1 teck-hua ho ch model march – june, 2003 a cognitive hierarchy theory of one-shot games teck h....

March – June, 20031

Teck-Hua HoCH Model

A Cognitive Hierarchy Theory of A Cognitive Hierarchy Theory of One-Shot GamesOne-Shot Games

Teck H. Ho

Haas School of Business

University of California, Berkeley

Joint work with Colin Camerer, Caltech

Juin-Kuan Chong, NUS


Teck-Hua HoCH Model

MotivationMotivation

Nash equilibrium and its refinements: Dominant theories in economics for predicting behaviors in games.

Subjects in experiments hardly play Nash in the first round but do often converge to it eventually.

Multiplicity problem (e.g., coordination games)

Modeling heterogeneity really matters in games.


Teck-Hua HoCH Model

Research GoalsResearch Goals

How to model bounded rationality (first-period behavior)? Cognitive Hierarchy (CH) model

How to model equilibration? EWA learning model (Camerer and Ho,

Econometrica, 1999; Ho, Camerer, and Chong, 2003)

How to model repeated game behavior? Teaching model (Camerer, Ho, and Chong,

Journal of Economic Theory, 2002)


Teck-Hua HoCH Model

Modeling PrinciplesModeling Principles

Principle Nash Thinking

Strategic Thinking

Best Response

Mutual Consistency


Teck-Hua HoCH Model

Modeling PhilosophyModeling Philosophy

General (Game Theory)Precise (Game Theory)Empirically disciplined (Experimental Econ)

“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)


Teck-Hua HoCH Model

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


Teck-Hua HoCH Model

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


Teck-Hua HoCH Model

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

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

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


Teck-Hua HoCH Model

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


Teck-Hua HoCH Model

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)

Modeling decision rule of each type

Guided by modeling philosophy (general, precise, and empirically disciplined)


Teck-Hua HoCH Model

Modeling Decision RuleModeling Decision Rule

f(0) step 0 choose randomly

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

Normalize and best-respond

1

1

'

'

)(

)()( K

h

hf

hfhg


Teck-Hua HoCH Model

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


Teck-Hua HoCH Model

ImplicationsImplications

Exhibits “increasingly rational expectations” Exhibits “increasingly rational expectations”

Normalized Normalized g(h)g(h) approximates approximates f(h)f(h) more closely more closely as as kk ∞∞ ((i.i.e., highest level types are e., highest level types are “sophisticated” (or ”worldly) and earn the most“sophisticated” (or ”worldly) and earn the most

Highest level type Highest level type actionsactions converge as converge as kk ∞∞

marginal benefit of thinking harder marginal benefit of thinking harder 00


Teck-Hua HoCH Model

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


Teck-Hua HoCH Model

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

A1:

sharp drop-off due to increasing working memory constraint

A2: f(1) is the mode

A3: f(0)=f(2) (partial symmetry)

A4a: f(0)+f(1)=f(2)+f(3)+f(4)… A4b: f(2)=f(3)+f(4)+f(5)…

kkf

kf

kkf

kf

)1(

)(1

)1(

)(


Teck-Hua HoCH Model

ImplicationsImplications

!)(

kekf

k A1 Poisson distribution with mean and variance =

A1,A2 Poisson distribution, 1<

A1,A3 Poisson,

ab) Poisson, golden ratio Φ)


Teck-Hua HoCH Model

Poisson DistributionPoisson Distribution

f(k) with mean step of thinking :!

)(k

ekfk

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


Teck-Hua HoCH Model

Historical RootsHistorical Roots

“Fictitious play” as an algorithm for computing Nash equilibrium (Brown, 1951; Robinson, 1951)

In our terminology, the fictitious play model is equivalent to one in which f(k) = 1/N for N steps of thinking and N ∞

Instead of a single player iterating repeatedly until a fixed point is reached and taking the player’s earlier tentative decisions as pseudo-data, we posit a population of players in which a fraction f(k) stop after k-steps of thinking


Teck-Hua HoCH Model

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

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

Solves refinement problems (all moves occur in equilibrium)

Sensible interpretation of mixed strategies (de facto purification)

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

dominance solvable gamesEqual splits in Nash demand games


Teck-Hua HoCH Model

Example 2: Entry gamesExample 2: Entry games

Market entry with many entrants:

Industry demand D (as % of # of players) is announced

Prefer to enter if expected %(entrants) < D;

Stay out if expected %(entrants) > D

All choose simultaneously

Experimental regularity in the 1st period: Consistent with Nash prediction, %(entrants) increases with D

“To a psychologist, it looks like magic”-- D. Kahneman ‘88


Teck-Hua HoCH Model

How entry varies with industry demand D, (Sundali, Seale & Rapoport, 2000)

0

0.1

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0.5

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0.8

0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Demand (as % of number of players )

% e

ntr

y

entry=demand

experimental data

Example 2: Entry games Example 2: Entry games (data)(data)


Teck-Hua HoCH Model

Behaviors of Level 0 and 1 Players (=1.25)

Level 0

Level 1

% o

f E

nt r

y

Demand (as % of # of players)


Teck-Hua HoCH Model

Behaviors of Level 0 and 1Players(=1.25)

Level 0 + Level 1

% o

f E

nt r

y



Teck-Hua HoCH Model

Behaviors of Level 2 Players(=1.25)

Level 2

Level 0 + Level 1

% o

f E

nt r

y



Teck-Hua HoCH Model

Behaviors of Level 0, 1, and 2 Players(=1.25)

Level 2

Level 0 +Level 1

Level 0 + Level 1 +Level 2

% o

f E

nt r

y



Teck-Hua HoCH Model

How entry varies with demand (D), experimental data and thinking model

0

0.1

0.2

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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


% e

ntr

y entry=demand

experimental data

Entry Games (Imposing Entry Games (Imposing Monotonicity on CH Model)Monotonicity on CH Model)


Teck-Hua HoCH Model

Estimates of Mean Thinking Estimates of Mean Thinking Step Step

Table 1: Parameter Estimate for Cognitive Hierarchy Models

Data set Stahl & Cooper & Costa-GomesWilson (1995) Van Huyck et al. Mixed Entry

Game-specific Game 1 2.93 16.02 2.16 0.98 0.69Game 2 0.00 1.04 2.05 1.71 0.83Game 3 1.35 0.18 2.29 0.86 -Game 4 2.34 1.22 1.31 3.85 0.73Game 5 2.01 0.50 1.71 1.08 0.69Game 6 0.00 0.78 1.52 1.13Game 7 5.37 0.98 0.85 3.29Game 8 0.00 1.42 1.99 1.84Game 9 1.35 1.91 1.06Game 10 11.33 2.30 2.26Game 11 6.48 1.23 0.87Game 12 1.71 0.98 2.06Game 13 2.40 1.88Game 14 9.07Game 15 3.49Game 16 2.07Game 17 1.14Game 18 1.14Game 19 1.55Game 20 1.95Game 21 1.68Game 22 3.06Median 1.86 1.01 1.91 1.77 0.71

Common 1.54 0.80 1.69 1.48 0.73


Teck-Hua HoCH Model

Table A1: 95% Confidence Interval for the Parameter Estimate of Cognitive Hierarchy Models

Data set

Lower Upper Lower Upper Lower Upper Lower Upper Lower UpperGame-specific Game 1 2.40 3.65 15.40 16.71 1.58 3.04 0.67 1.22 0.21 1.43Game 2 0.00 0.00 0.83 1.27 1.44 2.80 0.98 2.37 0.73 0.88Game 3 0.75 1.73 0.11 0.30 1.66 3.18 0.57 1.37 - -Game 4 2.34 2.45 1.01 1.48 0.91 1.84 2.65 4.26 0.56 1.09Game 5 1.61 2.45 0.36 0.67 1.22 2.30 0.70 1.62 0.26 1.58Game 6 0.00 0.00 0.64 0.94 0.89 2.26 0.87 1.77Game 7 5.20 5.62 0.75 1.23 0.40 1.41 2.45 3.85Game 8 0.00 0.00 1.16 1.72 1.48 2.67 1.21 2.09Game 9 1.06 1.69 1.28 2.68 0.62 1.64Game 10 11.29 11.37 1.67 3.06 1.34 3.58Game 11 5.81 7.56 0.75 1.85 0.64 1.23Game 12 1.49 2.02 0.55 1.46 1.40 2.35Game 13 1.75 3.16 1.64 2.15Game 14 6.61 10.84Game 15 2.46 5.25Game 16 1.45 2.64Game 17 0.82 1.52Game 18 0.78 1.60Game 19 1.00 2.15Game 20 1.28 2.59Game 21 0.95 2.21Game 22 1.70 3.63

Common 1.39 1.67 0.74 0.87 1.53 2.13 1.30 1.78 0.42 1.07

Stahl &Wilson (1995)

Cooper &Van Huyck

Costa-Gomeset al. Mixed Entry

CH Model: CI of Parameter Estimates


Teck-Hua HoCH Model

Table 2: Model Fit (Log Likelihood LL and Mean-squared Deviation MSD)

Stahl & Cooper & Costa-GomesData set Wilson (1995) Van Huyck et al. Mixed Entry

Cognitive Hierarchy (Game-specific ) 1

LL -721 -1690 -540 -824 -150MSD 0.0074 0.0079 0.0034 0.0097 0.0004Cognitive Hierarchy (Common )LL -918 -1743 -560 -872 -150MSD 0.0327 0.0136 0.0100 0.0179 0.0005

Cognitive Hierarchy (Common )LL -941 -1929 -599 -884 -153MSD 0.0425 0.0328 0.0257 0.0216 0.0034

Nash Equilibrium 2

LL -3657 -10921 -3684 -1641 -154MSD 0.0882 0.2040 0.1367 0.0521 0.0049

Note 1: The scale sensitivity parameter for the Cognitive Hierarchy models is set to infinity. The results reportedin Camerer, Ho & Chong(2001) presented at the Nobel Symposium 2001 are for models where is estimated.

Note 2: The Nash Equilibrium result is derived by allowing a non-zero mass of 0.0001 on non-equilibrium strategies.

Within-dataset Forecasting

Cross-dataset Forecasting

Nash versus CH Model: LL and MSD


Teck-Hua HoCH Model

Figure 2a: Predicted Frequencies of Cognitive Hierarchy Models

for Matrix Games (common )

y = 0.868x + 0.0499

R2 = 0.8203

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

CH Model: Theory vs. Data(Mixed Games)


Teck-Hua HoCH Model

Figure 3a: Predicted Frequencies of Nash Equilibrium for Matrix Games

y = 0.8273x + 0.0652

R2 = 0.3187

0

0.1

0.2

0.3

0.4

0.5

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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 (Mixed Games)


Teck-Hua HoCH Model

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

en

cy

CH Model: Theory vs. Data(Entry and Mixed Games)


Teck-Hua HoCH Model

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

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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 (Entry and Mixed Games)


Teck-Hua HoCH Model

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, would they have made a higher payoff?


Teck-Hua HoCH Model

Table 3: Economic Value for Cognitive Hierarchy and Nash Equilibrium

Stahl & Cooper & Costa-GomesData set Wilson (1995) Van Huyck et al. Mixed EntryTotal Payoff (% Improvement)

Actual Subject Choices 384 1169 530 328 118Ex-post Maximum 685 1322 615 708 176

79% 13% 16% 116% 49%Within-dataset EstimationCognitive Hierarchy (Game-specific ) 401 1277 573 471 128

4% 9% 8% 43% 8%Cognitive Hierarchy (Common ) 418 1277 573 471 128

9% 9% 8% 43% 8%

Cross-dataset EstimationCognitive Hierarchy (Common ) 418 1277 573 460 128

9% 9% 8% 40% 8%Nash Equilibrium 398 1230 556 274 112

4% 5% 5% -16% -5%

Note 1: The economic value is the total value (in USD) of all rounds that a "hypothetical" subject will earn using the respective modelto predict other's behavior and best responds with the strategy that yields the highest expected payoff in each round.

Nash versus CH Model: Economic Value


Teck-Hua HoCH Model

Example 3Example 3: 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$20


Teck-Hua HoCH Model

A Sample of Caltech Board of A Sample of Caltech Board of TrusteesTrustees

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


Teck-Hua HoCH Model

Results from Caltech Board of Results from Caltech Board of TrusteesTrustees

Caltech Board of TrusteesALL CEOs only

Mean 42.6 37.8Target 29.8 26.5Standard Deviation 23.4 18.9Sample Size 70 20


Teck-Hua HoCH Model

Results from Two Other Smart Subject Results from Two Other Smart Subject PoolsPools

Portfolio EconomicsManagers PhDs

Mean 24.3 27.4Target 17.0 19.2Standard Deviation 16.2 18.7Sample Size 26 16


Teck-Hua HoCH Model

Results from College StudentsResults from College Students

Caltech UCLA Wharton Germany Singapore

Mean 21.9 42.3 37.9 36.7 46.1Target 15.3 29.6 26.5 25.7 32.2Standard Deviation 10.4 18.0 18.8 20.2 28.0Sample Size 27 28 35 67 98


Teck-Hua HoCH Model

CH Model: Parameter EstimatesCH Model: Parameter EstimatesTable 1: Data and estimates of in pbc games(equilibrium = 0)

Steps ofsubjects/game Data CH Model Thinkinggame theorists 19.1 19.1 3.7Caltech 23.0 23.0 3.0newspaper 23.0 23.0 3.0portfolio mgrs 24.3 24.4 2.8econ PhD class 27.4 27.5 2.3Caltech g=3 21.5 21.5 1.8high school 32.5 32.7 1.61/2 mean 26.7 26.5 1.570 yr olds 37.0 36.9 1.1Germany 37.2 36.9 1.1CEOs 37.9 37.7 1.0game p=0.7 38.9 38.8 1.0Caltech g=2 21.7 22.2 0.8PCC g=3 47.5 47.5 0.1game p=0.9 49.4 49.5 0.1PCC g=2 54.2 49.5 0.0

mean 1.56median 1.30

Mean


Teck-Hua HoCH Model

SummarySummary

CH Model:

Discrete thinking steps

Frequency Poisson distributed

One-shot games

Fits better than Nash and adds more economic value

Explains “magic” of entry games

Sensible interpretation of mixed strategies

Can “solve” multiplicity problem

Initial conditions for learning

1 teck-hua ho ch model march – june, 2003 a cognitive hierarchy theory of one-shot games teck h....

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