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Behavioral game theory*
Colin F. Camerer, Caltech
[email protected]
Behavioral game theory:
– How people actually play games
– Uses concepts from psychology and data
– It is game theory: Has formal, replicable concepts
Framing:
Feeling:
Thinking:
Learning:
Teaching:
Mental representation
Social preferences
Cognitive hierarchy ()
Hybrid fEWA adaptive rule
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)
BGT modelling aesthetics
General
Precise
Progressive
Cognitively detailed
Empirically disciplined
(game theory)
(game theory)
(behavioral econ)
(behavioral econ)
(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)
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 diff’s & relation to cognitive measures
– *Q J Econ August ‘04
Unbundling equilibrium
Principle
Nash
Strategic Thinking

Best Response

Mutual Consistency 
CH
QRE




The cognitive hierarchy (CH) model (I)
Selten (1998):
– “The natural way of looking at game situations…is not based on circular
concepts, but rather on a step-by-step reasoning procedure”
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
Choose best responses
Exhibits “increasingly rational expectations”:
– Normalized beliefs approximate f(n) as n ∞
i.e., highest level types are “sophisticated”/”worldly and earn the most
Easy to calculate (see website “calculator”
http://groups.haas.berkeley.edu/simulations/ch/default.asp)
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-k/k!
mean, variance 
– A2: f(1) is modal
 1<  < 2
– A3: f(1) is a ‘maximal’ mode
or f(0)=f(2)
 t=2=1.414..
– 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)
Poisson distribution
Discrete, one parameter
– ( “spikes” in data)
Steps > 3 are rare (tight working memory bound)
Steps can be linked to cognitive measures
frequency
Poisson distributions for
various 
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
=1
=1.5
=2
0
1
2
3
4
number of steps
5
6
1. Limited equilibration
Beauty contest game
N players choose numbers xi in
[0,100]
Compute target (2/3)*( xi /N)
Closest to target wins $20
relative
frequencies
Beauty contest results (Expansion,
Financial Times, Spektrum)
average 23.07
0.20
0.15
0.10
0.05
0.00
numbers
22
100
50
33
num be r choice s
97
89
81
73
65
57
49
41
33
25
17
9
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
1
predicted frequency
0
Estimates of  in pBC games
Table 1: Data and estimates of  in pbc games
(equilibrium = 0)
Mean
Steps of
subjects/game
Data
CH Model
Thinking
game theorists
19.1
19.1
3.7
Caltech
23.0
23.0
3.0
newspaper
23.0
23.0
3.0
portfolio mgrs
24.3
24.4
2.8
econ PhD class
27.4
27.5
2.3
Caltech g=3
21.5
21.5
1.8
high school
32.5
32.7
1.6
1/2 mean
26.7
26.5
1.5
70 yr olds
37.0
36.9
1.1
Germany
37.2
36.9
1.1
CEOs
37.9
37.7
1.0
game p=0.7
38.9
38.8
1.0
Caltech g=2
21.7
22.2
0.8
PCC g=3
47.5
47.5
0.1
game p=0.9
49.4
49.5
0.1
PCC g=2
54.2
49.5
0.0
mean
1.56
median
1.30
2. Approximate equilibration in entry games
Entry games:
N entrants, capacity c
Entrants earn $1 if n(entrants)<c;
earn 0 if n(entrants)>c
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
frequency
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
total entry
Nash equilibrium
CH fit (tau=1.5)
2
4
6
8
capacity (out of 12)
10
0-Step and 1-Step Entry
100
90
Percentage Entry
80
70
60
Capacity
Capacity
0-Level
0+1 Level
1-Level
50
40
30
20
10
0
1
11
21
21
31
31
41
41
51
51
61
61
Percentage
Percentage Capacity
Capacity
71
71
81
81
91
91
101
101
0-Step
+ 1-Step
+ 2 Step
Entry
0-Step
and 1-Step
Entry
100
90
Percentage Entry
Entry
Percentage
80
70
60
60
50
50
` ``
40
40
30
30
20
20
10
10
0
0
0
0
10
10
10
20
20
20
30
30
30
40
40
50
100
40 50
50 6060
60 7070
70 8080
80 9090
90 100
100
Percentage
Percentage
Capacity
Capacity
Percentage
Capacity
Capacity
Capacity
Capacity
0+1
Level
0+1+2
Level
0+1
Level
2-Level
3. Purification and partial equilibration in
mixed-equilibrium games (=1.62)
row step thinker choices
T
B
0
1
2
3
4
5
L
2,0
0,1
.5
.5
0
0
0
0
R
0,1
1,0
.5
.5
1
1
1
1
0
.5
.5
1
1
0
2
1
0
3 4...
0 0
1 1
3. Purification and partial equilibration in
mixed-equilibrium games (=1.62)
row step thinker choices
L
T
2,0
B
0,1
0
.5
1
.5
2
0
3
0
4
0
5
0
CH
.26
mixed .33
data
.33
R
0,1
1,0
.5
.5
1
1
1
1
.74
.67
.67
0
.5
.5
1
1
0
2
1
0
3 4...
0 0
1 1
CH
mixed
pred’n equilm
.68 .50
.32 .50
data
.72
.28
Estimates of τ
game
Matrix games
Stahl, Wilson
Cooper, Van Huyck
Costa-Gomes et al
Mixed-equil. games
Entry games
Signaling games
specific τ
common τ
(0, 6.5) 1.86
(.5, 1.4) .80
(1, 2.3) 1.69
(.9,3.5)
--(.3,1.2)
1.48
.70
---
Fits consistently better than Nash, QRE
Unrestricted 6-parameter f(0),..f(6) fits only 1% better
CH fixes errors in Nash predictions
Figure 2: Mean Absolute Deviation for Matrix Games: Nash vs Cognitive
Hierarchy (Common )
1.00
0.90
MAD(Cognitive Hierarchy)
0.80
0.70
0.60
0.50
0.40
0.30
0.20
0.10
0.00
0.00
0.10
0.20
0.30
0.40
0.50
MAD(Nash)
0.60
0.70
0.80
0.90
1.00
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)
Table 7: Economic Value:
Cross-dataset Estimation
CH
580 1277
9% 9%
QRE
542 1277
2% 9%
Nash
513 1277
-3% 9%
573
8%
484
-9%
556
5%
460
40%
427
30%
355
8%
134
14%
98
-17%
121
2%
103
1%
111
9%
95
-7%
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
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
Thinking: Conclusions
Discrete thinking steps (mean τ ≈ 1.5)
Predicts one-shot games & initial conditions for
learning
Accounts for limited convergence in dominancesolvable 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
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 x>0
– Empirical:
Reject x=$2 half the time
What are the Responders thinking?
– Look inside their brains…
Feeling: This is your brain on unfairness
(Sanfey et al, Sci 13 March ’03)
Ultimatum offers of children who
failed/passed false belief test
Israeli subject (autistic?) complaining postexperiment (Zamir, 2000)
Ultimatum offer experimental sites
The Machiguenga
independent families
cash cropping
slash & burn
gathered foods
fishing
hunting
African pastoralists (Orma in Kenya)
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
Fair offers correlate with market integration (top),
cooperativeness in everyday life (bottom)
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
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!
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
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”
Framing enrichment:
Timing & virtual observability
Battle-of-sexes
row 1st
unobserved
B
G
Simul.
Seq’l
Unobs.
B
0,0
3,1
.62
.80
.70
G
1,3
0,0
.38
.20
.30
simul seq’l
.38
.10
.62
.90
seq’l
.20
.80
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 Ec’a; Sonsino, Erev, Gilat, unpub’d; Sovik,
unpub’d
Potential economic applications (cont’d)
A
B
I info
(A,B)
data
.77
CH (=1.5) .46
C
D
(C,D)
.53
.89
I payoffs +32
II info
A
data
.00
CH (=1.5) .12
-28
+20
(B,C)
.83
.72
-16
D
1.00
.89
II payoffs -32
+28
+16
-20
Potential economic applications (cont’d)
Prediction: Betting in (C,D) and (B,C) drops
when one number is changed
A
B
I info
(A,B)
data
?
CH (=1.5) .46
C
D
(C,D)
?
.46
I payoffs +32
II info
A
data
?
CH (=1.5) .12
-28
+32 -16
(B,C)
D
?
?
.12
.89
II payoffs -32
+28
-32
+16
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)
Framing: Limited planning in bargaining
(JEcThry ‘02; Science, ‘03)
Learning: fEWA
Attraction A
j
i
A ij (t) =(A
A ij (t) =(A
j)
(t-1) + (actual))/ ((1-)+1)
(chosen j)
(t-1) +   (foregone))/ ((1-  )+1) (unchosen
j
i
j
i
(t) for strategy j updated by
logit response function Pij(t)=exp(A
j
i
(t)/[Σkexp(A
k
i
(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
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
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)
Ultimatum offers across societies
(mean shaded, mode is largest circle…)
191~200
181~190
171~180
161~170
151~160
141~150
131~140
121~130
111~120
101~110
91~100
81~90
80
1
3
5
Period
7
9
Empirical Frequency
0.9
0.8
0.7
0.6
0.5
Prob
0.4
0.3
0.2
0.1
0
Strategy
191~200
181~190
171~180
161~170
151~160
141~150
131~140
121~130
111~120
101~110
191~200
181~190
171~180
161~170
151~160
141~150
131~140
9
0
91~100
81~90
80
1
3
5
Period
7
9
5
121~130
111~120
101~110
91~100
81~90
1
80
3
Period
7
Empirical Frequency
0.9
0.7
0.8
0.6
0.5
0.4
Prob
0.2
0.3
0.1
Thinking fEWA
Strategy
0.9
0.8
0.7
0.6
0.5
0.4
Prob
0.3
0.2
0.1
0
Strategy
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)
5. Automatic reduction of belief in noncredible threats
(subgame perfection)
T
B
4,4
L
6,3
R
0,1
0
.5
row level
1
2 3+
1
0
0
.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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