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Transcript of 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
March – June, 20032
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.
March – June, 20033
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)
March – June, 20034
Teck-Hua HoCH Model
Modeling PrinciplesModeling Principles
Principle Nash Thinking
Strategic Thinking
Best Response
Mutual Consistency
March – June, 20035
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)
March – June, 20036
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
March – June, 20038
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
March – June, 20039
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
March – June, 200310
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)
March – June, 200311
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
March – June, 200312
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
March – June, 200313
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
March – June, 200314
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
March – June, 200315
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(
)(
March – June, 200316
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)
March – June, 200317
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
March – June, 200318
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
March – June, 200319
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
March – June, 200320
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
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Teck-Hua HoCH Model
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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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)
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Teck-Hua HoCH Model
Behaviors of Level 0 and 1Players(=1.25)
Level 0 + Level 1
% o
f E
nt r
y
Demand (as % of # of players)
March – June, 200324
Teck-Hua HoCH Model
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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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
Demand (as % of # of players)
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Teck-Hua HoCH Model
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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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
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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
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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
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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)
March – June, 200331
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
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)
March – June, 200332
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)
March – June, 200333
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
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)
March – June, 200334
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?
March – June, 200335
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
March – June, 200336
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
March – June, 200337
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
March – June, 200338
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
March – June, 200339
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
March – June, 200340
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
March – June, 200341
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
March – June, 200342
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