1. Behavioral game theory* Colin F. Camerer, Caltech
BGT: How people actually play games
Framing: Mental representation
Feeling: Social preferences (Fehr et al)
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)
CEEL Summer Camp, Trento 07/07/03

2. Framing: Limited planning in bargaining (Science, 03)
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Thinking: A one-parameter cognitive hierarchy theory of one-shot games (with Teck Ho, Berkeley; Kuan Chong, N Univ Singapore)
Model of constrained strategic thinking
Model does several things:
1. Limited equilibration in some games (e.g., pBC)
2. Surprisingly fast 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 diff’s & relation to cognitive measures
CEEL Summer Camp, Trento 07/07/03

4. Unbundling equilibrium
Principle Nash CH QRE
Strategic Thinking   
Best Response  
Mutual Consistency  
CEEL Summer Camp, Trento 07/07/03

5. The cognitive hierarchy (CH) model (I)
Discrete steps of thinking:
Step 0’s choose randomly (nonstrategically)
K-step thinkers know proportions f(0),...f(K-1)
Calculate what 0, …K-1 step players will do
Normalize beliefs g(n)=f(n)/ h=0K-1 f(h).
Calculate expected payoffs and best-respond
Exhibits “increasingly rational expectations”:
Normalized g(n) approximates f(n) more closely as n ∞
i.e., highest level types are “sophisticated”/”worldly and earn the most
Also: highest level type actions converge as K ∞
( marginal benefit of thinking harder 0)
CEEL Summer Camp, Trento 07/07/03

6. The cognitive hierarchy (CH) model (II)
Two separate features:
Not imagining k+1 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 ∞
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)
CEEL Summer Camp, Trento 07/07/03

7. The cognitive hierarchy (CH) model (III)
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  1< t < 2
A3: f(1) is a ‘maximal’ mode
or f(0)=f(2)  t=2=
A4: f(0)+f(1)=2f(2)  t=1.618 (golden ratio Φ)
*Amount of working memory (digit span) correlated with steps of iterated deletion of dominated strategies (Devetag & Warglien, 03 J Ec Psych)
CEEL Summer Camp, Trento 07/07/03

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Poisson distribution
Discrete, one parameter
( “spikes” in data)
Steps > 3 are rare (working memory bound)
Steps can be linked to cognitive measures
CEEL Summer Camp, Trento 07/07/03

9. Limited equilibration Beauty contest game
N players choose numbers xi in [0,100]
Compute target (2/3)*( xi /N)
Closest to target wins $20
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1. Limited equilibration in p-BC: Pick [0,100]; closest to (2/3)*(average) wins
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12. Estimates of  in pBC games
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13. pBC estimation: Gory details
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14. 2. Approximate equilibration in entry games
N entrants, capacity c
Entrants earn $1 if n(entrants)<c;
earn 0 if n(entrants)>c
Earn $.50 by staying out
All choose simultaneously
Close to equilibrium in the 1st period:
Close to equilibrium prediction n(entrants) ≈ c
“To a psychologist, it looks like magic”-- D. Kahneman ’88
How? Pseudo-sequentiality of CH  “later” entrants smooth the entry function
CEEL Summer Camp, Trento 07/07/03

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17. 3. De facto purification in mixed-equilibrium games (t=1.62)
row step thinker choices CH mixed
L R pred’n equilm data
T 2,0 0,
B 0,1 1,
2 0 1
3 0 1
4 0 1
5 0 1 CH
mixed
data
CEEL Summer Camp, Trento 07/07/03

18. “typical” example 1: (median in LL(CH)/LL(Nash)
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“Typical” example 2
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5. Automatic reduction of belief in noncredible threats (subgame perfection)
row level
T ,
L R
B 6,3 0,
(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)
CEEL Summer Camp, Trento 07/07/03

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Estimates of τ
game
Matrix games specific τ common τ
Stahl, Wilson (0, 6.5)
Cooper, Van Huyck (.5, 1.4)
Costa-Gomes et al (1, 2.3)
Mixed-equil. games (.9,3.5)
Entry games
Signaling games (.3,1.2)
Fits consistently better than Nash, QRE
Unrestricted 6-parameter f(0),..f(6) fits only 1% better
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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
CEEL Summer Camp, Trento 07/07/03

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25. 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:
τ∞ 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 (τ=2.7  82% same-quadrant correspondence)
Equal splits in Nash demand games
Group size effects in stag hunt, beauty contest, centipede games
CEEL Summer Camp, Trento 07/07/03

26. 7. Preliminary findings on individual differences & response times
Caltech  is .53 higher than PCC
Persistent individual differences:
Estimated i (1st half) correlates .64 with i (2nd half)
Upward drift in estimated , .69 from 1st games to 2nd 11 games (no-feedback “learning”?)
One step adds .85 secs to response time
CEEL Summer Camp, Trento 07/07/03

27. 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)
CEEL Summer Camp, Trento 07/07/03

28. Thinking: Conclusions
Discrete thinking steps, frequency Poisson distributed
(mean number of steps τ ≈ 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
It is not ad hoc…
…quite the opposite
Definition of “ad hoc”: “temporary, for the purpose at hand, as in an ad hoc committee”
CEEL Summer Camp, Trento 07/07/03

29. 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
II info A (B,C) D
II payoffs
*Milgrom-Stokey ’82 Ec’a; Sonsino, Erev, Gilat, unpub’d; Sovik, unpub’d
CEEL Summer Camp, Trento 07/07/03

30. Potential economic applications (cont’d)
A B C D
I info (A,B) (C,D)
data
CH (=1.5)
I payoffs
II info A (B,C) D
data
CH (=1.5)
II payoffs
CEEL Summer Camp, Trento 07/07/03

31. Potential economic applications (cont’d)
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)
I payoffs
II info A (B,C) D
data ? ? ?
CH (=1.5)
II payoffs
CEEL Summer Camp, Trento 07/07/03

32. 3. De facto purification in mixed games: Battle of the sexes (t=1.62)
row step thinker choices CH mixed
L R pred’n equilm data
T 0,0 1,
B 3,1 0,
1 0 1
2 1 0
3 0 1
4 0 1
5 0 1 CH
mixed
data
CEEL Summer Camp, Trento 07/07/03

33. CEEL Summer Camp, Trento 07/07/03
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)/[Σkexp(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 Ec’a)
*Or divide by payoff variability (Erev et al ’99 JEBO); automatically “explores” when environment changes
CEEL Summer Camp, Trento 07/07/03

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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 φ
φ(i,t)=1-.5[k ( S-ik (t) - =1t S-ik()/t ) 2 ]
φ close to 1 when stable, dips to 0 when unstable
(i,t)=φ(i,t)/W (W=support of Nash equil’m)
CEEL Summer Camp, Trento 07/07/03

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Example: Price matching with loyalty rewards (Capra, Goeree, Gomez, Holt AER ‘99)
Players 1, 2 pick prices [80,200] ¢
Price is P=min(P1,,P2)
Low price firm earns P+R
High price firm earns P-R
What happens? (e.g., R=50)
CEEL Summer Camp, Trento 07/07/03

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38. 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)
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New frontiers
Field applications!
Imitation learning
Experimetrics (Wilcox, ’03 big downward bias in  due to heterogeneity)
No-feedback “learning”
Trifurcation:
Rational gt: Firms, expert players, long-run outcomes
Behavioral gt: Normal people, new games
Evolutionary gt: Animals, humans imitating
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Conclusions
Framing: Much to do…need help from cognitive science
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
challenge: endogenize number of steps (a la Gabaix-Laibson)
Learning ( response sensitivity)
Hybrid fits & predicts well (20+ games)
One-parameter fEWA fits well, easy to estimate
Teaching ( fraction of teaching)
Retains strategic foresight in repeated games with matching
Fits trust, entry deterrence better than QRE
Pedagogy: A radical new way teach game theory
Start with concept of a game. Mixing, dominance
Then teach cognitive hierarchy, learning…
end with equilibrium!
CEEL Summer Camp, Trento 07/07/03

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42. Behavior in “continental divide game”
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45. Simulations vs data: fEWA
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46. 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
CEEL Summer Camp, Trento 07/07/03

47. 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”
CEEL Summer Camp, Trento 07/07/03

48. Framing enrichment: Timing & virtual observability
Battle-of-sexes
row 1st unobserved
B G simul seq’l seq’l
B 0,0 1,
G 3,1 0,
Simul
Seq’l
Unobs
CEEL Summer Camp, Trento 07/07/03

49. Equilibrium and CH vs data in matrix games
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53. Cognitive hierarchy: Reprise
Model does several things:
1. Limited equilibration in some games (e.g., pBC)
2. Approximate 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 diff’s & relation to cognitive measures
CEEL Summer Camp, Trento 07/07/03

54. CEEL Summer Camp, Trento 07/07/03
Feeling: This is your brain on unfairness (Sanfey et al, Sci 13 March ’03)
CEEL Summer Camp, Trento 07/07/03