1 duke phd summer camp august 2007 outline motivation mutual consistency: ch model noisy...

August 2007 1Duke PhD Summer Camp

OutlineOutline

Motivation

Mutual Consistency: CH Model

Noisy Best-Response: QRE Model

Instant Convergence: EWA Learning


Standard Assumptions in Standard Assumptions in Equilibrium AnalysisEquilibrium Analysis

Assumptions Nash Cognitive QRE EWAEquilbirum Hierarchy Learning

Solution Method

Strategic Thinking X X X X

Best Response X X XX

Mutual Consistency X

Instant Convergence X X X


Example A: ExerciseExample A: Exercise

Consider matching pennies games 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


Bottom 40,80 80,40


Bottom 40,80 80,40

G1

G2


Example A: ExerciseExample A: Exercise

Consider matching pennies games in which the row player chooses between Top and Bottom and the column player simultaneously chooses between Left and Right, as shown below:


Bottom 40,80 80,40


Bottom 40,80 80,40


Bottom 40,80 80,40

G1

G2


Example A: DataExample A: Data


Example B: ExerciseExample B: Exercise

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.


Example B: ExerciseExample B: Exercise

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.


Example B: DataExample B: Data


Motivation: CHMotivation: CH

Model heterogeneity explicitly (people are not equally smart)

Introduce the word surprise into the game theory’s dictionary (e.g., Next movie)

Generate new predictions (reconcile various treatment effects in lab data not predicted by standard theory)

Camerer, Ho, and Chong (QJE, 2004)


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


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


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


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


The Cognitive Hierarchy (CH) ModelThe Cognitive Hierarchy (CH) Model

People 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.


Modeling Decision RuleModeling Decision Rule

Frequency 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


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


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


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


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(

)(


ImplicationsImplications

!)(

kekf

k A1 Poisson distribution with mean and variance =

A1,A2 Poisson, golden ratio Φ)


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


Existence and Uniqueness:Existence and Uniqueness:CH SolutionCH Solution

Existence: There is always a CH solution in any game

Uniqueness: It is always unique


Theoretical Properties of CH ModelTheoretical Properties of CH Model

Advantages 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


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


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 (data)Example 2: Entry games (data)


Behaviors of Level 0 and 1 Players ( =1.25)

Level 0

Level 1

% o

f E

nt r

y

Demand (as % of # of players)


Behaviors of Level 0 and 1 Players (=1.25)

Level 0 + Level 1

% o

f E

nt r

y



Behaviors of Level 2 Players(=1.25)

Level 2

Level 0 + Level 1

% o

f E

nt r

y



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



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


% e

ntr

y entry=demand

experimental data

CH Predictions in Entry Games CH Predictions in Entry Games (( = 1.25) = 1.25)


HomeworkHomework

What value of can help to explain the data in Example A?

How does CH model explain the data in Example B?


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


MLE EstimationMLE Estimation

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

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


Estimates of Mean Thinking Step Estimates of Mean Thinking 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


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


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


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)


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)


Nash vs. CH (Mixed Games)


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)


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)


CH vs. Nash (Entry and Mixed Games)


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


Nash versus CH Model: Economic Value


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


Results in various Results in various pp-BC games -BC games


Results in various Results in various pp-BC games -BC games

Subject Pool Group Size Sample Size Mean Error (Nash)

Error (CH)

CEOs 20 20 37.9 -37.9 -0.1 1.080 year olds 33 33 37.0 -37.0 -0.1 1.1Economics PhDs 16 16 27.4 -27.4 0.0 2.3Portfolio Managers 26 26 24.3 -24.3 0.1 2.8Game Theorists 27-54 136 19.1 -19.1 0.0 3.7


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


OutlineOutline

Motivation

Mutual Consistency: CH Model

Noisy Best-Response: QRE Model

Instant Convergence: EWA Learning

1 duke phd summer camp august 2007 outline motivation mutual consistency: ch model noisy...

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