) 30 Teck Hua Ho Player APlayer B (P, -, P, -)(-, P, -, P) (P, -, P, -)(-, P, -, T) (P, -, T, -)(-, P, -, T) (P, -, T, -)(-, T, -, T) (T, -, T, -)(-, T, -, T)

A 4-stage Centipede Game April 2011 Teck Hua Ho 31 A A B B

Player A in 4-Stage Centipede Game April 2011 Teck Hua Ho 32

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 ( and  ) April 2011 Teck Hua Ho 33

April 2011 Main Theoretical Results: Limited Induction 34 Teck Hua Ho Theorem 1: The dynamic level-k model implies that the limited induction property is satisfied. Specifically, we have:

April 2011 Main Theoretical Results: Time Unraveling 35 Teck Hua Ho Theorem 2: The dynamic level-k model implies that the time unraveling property is satisfied. Specifically, we have:

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 April 2011 Teck Hua Ho 36

4-Stage versus 6-Stage Centipede Games April 2011 Teck Hua Ho 37 A A B B A ABB AB

Caltech versus PCC Subjects April 2011 Teck Hua Ho 38

Caltech Subjects April 2011 Teck Hua Ho 39

Caltech Subjects: 6-Stage Centipede Game April 2011 Teck Hua Ho 40

Model Predictions; Caltech Subjects April 2011 Teck Hua Ho 41

Model Predictions: PCC subjects April 2011 Teck Hua Ho 42

Alternative 1: Reputation-based Model (Kreps, et al, 1982) April 2011 Teck Hua Ho 43 large  = proportion of altruistic players (level 0 players)

Alternative 1: Reputation-based Model April 2011 Teck Hua Ho 44

Alternative 2: Social Preferences April 2011 Teck Hua Ho 45

Alternative 2: Empirical Estimation April 2011 Teck Hua Ho 46

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) April 2011 Teck Hua Ho 47

April 2011 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)

Empirical Regularity 1: Groups with Smaller p Converge Faster April 2011 Teck Hua Ho 49

Empirical Regularity 2: Larger Groups Converge Faster April 2011 Teck Hua Ho 50

Dynamic Level-k Model Predictions April 2011 Teck Hua Ho 51

Teck H. Ho 52April 2011

Teck H. Ho 53 Modeling Philosophy Simple(Economics) General(Economics) Precise(Economics) Empirically disciplined(Psychology) “the empirical background of economic science is definitely inadequate...it would have been absurd in physics to expect Kepler and Newton without Tycho Brahe” (von Neumann & Morgenstern ‘44) “Without having a broad set of facts on which to theorize, there is a certain danger of spending too much time on models that are mathematically elegant, yet have little connection to actual behavior. At present our empirical knowledge is inadequate...” (Eric Van Damme ‘95) April 2011