Spring 2006 1Teck H. Ho

A Cognitive Hierarchy (A Cognitive Hierarchy (CCH) H) Model of GamesModel of Games

Teck H. Ho

Haas School of Business

University of California, Berkeley

Joint work with Colin Camerer, Caltech

Juin-Kuan Chong, NUS

Spring 2006 2Teck H. Ho

MotivationMotivationNash equilibrium and its refinements: Dominant

theories in economics and marketing for predicting behaviors in competitive situations.

Subjects do not play Nash in many one-shot games.

Behaviors do not converge to Nash with repeated interactions in some games.

Multiplicity problem (e.g., coordination games).

Modeling heterogeneity really matters in games.

Spring 2006 3Teck H. Ho

Main GoalsMain Goals

Provide a behavioral theory to explain and predict behaviors in any one-shot game.Normal-form games (e.g., zero-sum game, p-

beauty contest)Extensive-form games (e.g., centipede)

Provide an empirical alternative to Nash equilibrium (Camerer, Ho, and Chong, QJE, 2004) and backward induction principle (Ho, Camerer, and Chong, 2005)

Spring 2006 4Teck H. Ho

Modeling PrinciplesModeling Principles

Principle Nash CH

Strategic Thinking

Best Response

Mutual Consistency

Spring 2006 5Teck H. Ho

Modeling PhilosophyModeling 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)

Spring 2006 6Teck H. Ho

Example 1: “zero-sum game”Example 1: “zero-sum game”

COLUMNL C R

T 0,0 10,-10 -5,5

ROW M -15,15 15,-15 25,-25

B 5,-5 -10,10 0,0

Messick(1965), Behavioral Science

Spring 2006 7Teck H. Ho

Nash Prediction: Nash Prediction: “zero-sum game”“zero-sum game”

Nash COLUMN Equilibrium

L C RT 0,0 10,-10 -5,5 0.40

ROW M -15,15 15,-15 25,-25 0.11

B 5,-5 -10,10 0,0 0.49Nash

Equilibrium 0.56 0.20 0.24

Spring 2006 8Teck H. Ho

CH Prediction: CH Prediction: “zero-sum game”“zero-sum game”

Nash CH ModelCOLUMN Equilibrium ( = 1.55)

L C RT 0,0 10,-10 -5,5 0.40 0.07

ROW M -15,15 15,-15 25,-25 0.11 0.40

B 5,-5 -10,10 0,0 0.49 0.53Nash

Equilibrium 0.56 0.20 0.24CH Model( = 1.55) 0.86 0.07 0.07

Spring 2006 9Teck H. Ho

Empirical Frequency: Empirical Frequency: “zero-sum game”“zero-sum game”

http://groups.haas.berkeley.edu/simulations/CH/

Nash CH Model EmpiricalCOLUMN Equilibrium ( = 1.55) Frequency

L C RT 0,0 10,-10 -5,5 0.40 0.07 0.13

ROW M -15,15 15,-15 25,-25 0.11 0.40 0.33

B 5,-5 -10,10 0,0 0.49 0.53 0.54Nash

Equilibrium 0.56 0.20 0.24CH Model( = 1.55) 0.86 0.07 0.07Empirical

Frequency 0.88 0.08 0.04

Spring 2006 10Teck H. Ho

The Cognitive Hierarchy (CH) The Cognitive Hierarchy (CH) ModelModelPeople are different and have different decision rules.

Modeling heterogeneity (i.e., distribution of types of players). Types of players are denoted by levels 0, 1, 2, 3,…,

Modeling decision rule of each type.

Spring 2006 11Teck H. Ho

Modeling Decision RuleModeling Decision RuleFrequency of k-step is f(k)

Step 0 choose randomly

k-step thinkers know proportions f(0),...f(k-1)

Form beliefs and best-respond based on beliefs

Iterative and no need to solve a fixed point

gk (h) f (h)

f (h ' )h ' 1

K 1

Spring 2006 12Teck H. Ho

COLUMNL C R

T 0,0 10,-10 -5,5

ROW M -15,15 15,-15 25,-25

B 5,-5 -10,10 0,0

K's K+1's ROW COLLevel (K) Proportion Belief T M B L C R

0 0.212 1.00 0.33 0.33 0.33 0.33 0.33 0.33Aggregate 1.00 0.33 0.33 0.33 0.33 0.33 0.33

0 0.212 0.39 0.33 0.33 0.33 0.33 0.33 0.331 0.329 0.61 0 1 0 1 0 0

Aggregate 1.00 0.13 0.74 0.13 0.74 0.13 0.130 0.212 0.27 0.33 0.33 0.33 0.33 0.33 0.331 0.329 0.41 0 1 0 1 0 02 0.255 0.32 0 0 1 1 0 0

Aggregate 1.00 0.09 0.50 0.41 0.82 0.09 0.09

K Proportion, f(k)0 0.2121 0.3292 0.2553 0.132

>3 0.072

Spring 2006 13Teck H. Ho

Theoretical ImplicationsTheoretical Implications

Exhibits “increasingly rational expectations”

Normalized gK(h) approximates f(h) more closely as k ∞∞ (i.e., highest level types are

“sophisticated” (or "worldly") and earn the most.

Highest level type actions converge as k ∞∞

marginal benefit of thinking harder 00

Spring 2006 14Teck H. Ho

Modeling Heterogeneity, Modeling Heterogeneity, f(k)f(k)

A1:

sharp drop-off due to increasing difficulty in simulating others’ behaviors

A2: f(0) + f(1) = 2f(2)

kkf

kf

)1(

)(

Spring 2006 15Teck H. Ho

ImplicationsImplications

!)(

kekf

k A1 Poisson distribution with mean and variance =

A1,A2 Poisson, golden ratio Φ)

Spring 2006 16Teck H. Ho

Poisson DistributionPoisson Distribution f(k) with mean step of thinking :

!)(

kekf

k

Poisson distributions for various

00.05

0.10.15

0.20.25

0.30.35

0.4

0 1 2 3 4 5 6

number of steps

fre

qu

en

cy

Spring 2006 17Teck H. Ho

Theoretical Properties of CH Theoretical Properties of CH ModelModelAdvantages over Nash equilibrium

Can “solve” multiplicity problem (picks one statistical distribution)

Sensible interpretation of mixed strategies (de facto purification)

Theory: τ∞ converges to Nash equilibrium in (weakly)

dominance solvable games

Spring 2006 18Teck H. Ho

Estimates of Mean Thinking Estimates of Mean Thinking Step Step

Spring 2006 19Teck H. Ho

Figure 3b: Predicted Frequencies of Nash Equilibrium for Entry and Mixed Games

y = 0.707x + 0.1011

R2 = 0.4873

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Empirical Frequency

Pre

dic

ted

Fre

qu

en

cy

Nash: Theory vs. Data

Spring 2006 20Teck H. Ho

Figure 2b: Predicted Frequencies of Cognitive Hierarchy Models for Entry and Mixed Games (common )

y = 0.8785x + 0.0419

R2 = 0.8027

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Empirical Frequency

Pre

dic

ted

Fre

qu

ency

CH Model: Theory vs. Data

Spring 2006 21Teck H. Ho

Economic ValueEconomic Value

Evaluate models based on their value-added rather than statistical fit (Camerer and Ho, 2000)

Treat models like consultants

If players were to hire Mr. Nash and Ms. CH as consultants and listen to their advice (i.e., use the model to forecast what others will do and best-respond), would they have made a higher payoff?

Spring 2006 22Teck H. Ho

Nash versus CH Model: Economic Value

Spring 2006 23Teck H. Ho

Application: Strategic IQhttp://128.32.67.154/siq

A battery of 30 "well-known" games

Measure CH player and a subject's strategic IQ by how much money they make (matched against a defined pool of subjects)

Factor analysis + fMRI to figure out whether certain brain region accounts for superior performance in "similar" games

Specialized subject poolsSoldiers

Writers

Chess players

Patients with brain damages

Spring 2006 24Teck H. Ho

Example 2Example 2: P: P-Beauty Contest-Beauty Contest n players

Every player simultaneously chooses a number from 0

to 100

Compute the group average

Define Target Number to be 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$20Ho, Camerer, and Weigelt (AER, 1998)

Spring 2006 25Teck H. Ho

A Sample of CEOsA Sample of CEOs

David Baltimore President California Institute of Technology

Donald L. Bren

Chairman of the BoardThe Irvine Company

• Eli BroadChairmanSunAmerica Inc.

• Lounette M. Dyer Chairman Silk Route Technology

• David D. Ho Director The Aaron Diamond AIDS Research Center

• Gordon E. Moore Chairman Emeritus Intel Corporation

• Stephen A. Ross Co-Chairman, Roll and Ross Asset Mgt Corp

• Sally K. Ride President Imaginary Lines, Inc., and Hibben Professor of Physics

Spring 2006 26Teck H. Ho

Results in various p-BC gamesResults in various p-BC games

Subject Pool Group Size Sample Size Mean Error (Nash) Error (CH)

CEOs 20 20 37.9 -37.9 -0.1 1.0

80 year olds 33 33 37.0 -37.0 -0.1 1.1

Economics PhDs 16 16 27.4 -27.4 0.0 2.3

Portfolio Managers 26 26 24.3 -24.3 0.1 2.8

Game Theorists 27-54 136 19.1 -19.1 0.0 3.7

Spring 2006 27Teck H. Ho

Example 3: Centipede GameExample 3: Centipede Game

1 2 2 21 1

0.400.10

0.200.80

1.600.40

0.803.20

6.401.60

3.2012.80

25.606.40

Figure 1: Six-move Centipede Game

Spring 2006 28Teck H. Ho

CH vs. Backward Induction CH vs. Backward Induction Principle (BIP)Principle (BIP)

Can CH be extended (i.e., extensive CH) to provide an empirical alternative to BIP in predicting behavior in the Centipede?

Is there a difference between steps of thinking and look-ahead (planning)?

Spring 2006 29Teck H. Ho

The three underlying premises The three underlying premises

Rationality: Given a choice between two alternatives, a player chooses the most preferred.

Truncation consistency: Replacing a subgame with its equilibrium payoffs does not affect play elsewhere in the game.

Subgame consistency: Play in a subgame is independent of the subgame’s position in a larger game.

Binmore, McCarthy, Ponti, and Samuelson (JET, 2002) show violations of both truncation and subgame consistencies.

Spring 2006 30Teck H. Ho

Truncation ConsistencyTruncation Consistency

VS.

1 2 2 21 1

0.400.10

0.200.80

1.600.40

0.803.20

6.401.60

3.2012.80

25.606.40

Figure 1: Six-move Centipede game

1 2 21

0.400.10

0.200.80

1.600.40

0.803.20

6.401.60

Figure 2: Four-move Centipede game (Low-Stake)

Spring 2006 31Teck H. Ho

The three underlying premisesThe three underlying premises

Rationality: Given a choice between two alternatives, a player chooses the most preferred.

Truncation consistency: Replacing a subgame with its equilibrium payoffs does not affect play elsewhere in the game.

Subgame consistency: Play in a subgame is independent of the subgame’s position in a larger game.

Binmore, McCarthy, Ponti, and Samuelson (JET, 2002) show violations of both truncation and subgame consistencies.

Spring 2006 32Teck H. Ho

Subgame ConsistencySubgame Consistency

1 2 2 21 1

0.400.10

0.200.80

1.600.40

0.803.20

6.401.60

3.2012.80

25.606.40

VS.

2 21 1

1.600.40

0.803.20

6.401.60

3.2012.80

25.606.40

Figure 1: Six-move Centipede game

Figure 3: Four-move Centipede game (High-Stake (x4))

Spring 2006 33Teck H. Ho

Data: Truncation & Subgame Data: Truncation & Subgame ConsistenciesConsistencies

Data p1 p2 p3 p4 p5 p6

6-move 0.01 0.06 0.21 0.53 0.73 0.85

4-move(Low Stake) 0.07 0.38 0.65 0.75

4-move(High Stake) 0.15 0.44 0.67 0.69

*Data from McKelvey and Palfrey (1992)

Spring 2006 34Teck H. Ho

Limited thinking and PlanningLimited thinking and Planning

Steps of Reasoning(People)

Ste

ps

of P

lann

ing

(Tim

e)

0 1 2 3 4 5

1

2

3

4

5

Bivariate Poisson

Spring 2006 35Teck H. Ho

Estimation Results (out-of-Estimation Results (out-of-sample)sample)

Thinking steps and steps of planning are perfectly correlated

6 stages All sessionsLow-Stake High-Stake

Sample Size 281 100 281 662

CalibrationSample Size 197 70 197 464

Agent Quantal Response Eqlbm (AQRE) -287.0 -106.8 -409.8 -848.2

Extensive Cognitive Hierarchy (xCH) -276.1 -105.9 -341.2 -753.0xCH (Step of Thinking = Steps of Planning) -276.1 -105.9 -341.2 -753.0

4 stages

Spring 2006 36Teck H. Ho

Data and xCH Prediction: Data and xCH Prediction: Truncation & Subgame ConsistenciesTruncation & Subgame Consistencies

xCH Prediction

6-move 0.06 0.16 0.15 0.48 0.90 0.99

4-move(Low-Stake) 0.15 0.31 0.76 0.97

4-move(High-Stake) 0.21 0.34 0.71 0.95

Data p1 p2 p3 p4 p5 p6

6-move 0.01 0.06 0.21 0.53 0.73 0.85

4-move(Low Stake) 0.07 0.38 0.65 0.75

4-move(High Stake) 0.15 0.44 0.67 0.69

Spring 2006 37Teck H. Ho

SummarySummaryCH Model:

Discrete thinking stepsFrequency Poisson distributed

One-shot gamesFits better than Nash and adds more economic valueSensible interpretation of mixed strategies Can “solve” multiplicity problem

xCH:Provides an empirical alternative to BIPLimited steps of thinking and planning are perfectly

correlated

Spring 2006 38Teck H. Ho

Spring 2006 39Teck H. Ho

Bounded Rationality in Markets: Revised Utility Function

Behavioral Regularities Standard Assumption New Model Specification Marketing ApplicationReference Example Example

1. Revised Utility Function

- Reference point and - Expected Utility Theory - Prospect Theory - Two-part tariff - double loss aversion Kahneman and Tversky (1979) marginalization problem

- Fairness - Self-interested - Inequality aversion - Price discrimination Fehr and Schmidt (1999)

- Impatience - Exponential discounting - Hyperbolic Discounting - Price promotion and Ainslie (1975) packaging size design

Spring 2006 40Teck H. Ho

Bounded Rationality in Markets: Alternative Solution Concepts

Behavioral Regularities Standard Assumption New Model Specification Marketing ApplicationExample Example

2. Bounded Computation Ability

- Nosiy Best Response - Best Response - Quantal Best Response - NEIO McKelvey and Palfrey (1995)

- Limited Thinking Steps - Rational expectation - Cognitive hierarchy - Market entry competition Camerer, Ho, Chong (2004)

- Myopic and learn - Instant equilibration - Experience weighted attraction - Lowest price guarantee Camerer and Ho (1999) competition

Spring 2006 41Teck H. Ho

Neural Foundations of Game TheoryNeural foundation of game theory

Spring 2006 42Teck H. Ho

Strategic IQ: A New Taxonomy of Games

Spring 2006 43Teck H. Ho

Spring 2006 44Teck H. Ho

First-Shot GamesFirst-Shot Games

The FCC license auctions, elections, military campaigns, legal disputes

Many marketing/IO models

Simple gane experiments in economics and marketing

Spring 2006 45Teck H. Ho

Nash versus CH Model: LL and MSD (in-sample)

Spring 2006 46Teck H. Ho

Economic Value:Economic Value:Definition and MotivationDefinition and Motivation

“A normative model must produce strategies that are at least as good as what people can do without them.” (Schelling, 1960)

A measure of degree of disequilibrium, in dollars.

If players are in equilibrium, then an equilibrium theory will advise them to make the same choices they would make anyway, and hence will have zero economic value

If players are not in equilibrium, then players are mis-forecasting what others will do. A theory with more accurate beliefs will have positive economic value (and an equilibrium theory can have negative economic value if it misleads players)

Spring 2006 47Teck H. Ho

Alternative SpecificationsAlternative Specifications

Overconfidence:

k-steps think others are all one step lower (k-1) (Stahl, GEB, 1995; Nagel, AER, 1995; Ho, Camerer and Weigelt, AER, 1998)

“Increasingly irrational expectations” as K ∞

Has some odd properties (e.g., cycles in entry games)

Self-conscious:

k-steps think there are other k-step thinkers

Similar to Quantal Response Equilibrium/Nash

Fits worse

Spring 2006 48Teck H. Ho

Implied Take ProbabilityImplied Take ProbabilityImplied take probability at each stage, pj

Truncation consistency: For a given j, pj is identical in both 4-move (low-stake) and 6-move games.

Subgame consistency: For a given j, pn-j (n=4 or 6)

is identical in both 4-move (high-stake) and 6-move games.

Spring 2006 49Teck H. Ho

KK-Step Look-ahead (Planning)-Step Look-ahead (Planning)

1 2 2 21 1

0.400.10

0.200.80

1.600.40

0.803.20

6.401.60

3.2012.80

25.606.40

1 2

0.400.10

0.200.80

V1

V2

Example: 1-step look-ahead