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1 University of Michigan, Ann Arbor A Cognitive Hierarchy (CH) A Cognitive Hierarchy (CH) Model of Games Model of Games

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A Cognitive Hierarchy (CH) Model of Games. Motivation. Nash equilibrium and its refinements: Dominant theories in economics for predicting behaviors in competitive situations. Subjects do not play Nash in many one-shot games. - PowerPoint PPT Presentation

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Page 1: A Cognitive Hierarchy (CH)  Model of Games

1University of Michigan, Ann Arbor

A Cognitive Hierarchy (CH) A Cognitive Hierarchy (CH) Model of GamesModel of Games

Page 2: A Cognitive Hierarchy (CH)  Model of Games

2University of Michigan, Ann Arbor

MotivationMotivationNash equilibrium and its refinements: Dominant

theories in economics 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.

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Main GoalsMain Goals

Provide a behavioral theory to explain and predict behaviors in any one-shot gameNormal-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)

Page 4: A Cognitive Hierarchy (CH)  Model of Games

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Modeling PrinciplesModeling Principles

Principle Nash CH

Strategic Thinking

Best Response

Mutual Consistency

Page 5: A Cognitive Hierarchy (CH)  Model of Games

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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)

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6University of Michigan, Ann Arbor

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7University of Michigan, Ann Arbor

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

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8University of Michigan, Ann Arbor

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

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9University of Michigan, Ann Arbor

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

Page 10: A Cognitive Hierarchy (CH)  Model of Games

10University of Michigan, Ann Arbor

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

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

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

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11University of Michigan, Ann Arbor

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

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12University of Michigan, Ann Arbor

Modeling Decision RuleModeling Decision RuleProportion 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

Page 13: A Cognitive Hierarchy (CH)  Model of Games

13University of Michigan, Ann Arbor

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

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14University of Michigan, Ann Arbor

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

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15University of Michigan, Ann Arbor

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(

)(

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16University of Michigan, Ann Arbor

ImplicationsImplications

!)(

kekf

k A1 Poisson distribution with mean and variance =

A1,A2 Poisson, golden ratio Φ)

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La loi de Poisson a été introduite en 1838 par Siméon Denis Poisson (1781–1840), dans son ouvrage Recherches sur la probabilité des jugements en matière criminelle et en matière civile. Le sujet principal de cet ouvrage consiste en certaines variables aléatoires N qui dénombrent, entre autres choses, le nombre d'occurrences (parfois appelées « arrivées ») qui prennent place pendant un laps de temps donné.Si le nombre moyen d'occurrences dans cet intervalle est λ, alors la probabilité qu'il existe exactement k occurrences (k étant un entier naturel, k = 0, 1, 2, ...) est:

Où:e est la base de l'exponentielle (2,718...)k! est la factorielle de kλ est un nombre réel strictement positif.On dit alors que X suit la loi de Poisson de paramètre λ.Par exemple, si un certain type d'évènements se produit en moyenne 4 fois par minute, pour étudier le nombre d'évènements se produisant dans un laps de temps de 10 minutes, on choisit comme modèle une loi de Poisson de paramètre λ = 10×4 = 40.

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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

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19University of Michigan, Ann Arbor

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 0.090 0.212 0.23 0.33 0.33 0.33 0.33 0.33 0.331 0.329 0.35 0 1 0 1 0 02 0.255 0.28 0 0 1 1 0 03 0.132 0.14 0 0 1 1 0 0

Aggregate 1.00 0.08 0.43 0.50 0.85 0.08 0.08

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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

Page 21: A Cognitive Hierarchy (CH)  Model of Games

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Estimates of Mean Thinking Estimates of Mean Thinking Step Step

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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

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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

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24University of Michigan, Ann Arbor

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?

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Nash versus CH Model: Economic Value

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Application: Strategic IQhttp://128.32.67.154/siq13/default1.asp

A battery of 30 "well-known" games

Measure a subject's strategic IQ by how much money she makes (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

Page 27: A Cognitive Hierarchy (CH)  Model of Games

27University of Michigan, Ann Arbor

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 closest to

the Target Number

The prize to the winner is US$20Ho, Camerer, and Weigelt (AER, 1998)

Page 28: A Cognitive Hierarchy (CH)  Model of Games

28University of Michigan, Ann Arbor

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

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29University of Michigan, Ann Arbor

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

Page 30: A Cognitive Hierarchy (CH)  Model of Games

30University of Michigan, Ann Arbor

SummarySummary

CH Model:

Discrete thinking steps

Frequency Poisson distributed

One-shot games

Fits better than Nash and adds more economic value

Sensible interpretation of mixed strategies

Can “solve” multiplicity problem

Application: Measurement of Strategic IQ

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31University of Michigan, Ann Arbor

Research AgendaResearch AgendaBounded Rationality in Markets

Revised Utility Functions

Empirical Alternatives to Nash Equilibrium

A New Taxonomy of Games

Neural Foundation of Game Theory

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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

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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

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Neural Foundations of Game Theory

Neural foundation of game theory

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Strategic IQ: A New Taxonomy of Games

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

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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)

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38University of Michigan, Ann Arbor

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

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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

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

Is extensive CH (xCH) a sensible empirical alternative to BIP in predicting behavior in an extensive-form game like the Centipede?

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

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BIP consists of three premisesBIP consists of three 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.

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42University of Michigan, Ann Arbor

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)

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43University of Michigan, Ann Arbor

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))

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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.

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Prediction on Implied Take Prediction on Implied Take ProbabilityProbability

Implied 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.

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46University of Michigan, Ann Arbor

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

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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

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Limited thinking and PlanningLimited thinking and PlanningXk (k), k=1,2,3 follow independent Poisson

distributions

X3=common thinking/planning; X1=extra thinking, X2=extra planning

X (thinking) =X1+X3 ; Y (planning) =X2 +X3

follow jointly a bivariate Poisson distribution BP(1, 2, 3)

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Estimation ResultsEstimation Results6 stages All sessions

Low-Stake High-StakeSample 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 (1=2=0) -276.1 -105.9 -341.2 -753.0

ValidationSample Size 84 30 84 198

Agent Quantal Response Eqlbm (AQRE) 281.0 100.0 281.0 662.0

Extensive Cognitive Hierarchy (xCH) -132.8 -41.5 -120.7 -293.9xCH (1=2=0) -132.8 -41.5 -121.1 -293.9

4 stages

Thinking steps and steps of planning are perfectly correlated

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

CH 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