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Behavioral game theory* Colin F. Camerer, Caltech camerer@hss.caltech.edu

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Behavioral game theory* Colin F. Camerer, Caltech camerer_at_hss.caltech.edu Behavioral game theory: How people actually play games Uses concepts from psychology and data – PowerPoint PPT presentation

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Title: Behavioral game theory* Colin F. Camerer, Caltech camerer@hss.caltech.edu


1
Behavioral game theory Colin F. Camerer, Caltech
camerer_at_hss.caltech.edu
  • Behavioral game theory
  • How people actually play games
  • Uses concepts from psychology and data
  • It is game theory Has formal, replicable
    concepts
  • Framing Mental representation
  • Feeling Social preferences
  • Thinking Cognitive hierarchy (?)
  • Learning Hybrid fEWA adaptive rule
  • Teaching Bounded rationality in repeated games
  • Behavioral Game Theory, Princeton Press 03 (550
    pp) Trends in Cog Sci, May 03 (10 pp)
    AmerEcRev, May 03 (5 pp) Science, 13 June 03 (2
    pp)

2
BGT modelling aesthetics
  • General (game theory)
  • Precise (game theory)
  • Progressive (behavioral econ)
  • Cognitively detailed (behavioral econ)
  • 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)

3
Thinking A one-parameter cognitive hierarchy
theory of one-shot games (with Teck Ho,
Berkeley Kuan Chong, NUSingapore)
  • Model of constrained strategic thinking
  • Model does several things
  • 1. Limited equilibration in some games (e.g.,
    pBC)
  • 2. Instant equilibration in some games (e.g.
    entry)
  • 3. De facto purification in mixed games
  • 4. Limited belief in noncredible threats
  • 5. Has economic value
  • 6. Can prove theorems
  • e.g. risk-dominance in 2x2 symmetric games
  • 7. Permits individual diffs relation to
    cognitive measures
  • Q J Econ August 04

4
Unbundling equilibrium
  • Principle Nash CH QRE
  • Strategic Thinking ? ? ?
  • Best Response ? ?
  • Mutual Consistency ? ?

5
The cognitive hierarchy (CH) model (I)
  • Selten (1998)
  • The natural way of looking at game situationsis
    not based on circular concepts, but rather on a
    step-by-step reasoning procedure
  • Discrete steps of thinking
  • Step 0s choose randomly (nonstrategically)
  • K-step thinkers know proportions f(0),...f(K-1)
  • Calculate what 0, K-1 step players will do
  • Choose best responses
  • Exhibits increasingly rational expectations
  • Normalized beliefs approximate f(n) as n? 8
  • i.e., highest level types are sophisticated/wor
    ldly and earn the most
  • Easy to calculate (see website calculator
    http//groups.haas.berkeley.edu/simulations/ch/def
    ault.asp)

6
The cognitive hierarchy (CH) model (II)
  • What is a reasonable simple f(K)?
  • A1 f(k)/f(k-1) ?1/k
  • ? Poisson f(k)e-ttk/k! mean, variance t
  • A2 f(1) is modal ? 1lt t lt 2
  • A3 f(1) is a maximal mode
  • or f(0)f(2) ? t?21.414..
  • A4 f(0)f(1)2f(2) ? t1.618 (golden ratio F)
  • Amount of working memory (digit span) correlated
    with steps of iterated deletion of dominated
    strategies (Devetag Warglien, 03 J Ec Psych)

7
Poisson distribution
  • Discrete, one parameter
  • (? spikes in data)
  • Steps gt 3 are rare (tight working memory bound)
  • Steps can be linked to cognitive measures

8
Limited equilibration Beauty contest game
  • N players choose numbers xi in 0,100
  • Compute target (2/3)(? xi /N)
  • Closest to target wins 20

9
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10
Estimates of ? in pBC games
11
2. Approximate equilibration in entry games
  • Entry games
  • N entrants, capacity c
  • Entrants earn 1 if n(entrants)ltc
  • earn 0 if n(entrants)gtc
  • Earn .50 by staying out
  • n(entrants) c in the 1st period
  • To a psychologist, it looks like magic--
    D. Kahneman 88
  • How? Pseudo-sequentiality of CH ?
    later-thinking entrants smooth the entry
    function

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15
3. Purification and partial equilibration in
mixed-equilibrium games (t1.62)
  • row step thinker choices
  • L R 0 1 2 3 4...
  • T 2,0 0,1 .5 1 1 0 0
  • B 0,1 1,0 .5 0 0 1 1
  • 0 .5 .5
  • 1 .5 .5
  • 2 0 1
  • 3 0 1
  • 4 0 1
  • 5 0 1

16
3. Purification and partial equilibration in
mixed-equilibrium games (t1.62)
  • row step thinker choices CH mixed
  • L R 0 1 2 3 4... predn equilm
    data
  • T 2,0 0,1 .5 1 1 0 0 .68
    .50 .72
  • B 0,1 1,0 .5 0 0 1 1 .32
    .50 .28
  • 0 .5 .5
  • 1 .5 .5
  • 2 0 1
  • 3 0 1
  • 4 0 1
  • 5 0 1
  • CH .26 .74
  • mixed .33 .67
  • data .33 .67

17
Estimates of t
  • game
  • Matrix games specific t common t
  • Stahl, Wilson (0, 6.5) 1.86
  • Cooper, Van Huyck (.5, 1.4) .80
  • Costa-Gomes et al (1, 2.3)
    1.69
  • Mixed-equil. games (.9,3.5) 1.48
  • Entry games --- .70
  • Signaling games (.3,1.2) ---
  • Fits consistently better than Nash, QRE
  • Unrestricted 6-parameter f(0),..f(6) fits only
    1 better

18
CH fixes errors in Nash predictions
19
4. Economic Value
  • Treat models like consultants
  • If players were to hire Mr. Nash and Mr.
    Camhocho as consultants and listen to their
    advice, would they have made a higher payoff?
  • If players are in equilibrium, Nash advice will
    have zero value
  • ?if theories have economic value, players are not
    in equilibrium
  • Advised strategy is what highest-level players
    choose
  • ? economic value is the payoff advantage of
    thinking harder
  • (selection pressure in replicator dynamics)

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21
6. Other theoretical properties of CH model
  • Advantages over Nash equilibrium
  • No multiplicity problem (picks one distribution)
  • No weird beliefs in games of incomplete info.
  • Theory
  • t?8 converges to Nash equilibrium in (weakly)
    dominance solvable games
  • Coincides with risk dominant equilibrium in
    symmetric 2x2 games
  • Close to Nash in 2x2 mixed games (t2.7 ? 82
    same-quadrant correspondence)
  • Equal splits in Nash demand games
  • Group size effects in stag hunt, beauty contest,
    centipede games

22
7. Preliminary findings on individual differences
response times
  • Caltech ? is .53 higher than PCC
  • Individual differences
  • Estimated ?i (1st half) correlates .64
  • with ?i (2nd half)
  • Upward drift in ?, .69 from 1st half to 2nd half
    of game (no-feedback learning ala Weber ExEc
    03?)
  • One step adds .85 secs to response time

23
Thinking Conclusions
  • Discrete thinking steps (mean t 1.5)
  • Predicts one-shot games initial conditions for
    learning
  • Accounts for limited convergence in
    dominance-solvable games and approximate
    convergence in mixed entry games
  • Advantages
  • More precise than Nash Can solve
    multiplicity problem
  • Has economic value
  • Can be tied to cognitive measures
  • Important! This is game theory
  • It is a formal specification which makes
    predictions

24
Feeling in ultimatum games How much do you offer
out of 10?
  • Proposer has 10
  • Offers x to Responder (keeps 10-x)
  • What should the Responder do?
  • Self-interest Take any xgt0
  • Empirical Reject x2 half the time
  • What are the Responders thinking?
  • Look inside their brains

25
Feeling This is your brain on unfairness (Sanfey
et al, Sci 13 March 03)
26
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27
Ultimatum offers of children who failed/passed
false belief test
28
Israeli subject (autistic?) complaining
post-experiment (Zamir, 2000)
29
Ultimatum offer experimental sites
30
The Machiguenga independent families cash
cropping
slash burn gathered foods fishing hunting
31
African pastoralists (Orma in Kenya)
32
Whale Hunters of Lamalera, Indonesia
High levels of cooperation among hunters of
whales, sharks, dolphins and rays. Protein for
carbs, trade with inlanders. Carefully regulated
division of whale meat
Researcher Mike Alvard
33
Fair offers correlate with market integration
(top), cooperativeness in everyday life (bottom)
34
New frontiers
  • Field applications!
  • Imitation learning
  • Trifurcation
  • Rational gt Firms, expert players, long-run
    outcomes
  • Behavioral gt Normal people, new games
  • Evolutionary gt Animals, humans imitating

35
Conclusions
  • Thinking CH model (? mean number of steps)
  • ? is similar (1.5) in many games Explains
    limited and surprising equilibration
  • Easy to use empirically do theory
  • Feeling
  • Ultimatum rejections are common, vary across
    culture
  • fairness correlated with market integration
    (cf. Adam Smith)
  • Unfair offers activate insula, ACC, DLPFC
  • U-shaped rejections common
  • Dictators offer less when threatened with
    3rd-party punishment
  • Pedagogy A radical new way to teach game theory
  • Start with concept of a game.
  • Building blocks Mixing, dominance, foresight.
  • Then teach cognitive hierarchy, learning
  • end with equilibrium!

36
Potential applications
  • Thinking
  • price bubbles, speculation, competition neglect
  • Learning
  • evolution of institutions, new industries
  • Neo-Keynesian macroeconomic coordination
  • bidding, consumer choice
  • Teaching
  • contracting, collusion, inflation policy

37
Framing How are games represented?
  • Invisible assumption
  • People represent games in matrix/tree form
  • Mental representations may be simplified
  • analogies Iraq war is Afghanistan, not
    Vietnam
  • shrinking-pie bargaining
  • or enriched
  • Schelling matching games
  • timing virtual observability

38
Framing enrichment Timing virtual
observability
  • Battle-of-sexes
  • row 1st
    unobserved
  • B G simul seql seql
  • B 0,0 1,3 .38 .10 .20
  • G 3,1 0,0 .62 .90 .80
  • Simul. .62 .38
  • Seql .80 .20
  • Unobs. .70 .30

39
Potential economic applications
  • Price bubbles
  • thinking steps correspond to timing of selling
    before a crash
  • Speculation
  • Violates Groucho Marx no-bet theorem
  • A B C D
  • I info (A,B) (C,D)
  • I payoffs 32 -28 20 -16
  • II info A (B,C) D
  • II payoffs -32 28 -20 16
  • Milgrom-Stokey 82 Eca Sonsino, Erev, Gilat,
    unpubd Sovik, unpubd

40
Potential economic applications (contd)
  • A B C D
  • I info (A,B) (C,D)
  • data .77 .53
  • CH (?1.5) .46 .89
  • I payoffs 32 -28 20 -16
  • II info A (B,C) D
  • data .00 .83 1.00
  • CH (?1.5) .12 .72 .89
  • II payoffs -32 28 -20 16

41
Potential economic applications (contd)
  • Prediction Betting in (C,D) and (B,C) drops when
    one number is changed
  • A B C D
  • I info (A,B) (C,D)
  • data ? ?
  • CH (?1.5) .46 .46
  • I payoffs 32 -28 32 -16
  • II info A (B,C) D
  • data ? ? ?
  • CH (?1.5) .12 .12 .89
  • II payoffs -32 28 -32 16

42
The cognitive hierarchy (CH) model (II)
  • Two separate features
  • Not imagining k1 types
  • Not believing there are other k types
  • Overconfidence
  • K-steps think others are all one step lower
    (K-1)
  • (Nagel-Stahl-CCGB)
  • Increasingly irrational expectations as K? 8
  • Has some odd properties (cycles in entry
    games)
  • Self-conscious
  • K-steps believe there are other K-step thinkers
  • Too similar to quantal response
    equilibrium/Nash
  • ( fits worse)

43
Framing Limited planning in bargaining (JEcThry
02 Science, 03)
44
Learning fEWA
  • Attraction A ij (t) for strategy j updated by
  • A ij (t) (?A ij (t-1) ?(actual))/ (?(1-?)1)
    (chosen j)
  • A ij (t) (?A ij (t-1) ? ? (foregone))/ (?(1-
    ? )1) (unchosen j)
  • logit response function Pij(t)exp(?A ij
    (t)/Skexp(?A ik (t)
  • key parameters
  • ? imagination, ? decay/change-detection
  • In nature a hybrid species is usually sterile,
    but in science the opposite is often true--
    Francis Crick 88
  • Special cases
  • Weighted fictitious play (?1, ?0)
  • Choice reinforcement (?0)
  • EWA estimates parameters ?, ?, ? (Cam.-Ho 99
    Eca)
  • Or divide by payoff variability (Erev et al 99
    JEBO) automatically explores when environment
    changes

45
Functional fEWA
  • Substitute functions for parameters
  • Easy to estimate (only ?)
  • Tracks parameter differences across games
  • Allows change within a game
  • Change detector for decay rate f
  • f(i,t)1-.5?k ( S-ik (t) - ??1t S-ik(?)/t ) 2
  • f close to 1 when stable, dips to 0 when
    unstable

46
Example Price matching with loyalty rewards
(Capra, Goeree, Gomez, Holt AER 99)
  • Players 1, 2 pick prices 80,200
  • Price is Pmin(P1,,P2)
  • Low price firm earns PR
  • High price firm earns P-R
  • What happens? (e.g., R50)

47
Ultimatum offers across societies (mean shaded,
mode is largest circle)
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50
A decade of empirical studies of learning Taking
stock
  • Early studies show models can track basic
    features of learning paths
  • McAllister, 91 Annals OR Cheung-Friedman 94
    GEB Roth-Erev 95 GEB,98 AER
  • Is one model generally better? Horse races
  • Speeds up process of single-model exploration
  • Fair tests Common games empirical methods
  • match races in horse racing Champions forced
    to compete
  • Development of hybrids which are robust (improve
    on failures of specific models)
  • EWA (Camerer-Ho 99, Anderson-Camerer 00 Ec Thy)
  • fEWA (Camerer-Ho, 0?)
  • Rule learning (Stahl, 01 GEB)

51
5. Automatic reduction of belief in noncredible
threats (subgame perfection)
  • row level
  • 0 1 2 3
  • T 4,4 .5 1 0 0
  • L R
  • B 6,3 0,1 .5 0 1 1
  • (T,R) Nash, (B,L) subgame perfect
  • CH Prediction (?1.5)
  • 89 play L
  • 56 play B
  • ? (Level 1) players do not have enough faith in
    rationality of others
  • (Beard Beil, 90 Mgt Sci Weiszacker 03 GEB)
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