April 4, 20041

Teck-Hua HoCH Model

OutlineOutline

In-class Experiment and Examples

Empirical Alternative I: Model of Cognitive Hierarchy (Camerer, Ho, and Chong, QJE, 2004)

Empirical Alternative II: Quantal Response Equilibrium (McKelvey and Palfrey, GEB, 1995)

Empirical Alternative III: Model of Noisy Introspection (Goeree and Holt, AER, 2001)

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Teck-Hua HoCH Model

Example 1:Example 1:

Consider the game in which two players independently and simultaneously choose integer numbers between and including 180 and 300. Both players are paid the lower of the two numbers, and, in addition, an amount R > 1 is transferred from the player with the higher number to the player with the lower number.

R = 5 versus R = 180

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Teck-Hua HoCH Model

Example 1:Example 1:

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Teck-Hua HoCH Model

Example 2:Example 2: Consider a symmetric matching pennies game in which the

row player chooses between Top and Bottom and the column player simultaneously chooses between Left and Right, as shown below:

Left RightTop 80,40 40,80

Bottom 40,80 80,40

Left RightTop 320,40 40,80

Bottom 40,80 80,40

Left RightTop 44,40 40,80

Bottom 40,80 80,40

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Teck-Hua HoCH Model

Example 2:Example 2:

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Example 3:Example 3:The two players choose “effort” levels

simultaneously, and the payoff of each player is given by i = min (e1, e2) – c x ei

Efforts are integer from 110 to 170.

C = 0.1 or 0.9.

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Teck-Hua HoCH Model

Example 3:Example 3:

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Teck-Hua HoCH Model

MotivationMotivation

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

Subjects in experiments do not play Nash in most one-shot games and in some repeated games (i.e., behavior do not converge to Nash).

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

Modeling heterogeneity really matters in games.

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Teck-Hua HoCH Model

First-Shot GamesFirst-Shot Games

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

Many IO models and simple game experiments

Initial conditions for learning

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Teck-Hua HoCH Model

Behavioral Game TheoryBehavioral Game Theory

How to model bounded rationality (one-shot game)? Cognitive Hierarchy (CH) model (Camerer, Ho,

and Chong, QJE, 2004)

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, JET,

2002)

NextWeek

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Teck-Hua HoCH Model

Modeling PrinciplesModeling Principles

Principle Nash CH QRE NI

Strategic Thinking

Best Response

Mutual Consistency

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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)

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Teck-Hua HoCH Model

OutlineOutline

In-class Experiment and Examples

Empirical Alternative I: Model of Cognitive Hierarchy (Camerer, Ho, and Chong, QJE, 2004)

Empirical Alternative II: Quantal Response Equilibrium (McKelvey and Palfrey, GEB, 1995)

Empirical Alternative III: Model of Noisy Introspection (Goeree and Holt, AER, 2001)

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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

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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

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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

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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

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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)

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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

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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

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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

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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)

A4: f(0) + f(1) = 2f(2)

kkf

kf

kkf

kf

)1(

)(1

)1(

)(

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ImplicationsImplications

!)(

kekf

k A1 Poisson distribution with mean and variance =

A1,A2 Poisson distribution, 1<

A1,A3 Poisson,

) Poisson, golden ratio Φ)

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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

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

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

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

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

ROW COLLevel Frequency T M B L C R

0 0.212 0.333 0.333 0.333 0.333 0.333 0.3331 0.329 0 1 0 1 0 02 0.255 0 0 1 1 0 03 0.132 0 0 1 1 0 04 0.051 0 0 1 1 0 05 0.016 0 0 1 1 0 06 0.004 0 0 1 1 0 0

AGGREGATE 1.000 0.07 0.40 0.53 0.86 0.07 0.07

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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

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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

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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

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

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

Demand (as % of number of players )

% e

ntr

y

entry=demand

experimental data

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

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Behaviors of Level 0 and 1 Players (=1.25)

Level 0

Level 1

% o

f E

nt r

y

Demand (as % of # of players)

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Behaviors of Level 0 and 1Players(=1.25)

Level 0 + Level 1

% o

f E

nt r

y

Demand (as % of # of players)

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Behaviors of Level 2 Players(=1.25)

Level 2

Level 0 + Level 1

% o

f E

nt r

y

Demand (as % of # of players)

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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

Demand (as % of # of players)

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

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

Demand (as % of # of players)

% e

ntr

y entry=demand

experimental data

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

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

COLUMN FrequencyL C R

T 0,0 10,-10 -5,5 0.125

ROW M -15,15 15,-15 25,-25 0.333

B 5,-5 -10,10 0,0 0.542Empirical

Frequency 0.875 0.083 0.042

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MLE EstimationMLE Estimation Count LabelT 13 N1M 33 N2B 54 N3L 88 M1C 8 M2R 4 M3

321321 ))()(1()()())()(1()()()( 21212121MMMNNN qqqqppppLL

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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

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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

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

Note 1: 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

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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)

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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

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 (Mixed Games)

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Nash vs CH (Mixed Games)

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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

en

cy

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

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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 (Entry and Mixed Games)

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Teck-Hua HoCH Model

CH vs. Nash (Entry and Mixed Games)

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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 (i.e., use the model to forecast what others will do and best-respond), would they have made a higher payoff?

A measure of disequilibrium

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Teck-Hua HoCH Model

Nash versus CH Model: Economic Value

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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$10

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Teck-Hua HoCH Model

CH Model: Parameter EstimatesCH Model: Parameter Estimates

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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