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					A Cognitive Hierarchy Theory of
       One-Shot Games
                   Teck H. Ho
           Haas School of Business
       University of California, Berkeley



        Joint work with Colin Camerer, Caltech
               Juin-Kuan Chong, NUS




                      CH Model                   Teck-Hua Ho
                                                               1
Motivation
q Nash equilibrium and its refinements: Dominant theories
  in economics for predicting behaviors in games.


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


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

q Modeling heterogeneity really matters in games.


                        CH Model                    Teck-Hua Ho
                                                                  2
Research Goals
qHow to model bounded rationality (first-period
  behavior)?
  qCognitive Hierarchy (CH) model
qHow to model equilibration?
  qEWA learning model (Camerer and Ho,
   Econometrica, 1999; Ho, Camerer, and Chong,
   2003)
qHow to model repeated game behavior?
  qTeaching model (Camerer, Ho, and Chong,
   Journal of Economic Theory, 2002)
                    CH Model                 Teck-Hua Ho
                                                           3
Modeling Principles
Principle                Nash   Thinking


Strategic Thinking       í      í

Best Response            í      í


Mutual Consistency       í




                     CH Model       Teck-Hua Ho
                                                  4
Modeling 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)
                               CH Model                             Teck-Hua Ho
                                                                                  5
Example 1: “zero-sum game”




 Messick(1965), Behavioral Science


                                CH Model   Teck-Hua Ho
                                                         6
Nash Prediction:
“zero-sum game”




           CH Model   Teck-Hua Ho
                                    7
CH Prediction:
“zero-sum game”




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


                                 CH Model          Teck-Hua Ho
                                                                 8
Empirical Frequency:
“zero-sum game”




           CH Model    Teck-Hua Ho
                                     9
The Cognitive Hierarchy (CH)
Model
qPeople are different and have different decision rules

qModeling heterogeneity (i.e., distribution of types of
 players)


qModeling decision rule of each type

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


                        CH Model                     Teck-Hua Ho
                                                                   10
Modeling Decision Rule
q f(0) step 0 choose randomly



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



q Normalize                      and best-respond




                           CH Model                      Teck-Hua Ho
                                                                       11
Example 1: “zero-sum game”




           CH Model    Teck-Hua Ho
                                     12
Implications
qExhibits “increasingly rational expectations”
   q Normalized g(h) approximates f(h) more closely
    as kà ∞ (i.e., highest level types are
    “sophisticated” (or ”worldly) and earn the most


qHighest level type actions converge as kà ∞
   à marginal benefit of thinking harder à0




                      CH Model                   Teck-Hua Ho
                                                               13
Alternative Specifications
qOverconfidence:
   qk-steps think others are all one step lower (k-1) (Stahl, GEB,
   1995; Nagel, AER, 1995; Ho, Camerer and Weigelt, AER, 1998)
   q“Increasingly irrational expectations” as Kà ∞
   qHas some odd properties (e.g., cycles in entry games)


qSelf-conscious:
   qk-steps think there are other k-step thinkers
   qSimilar to Quantal Response Equilibrium/Nash
   qFits worse



                           CH Model                         Teck-Hua Ho
                                                                          14
Modeling Heterogeneity, f(k)

q A1:


    q sharp drop-off due to increasing working memory constraint



q A2: f(1) is the mode

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

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

                            CH Model                       Teck-Hua Ho
                                                                         15
Implications

 q A1à Poisson distribution               with mean
  and variance = t


 qA1,A2à Poisson distribution, 1< t < 2


 qA1,A3 à Poisson, t=Ö2=1.414..


 q(A1,A4a,A4b) à Poisson, t=1.618..(golden ratio Φ)



                      CH Model                  Teck-Hua Ho
                                                              16
Poisson Distribution

q f(k) with mean step of thinking t:




                          CH Model     Teck-Hua Ho
                                                     17
 Historical Roots
q “Fictitious play” as an algorithm for computing Nash
  equilibrium (Brown, 1951; Robinson, 1951)


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


q 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


                          CH Model                      Teck-Hua Ho
                                                                      18
Theoretical Properties of
CH Model
qAdvantages over Nash equilibrium
  qCan “solve” multiplicity problem (picks one statistical
   distribution)
  qSolves refinement problems (all moves occur in
   equilibrium)
  qSensible interpretation of mixed strategies (de facto
   purification)
qTheory:
  qτà∞ converges to Nash equilibrium in (weakly)
   dominance solvable games
  qEqual splits in Nash demand games


                       CH Model                    Teck-Hua Ho
                                                                 19
 Example 2: Entry games
q 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
q Experimental regularity in the 1st period:
   q Consistent with Nash prediction, %(entrants) increases with D
   q “To a psychologist, it looks like magic”-- D. Kahneman ‘88




                             CH Model                         Teck-Hua Ho
                                                                            20
Example 2: Entry games
(data)




          CH Model       Teck-Hua Ho
                                       21
Behaviors of Level 0 and 1
Players (t =1.25)
                                              Level 1
 % of Entry




                                               Level 0




              Demand (as % of # of players)

                            CH Model                     Teck-Hua Ho
                                                                       22
Behaviors of Level 0 and 1
Players(t =1.25)

                                              Level 0 + Level 1
 % of Entry




              Demand (as % of # of players)

                            CH Model                      Teck-Hua Ho
                                                                        23
Behaviors of Level 2 Players
(t =1.25)
                                              Level 2


                                              Level 0 + Level 1
 % of Entry




              Demand (as % of # of players)

                            CH Model                     Teck-Hua Ho
                                                                       24
Behaviors of Level 0, 1, and
2 Players(t =1.25)
                                                 Level 2
                                                 Level 0 + Level 1 +
                                                 Level 2
    % of Entry




Level 0 +
Level 1



                 Demand (as % of # of players)

                               CH Model                      Teck-Hua Ho
                                                                           25
Entry Games (Imposing
Monotonicity on CH Model)




           CH Model    Teck-Hua Ho
                                     26
Estimates of Mean Thinking
Step t




           CH Model     Teck-Hua Ho
                                      27
CH Model: CI of Parameter
Estimates




           CH Model     Teck-Hua Ho
                                      28
Nash versus CH Model:
LL and MSD




          CH Model      Teck-Hua Ho
                                      29
CH Model: Theory vs. Data
(Mixed Games)




           CH Model    Teck-Hua Ho
                                     30
Nash: Theory vs. Data
(Mixed Games)




           CH Model     Teck-Hua Ho
                                      31
CH Model: Theory vs. Data
(Entry and Mixed Games)




          CH Model     Teck-Hua Ho
                                     32
Nash: Theory vs. Data
(Entry and Mixed Games)




          CH Model        Teck-Hua Ho
                                        33
Economic Value
q Evaluate models based on their value-added rather than statistical
  fit (Camerer and Ho, 2000)



q Treat models like consultants



q If players were to hire Mr. Nash and Ms. CH as consultants and
  listen to their advice, would they have made a higher payoff?




                            CH Model                         Teck-Hua Ho
                                                                           34
Nash versus CH Model:
Economic Value




          CH Model      Teck-Hua Ho
                                      35
Example 3: P-Beauty
Contest
q n players
q Every player simultaneously chooses a number from 0
    to 100
q Compute the group average
q Define Target Number to be 0.7 times the group
    average
q The winner is the player whose number is the closet to
    the Target Number
q The prize to the winner is US$20
                        CH Model                 Teck-Hua Ho
                                                               36
A Sample of Caltech Board
of Trustees
§ David Baltimore                • David D. Ho
  President                       Director
  California Institute of         The Aaron Diamond AIDS Research Center
  Technology
§ Donald L. Bren                 • Gordon E. Moore
                                  Chairman Emeritus
  Chairman of the Board           Intel Corporation
  The Irvine Company
                                 • Stephen A. Ross
• Eli Broad                       Co-Chairman, Roll and Ross Asset Mgt Corp
  Chairman
  SunAmerica Inc.
• Lounette M. Dyer               • Sally K. Ride
  Chairman                         President Imaginary Lines, Inc., and
  Silk Route Technology            Hibben Professor of Physics



                            CH Model                             Teck-Hua Ho
                                                                               37
Results from Caltech Board
of Trustees




          CH Model     Teck-Hua Ho
                                     38
Results from Two Other Smart
Subject Pools




             CH Model          Teck-Hua Ho
                                             39
Results from College
Students




            CH Model   Teck-Hua Ho
                                     40
CH Model: Parameter
Estimates




          CH Model    Teck-Hua Ho
                                    41
Summary
q CH Model:
   qDiscrete thinking steps
   qFrequency Poisson distributed
q One-shot games
   qFits better than Nash and adds more economic value
   qExplains “magic” of entry games
   qSensible interpretation of mixed strategies
   qCan “solve” multiplicity problem
q Initial conditions for learning

                         CH Model                 Teck-Hua Ho
                                                                42

				
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posted:9/26/2013
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