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PPT – Behavioral game theory* Colin F. Camerer, Caltech camerer@hss.caltech.edu PowerPoint presentation | free to download - id: 6b69ca-NTNlZ

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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
- 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 diffs relation to

cognitive measures - Q J Econ August 04

Unbundling equilibrium

- Principle Nash CH QRE
- Strategic Thinking ? ? ?
- Best Response ? ?
- Mutual Consistency ? ?

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)

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)

Poisson distribution

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

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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Estimates of ? in pBC games

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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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

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

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

CH fixes errors in Nash predictions

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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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

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 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

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

Feeling This is your brain on unfairness (Sanfey

et al, Sci 13 March 03)

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Ultimatum offers of children who failed/passed

false belief test

Israeli subject (autistic?) complaining

post-experiment (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 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

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

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

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

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)

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)/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

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

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)

Ultimatum offers across societies (mean shaded,

mode is largest circle)

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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)