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Behavioral game theory* Colin F. Camerer, Caltech camerer@hssltech

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## Behavioral game theory* Colin F. Camerer, Caltech camerer@hssltech

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### Fair offers correlate with market integration (top), cooperativeness in everyday life (bottom)

Behavioral game theory*Colin F. Camerer, Caltech camerer@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)

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)

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

PrincipleNashCHQRE

Strategic Thinking

Best Response

Mutual Consistency

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-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=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

Limited equilibrationBeauty contest game

- N players choose numbers xi in [0,100]
- Compute target (2/3)*( xi /N)
- Closest to target wins $20

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

3. Purification and partial equilibration in mixed-equilibrium games (t=1.62)

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

3. Purification and partial equilibration in mixed-equilibrium games (t=1.62)

row step thinker choices CH mixed

L R 0 1 2 3 4... pred’n 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

Estimates of τ

game

- Matrix gamesspecificτ common τ

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

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)

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 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

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…

Israeli subject (autistic?) complaining post-experiment (Zamir, 2000)

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

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 1stunobserved

B G simul seq’l seq’l

B 0,0 1,3 .38 .10 .20

G 3,1 0,0 .62 .90 .80

Simul. .62 .38

Seq’l .80 .20

Unobs. .70 .30

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 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

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) .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

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 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

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…)

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)

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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