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

Behavioral game theory:Behavioral game theory:– How people actually play gamesHow people actually play games– Uses concepts from psychology and dataUses concepts from psychology and data– It It isis game theory: Has formal, replicable concepts game theory: Has formal, replicable concepts

Framing:Framing: Mental representation Mental representationFeelingFeeling: Social preferences : Social preferences ThinkingThinking: Cognitive hierarchy (: Cognitive hierarchy ())LearningLearning: Hybrid fEWA adaptive rule: Hybrid fEWA adaptive ruleTeachingTeaching: Bounded rationality in repeated games : Bounded rationality in repeated games

**Behavioral Game TheoryBehavioral Game Theory, Princeton Press 03 (550 pp); , Princeton Press 03 (550 pp); Trends in Cog SciTrends in Cog Sci, May 03 (10 pp); , May 03 (10 pp); AmerEcRevAmerEcRev, May 03 (5 pp); , May 03 (5 pp); ScienceScience, 13 June 03 (2 pp), 13 June 03 (2 pp)

BGT modelling aestheticsBGT modelling aestheticsGeneralGeneral (game theory)(game theory)

PrecisePrecise (game theory)(game theory)

ProgressiveProgressive (behavioral econ) (behavioral econ)

Cognitively detailedCognitively detailed (behavioral econ)(behavioral econ)

Empirically disciplinedEmpirically disciplined (experimental econ)(experimental econ)““...the empirical background of economic science is ...the empirical background of economic science is definitely definitely inadequateinadequate...it would have been absurd in physics ...it would have been absurd in physics to expect Kepler and Newton without Tycho Brahe” (von to expect Kepler and Newton without Tycho Brahe” (von Neumann & Morgenstern ‘44)Neumann & Morgenstern ‘44)

““Without having a broad set of facts on which to theorize, Without having a broad set of facts on which to theorize, there is a certain danger of spending too much time on there is a certain danger of spending too much time on models that are mathematically elegant, yet have little models that are mathematically elegant, yet have little connection to actual behavior. At present our empirical connection to actual behavior. At present our empirical knowledge is knowledge is inadequateinadequate...” (Eric Van Damme ‘95)...” (Eric Van Damme ‘95)

Thinking: A one-parameter cThinking: A one-parameter cognitive ognitive hierarchy theory of one-shot games* hierarchy theory of one-shot games*

(with Teck Ho, Berkeley; Kuan Chong, NUSingapore)(with Teck Ho, Berkeley; Kuan Chong, NUSingapore)

Model of constrained strategic thinking Model of constrained strategic thinking Model does several things:Model does several things:– 1. Limited equilibration in some games (e.g., pBC) 1. Limited equilibration in some games (e.g., pBC) – 2. Instant equilibration in some games (e.g. entry)2. Instant equilibration in some games (e.g. entry) – 3. De facto purification in mixed games3. De facto purification in mixed games– 4. Limited belief in noncredible threats4. Limited belief in noncredible threats– 5. Has “economic value”5. Has “economic value”– 6. Can prove theorems6. Can prove theorems

e.g. risk-dominance in 2x2 symmetric gamese.g. risk-dominance in 2x2 symmetric games– 7. Permits individual diff’s & relation to cognitive measures7. Permits individual diff’s & relation to cognitive measures

– **Q J EconQ J Econ August ‘04 August ‘04

Unbundling equilibriumUnbundling equilibriumPrinciplePrinciple NashNash CHCH QREQRE

Strategic ThinkingStrategic Thinking Best Response Best Response Mutual ConsistencyMutual Consistency

The cognitive hierarchy (CH) model (I)The cognitive hierarchy (CH) model (I)Selten (1998):Selten (1998):– ““The natural way of looking at game situations…is not based on circular The natural way of looking at game situations…is not based on circular

concepts, but rather on a step-by-step reasoning procedure”concepts, but rather on a step-by-step reasoning procedure”

Discrete steps of thinking:Discrete steps of thinking:

Step 0’s choose randomly (nonstrategically)Step 0’s choose randomly (nonstrategically)

K-step thinkers know proportions f(0),...f(K-1)K-step thinkers know proportions f(0),...f(K-1) Calculate what 0, …K-1 step players will do Calculate what 0, …K-1 step players will do

Choose best responsesChoose best responses

Exhibits “increasingly rational expectations”:Exhibits “increasingly rational expectations”:– Normalized beliefs approximate f(n) as nNormalized beliefs approximate f(n) as n ∞∞

i.e., highest level types are “sophisticated”/”worldly and earn the mosti.e., highest level types are “sophisticated”/”worldly and earn the most

Easy to calculate Easy to calculate (see website “calculator” (see website “calculator” http://groups.haas.berkeley.edu/simulations/ch/default.asphttp://groups.haas.berkeley.edu/simulations/ch/default.asp) )

The cognitive hierarchy (CH) model (II)The cognitive hierarchy (CH) model (II)

What is a reasonable simple f(K)?What is a reasonable simple f(K)?– A1*: f(k)/f(k-1) A1*: f(k)/f(k-1) ∝1/∝1/kk

Poisson f(k)=ePoisson f(k)=e/k! mean, variance /k! mean, variance

– A2: f(1) is modal A2: f(1) is modal 1< 1< – A3: f(1) is a ‘maximal’ mode A3: f(1) is a ‘maximal’ mode

or f(0)=f(2) or f(0)=f(2) t= t=2=1.414..2=1.414..

– A4: f(0)+f(1)=2f(2) A4: f(0)+f(1)=2f(2) t=1.618 (golden ratio t=1.618 (golden ratio ΦΦ))

*Amount of working memory (digit span) correlated with steps of iterated deletion *Amount of working memory (digit span) correlated with steps of iterated deletion of dominated strategies (Devetag & Warglien, 03 J Ec Psych)of dominated strategies (Devetag & Warglien, 03 J Ec Psych)

Poisson distributionPoisson distributionDiscrete, one parameterDiscrete, one parameter– (( “spikes” in data) “spikes” in data)

Steps > 3 are rare (tight working memory bound)Steps > 3 are rare (tight working memory bound)Steps can be linked to cognitive measuresSteps can be linked to cognitive measures

Poisson distributions for various

00.05

0.10.15

0.20.25

0.30.35

0.4

0 1 2 3 4 5 6

number of steps

fre

qu

en

cy

1.1. Limited equilibrationLimited equilibrationBeauty contest gameBeauty contest game

N players choose numbers xN players choose numbers xii in in [0,100][0,100]

Compute target (2/3)*(Compute target (2/3)*( x xii /N) /N)

Closest to target wins $20Closest to target wins $20

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

1 9

17

25

33

41

49

57

65

73

81

89

97

number choices

pre

dic

ted

fre

qu

en

cy

Beauty contest results (Expansion, Financial Times, Spektrum)

0.00

0.05

0.10

0.15

0.20

numbers

rela

tive

fr

eq

ue

nci

es

22 50 10033

average 23.07

0

Estimates of Estimates of in pBC games in pBC games Table 1: Data and estimates of in pbc games(equilibrium = 0)

Steps ofsubjects/game Data CH Model Thinkinggame theorists 19.1 19.1 3.7Caltech 23.0 23.0 3.0newspaper 23.0 23.0 3.0portfolio mgrs 24.3 24.4 2.8econ PhD class 27.4 27.5 2.3Caltech g=3 21.5 21.5 1.8high school 32.5 32.7 1.61/2 mean 26.7 26.5 1.570 yr olds 37.0 36.9 1.1Germany 37.2 36.9 1.1CEOs 37.9 37.7 1.0game p=0.7 38.9 38.8 1.0Caltech g=2 21.7 22.2 0.8PCC g=3 47.5 47.5 0.1game p=0.9 49.4 49.5 0.1PCC g=2 54.2 49.5 0.0

mean 1.56median 1.30

Mean

2. Approximate equilibration in entry games2. Approximate equilibration in entry games

Entry games: Entry games: N entrants, capacity c N entrants, capacity c Entrants earn $1 if n(entrants)Entrants earn $1 if n(entrants)<<c; c;

earn 0 if n(entrants)>cearn 0 if n(entrants)>cEarn $.50 by staying outEarn $.50 by staying out

n(entrants) n(entrants) ≈ ≈ c in the 1st period:c in the 1st period: “ “To a psychologist, it looks like magic”-- D. To a psychologist, it looks like magic”-- D. Kahneman ’88Kahneman ’88How? Pseudo-sequentiality of CH How? Pseudo-sequentiality of CH “later”-thinking “later”-thinking entrants smooth the entry functionentrants smooth the entry function

0

0.1

0.20.3

0.4

0.5

0.6

0.70.8

0.9

1

2 4 6 8 10

capacity (out of 12)

frequ

ency

total entry

Nash equilibrium

CH fit (tau=1.5)

0-Step and 1-Step Entry

0

10

20

30

40

50

60

70

80

90

100

1 11 21 31 41 51 61 71 81 91 101

Percentage Capacity

Pe

rce

nta

ge

En

try

Capacity

0-Level

1-Level

0-Step and 1-Step Entry

0

10

20

30

40

50

60

70

80

90

100

1 11 21 31 41 51 61 71 81 91 101

Percentage Capacity

Pe

rce

nta

ge

En

try

Capacity

0+1 Level

0-Step and 1-Step Entry

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

Percentage Capacity

Pe

rce

nta

ge

En

try

Capacity

0+1 Level`

0-Step + 1-Step + 2 Step Entry

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

Percentage Capacity

Pe

rce

nta

ge

En

try

Capacity

0+1 Level

2-Level`

0-Step + 1-Step + 2 Step Entry

0

10

20

30

40

50

60

70

80

90

100

0 10 20 30 40 50 60 70 80 90 100

Percentage Capacity

Pe

rce

nta

ge

En

try

Capacity

0+1+2 Level`

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

row step thinker choicesrow step thinker choicesLL RR 0 1 2 30 1 2 3 4... 4...

TT 2,02,0 0,10,1 .5 1 1 0 0.5 1 1 0 0BB 0,10,1 1,01,0 .5 0.5 0 0 1 0 1 1 100 .5.5 .5.511 .5.5 .5.522 00 1133 00 1144 00 1155 00 11

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

row step thinker choices CH row step thinker choices CH mixedmixedLL RR 0 1 2 30 1 2 3 4... pred’n equilm data 4... pred’n equilm data

TT 2,02,0 0,10,1 .5 1 1 0 0 .68 .50 .5 1 1 0 0 .68 .50 .72.72BB 0,10,1 1,01,0 .5 0.5 0 0 1 0 1 1 .32 .50 .28 1 .32 .50 .2800 .5.5 .5.511 .5.5 .5.522 00 1133 00 1144 00 1155 00 11

CHCH .26.26 .74.74mixed .33mixed .33 .67.67datadata .33.33 .67.67

Estimates of Estimates of ττ gamegame

Matrix gamesMatrix games specific specific ττ common common ττ

Stahl, Wilson Stahl, Wilson (0, 6.5) 1.86(0, 6.5) 1.86Cooper, Van Huyck Cooper, Van Huyck (.5, 1.4) .80(.5, 1.4) .80Costa-Gomes et alCosta-Gomes et al (1, 2.3) 1.69 (1, 2.3) 1.69

Mixed-equil. gamesMixed-equil. games (.9,3.5) 1.48(.9,3.5) 1.48

Entry games Entry games --- --- .70 .70

Signaling gamesSignaling games (.3,1.2) ---(.3,1.2) ---

Fits consistently better than Nash, QREFits consistently better than Nash, QREUnrestricted 6-parameter f(0),..f(6) fits only 1% better Unrestricted 6-parameter f(0),..f(6) fits only 1% better

CH fixes errors in Nash predictionsCH fixes errors in Nash predictionsFigure 2: Mean Absolute Deviation for Matrix Games: Nash vs Cognitive

Hierarchy (Common )

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

MAD(Nash)

MA

D(C

og

nit

ive H

iera

rch

y)

4. Economic Value4. Economic Value Treat models like consultantsTreat models like consultants

If players were to hire Mr. Nash and Mr. Camhocho as If players were to hire Mr. Nash and Mr. Camhocho as consultants and listen to their advice, would they have made a consultants and listen to their advice, would they have made a higher payoff? higher payoff?

If players are in equilibrium, Nash advice will have If players are in equilibrium, Nash advice will have zero valuezero value if theories have economic value, players are if theories have economic value, players are notnot in in

equilibriumequilibrium

Advised strategy is what highest-level players chooseAdvised strategy is what highest-level players choose economic value is the payoff advantage of thinking hardereconomic value is the payoff advantage of thinking harder

(selection pressure in replicator dynamics)(selection pressure in replicator dynamics)

Table 7: Economic Value:Cross-dataset EstimationCH 580 1277 573 460 134 103

9% 9% 8% 40% 14% 1%QRE 542 1277 484 427 98 111

2% 9% -9% 30% -17% 9%Nash 513 1277 556 355 121 95

-3% 9% 5% 8% 2% -7%

6. Other theoretical properties of CH model6. Other theoretical properties of CH model Advantages over Nash equilibriumAdvantages over Nash equilibrium

No multiplicity problem (picks No multiplicity problem (picks oneone distribution) distribution) No “weird” beliefs in games of incomplete info.No “weird” beliefs in games of incomplete info.

Theory: Theory: ττ∞ converges to ∞ converges to Nash equilibrium in (weakly) Nash equilibrium in (weakly) dominance solvable gamesdominance solvable games

Coincides with “risk dominant” equilibrium in Coincides with “risk dominant” equilibrium in symmetric 2x2 gamessymmetric 2x2 games

“ “Close” to Nash in 2x2 mixed games (Close” to Nash in 2x2 mixed games (ττ=2.7 =2.7 82% 82% same-quadrant correspondence)same-quadrant correspondence)

Equal splits in Nash demand games Equal splits in Nash demand games Group size effects in stag hunt, beauty contest, Group size effects in stag hunt, beauty contest, centipede gamescentipede games

7. Preliminary findings on individual 7. Preliminary findings on individual differences & response timesdifferences & response times

Caltech Caltech is .53 higher than PCC is .53 higher than PCC

Individual differences:Individual differences:– Estimated Estimated i i (1st half) correlates .64 (1st half) correlates .64

with with i i (2nd half)(2nd half)

Upward drift in Upward drift in , .69 from 1, .69 from 1stst half to 2 half to 2ndnd half of game half of game (no-feedback “learning” ala Weber ExEc 03?)(no-feedback “learning” ala Weber ExEc 03?)

One step adds .85 secs to response timeOne step adds .85 secs to response time

Thinking: ConclusionsThinking: ConclusionsDiscrete thinking steps (mean Discrete thinking steps (mean ττ ≈ ≈ 1.5) 1.5)

Predicts one-shot games & initial conditions for Predicts one-shot games & initial conditions for learninglearning

Accounts for limited convergence in dominance-Accounts for limited convergence in dominance-solvable games solvable games andand approximate convergence in approximate convergence in mixed & entry gamesmixed & entry gamesAdvantages:Advantages:

MoreMore precise than Nash: Can “solve” multiplicity precise than Nash: Can “solve” multiplicity problemproblem

Has economic valueHas economic valueCan be tied to cognitive measuresCan be tied to cognitive measures

Important! This Important! This isis game theory game theory It is a formal specification which makes predictionsIt is a formal specification which makes predictions

Feeling in ultimatum games: How Feeling in ultimatum games: How much do you offer out of $10? much do you offer out of $10?

Proposer has $10Proposer has $10

Offers x to Responder (keeps $10-x)Offers x to Responder (keeps $10-x)

What should the Responder do? What should the Responder do? – Self-interest: Take any x>0Self-interest: Take any x>0– Empirical: Empirical: Reject x=$2 half the Reject x=$2 half the

timetime

What are the Responders thinking? What are the Responders thinking? – Look inside their brains…Look inside their brains…

Feeling: This is your brain on unfairnessFeeling: This is your brain on unfairness(Sanfey et al, Sci 13 March ’03)(Sanfey et al, Sci 13 March ’03)

Ultimatum offers of children who Ultimatum offers of children who failed/passed false belief testfailed/passed false belief test

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

Ultimatum offer experimental sitesUltimatum offer experimental sites

slash & burngathered foods

fishinghunting

The Machiguengaindependent families

cash cropping

African pastoralists (Orma in Kenya)

Whale Huntersof

Lamalera, Indonesia

High levels ofcooperation 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), Fair offers correlate with market integration (top), cooperativeness in everyday life (bottom)cooperativeness in everyday life (bottom)

New frontiersNew frontiersField applications!Field applications!

Imitation learningImitation learningTrifurcation:Trifurcation:– Rational gt: Firms, expert players, long-run Rational gt: Firms, expert players, long-run

outcomesoutcomes– Behavioral gt: Normal people, new gamesBehavioral gt: Normal people, new games– Evolutionary gt: Animals, humans imitating Evolutionary gt: Animals, humans imitating

ConclusionsConclusionsThinkingThinking CH model ( CH model ( mean number of steps) mean number of steps) is similar (is similar (≈≈1.5) in many games: Explains limited 1.5) in many games: Explains limited and and surprising surprising equilibrationequilibrationEasy to use empirically & do theoryEasy to use empirically & do theory

Feeling Feeling Ultimatum rejections are common, vary across cultureUltimatum rejections are common, vary across culture

fairness correlated with market integration (cf. Adam fairness correlated with market integration (cf. Adam Smith)Smith)Unfair offers activate insula, ACC, DLPFCUnfair offers activate insula, ACC, DLPFCU-shaped rejections commonU-shaped rejections commonDictators offer less when threatened with 3Dictators offer less when threatened with 3rdrd-party punishment-party punishment

PedagogyPedagogy: A radical new way to teach game theory: A radical new way to teach game theory– Start with concept of a game. Start with concept of a game. – Building blocks: Mixing, dominance, foresight.Building blocks: Mixing, dominance, foresight.– Then teach cognitive hierarchy, learning…Then teach cognitive hierarchy, learning…– endend with equilibrium! with equilibrium!

Potential applicationsPotential applications

ThinkingThinking– price bubbles, speculation, competition price bubbles, speculation, competition

neglect neglect

LearningLearning– evolution of institutions, new industriesevolution of institutions, new industries– Neo-Keynesian macroeconomic Neo-Keynesian macroeconomic

coordination coordination – bidding, consumer choicebidding, consumer choice

TeachingTeaching– contracting, collusion, inflation policycontracting, collusion, inflation policy

FramingFraming: How are games represented?: How are games represented?

Invisible assumption: Invisible assumption: – People represent games in matrix/tree formPeople represent games in matrix/tree form

Mental representations may be Mental representations may be simplified…simplified…– analogies: `Iraq war is Afghanistan, not analogies: `Iraq war is Afghanistan, not

Vietnam’Vietnam’– shrinking-pie bargainingshrinking-pie bargaining

……or enrichedor enriched– Schelling matching gamesSchelling matching games– timing & “virtual observability”timing & “virtual observability”

Framing enrichment: Framing enrichment: Timing & virtual observabilityTiming & virtual observability

Battle-of-sexesBattle-of-sexes row 1strow 1st

unobservedunobserved

BB GG simulsimul seq’lseq’l seq’l seq’l

B B 0,00,0 1,31,3 .38.38 .10 .10 .20 .20

G G 3,13,1 0,00,0 .62.62 .90.90 .80 .80

Simul.Simul. .62.62 .38.38

Seq’lSeq’l .80.80 .20.20

Unobs. .70Unobs. .70 .30.30

Potential economic applicationsPotential economic applications

– Price bubbles Price bubbles

thinking steps correspond to timing of thinking steps correspond to timing of selling before a crashselling before a crash

– SpeculationSpeculation

Violates “Groucho Marx” no-bet theorem* Violates “Groucho Marx” no-bet theorem* AA BB CC DD

I info I info (A,B) (A,B) (C,D) (C,D)

I payoffs +32I payoffs +32 -28-28 +20+20 -16-16

II infoII info AA (B,C) (B,C) D D

II payoffs -32 +28 -20 +16II payoffs -32 +28 -20 +16

*Milgrom-Stokey ’82 Ec’a; Sonsino, Erev, Gilat, unpub’d; Sovik, *Milgrom-Stokey ’82 Ec’a; Sonsino, Erev, Gilat, unpub’d; Sovik, unpub’dunpub’d

Potential economic applications (cont’d)Potential economic applications (cont’d)

AA BB CC DD

I info I info (A,B) (A,B) (C,D) (C,D)

data .77 .53data .77 .53

CH (CH (=1.5) .46 .89=1.5) .46 .89

I payoffs +32I payoffs +32 -28-28 +20+20 -16-16

II infoII info A A (B,C) (B,C) D D

data .00data .00 .83 .83 1.001.00

CH (CH (=1.5) .12 .72 .89=1.5) .12 .72 .89

II payoffs -32 +28 -20 +16II payoffs -32 +28 -20 +16

Potential economic applications (cont’d)Potential economic applications (cont’d)Prediction: Betting in (C,D) and (B,C) drops Prediction: Betting in (C,D) and (B,C) drops

when one number is changedwhen one number is changed AA BB CC DD

I info I info (A,B) (A,B) (C,D) (C,D)

data ? ?data ? ?

CH (CH (=1.5) .46 =1.5) .46 .46.46

I payoffs +32I payoffs +32 -28-28 +32+32 -16-16

II infoII info A A (B,C) (B,C) D D

data ?data ? ? ? ? ?

CH (CH (=1.5) .12 =1.5) .12 .12.12 .89 .89

II payoffs -32 +28 II payoffs -32 +28 -32-32 +16 +16

The cognitive hierarchy (CH) model (II)The cognitive hierarchy (CH) model (II)

Two separate features:Two separate features:– Not imagining k+1 typesNot imagining k+1 types– Not believing there are other k typesNot believing there are other k types

OverconfidenceOverconfidence: : K-steps think others are all one step lower (K-1) K-steps think others are all one step lower (K-1)

(Nagel-Stahl-CCGB)(Nagel-Stahl-CCGB)““Increasingly Increasingly irirrational expectations” as Krational expectations” as K ∞∞ Has some odd properties (cycles in entry games…)Has some odd properties (cycles in entry games…)

Self-consciousSelf-conscious:: K-steps believe there are other K-step thinkersK-steps believe there are other K-step thinkers““Too similar” to quantal response equilibrium/NashToo similar” to quantal response equilibrium/Nash(& fits worse)(& fits worse)

Framing: Limited planning in bargaining Framing: Limited planning in bargaining (JEcThry ‘02; Science, ‘03)(JEcThry ‘02; Science, ‘03)

LearningLearning: fEWA: fEWAAttraction A Attraction A ii

jj (t) for strategy j updated by (t) for strategy j updated by A A ii

jj (t) =( (t) =(A A iijj (t-1) + (t-1) + (actual))/ ((actual))/ ((1-(1-)+1) (chosen j))+1) (chosen j)

A A iijj (t) =( (t) =(A A ii

jj (t-1) + (t-1) + (foregone))/ ( (foregone))/ ((1- (1- )+1) (unchosen j) )+1) (unchosen j)logit response function Plogit response function Pii

jj(t)=exp((t)=exp(A A iijj (t)/[ (t)/[ΣΣkkexpexp((A A ii

kk (t)]* (t)]*

key parameters:key parameters: imagination, imagination, decay/change-detection decay/change-detection

““In nature a hybrid [species] is usually sterile, but in science the In nature a hybrid [species] is usually sterile, but in science the opposite is often true”-- Francis Crick ’88opposite is often true”-- Francis Crick ’88

Special cases:Special cases:– Weighted fictitious play (Weighted fictitious play (=1, =1, =0)=0)– CChoice reinforcement (hoice reinforcement (=0) =0)

EWA estimates parameters EWA estimates parameters , , , , (Cam.-Ho ’99 Ec’a)(Cam.-Ho ’99 Ec’a)

*Or divide by payoff variability (Erev et al ’99 JEBO); automatically “explores” when *Or divide by payoff variability (Erev et al ’99 JEBO); automatically “explores” when environment changesenvironment changes

Functional fEWAFunctional fEWA

Substitute Substitute functionsfunctions for parameters for parameters

Easy to estimate (only Easy to estimate (only ))

Tracks parameter differences across gamesTracks parameter differences across games

Allows change Allows change withinwithin a game a game

““Change detector” for decay rate Change detector” for decay rate φφ

φφ(i,t)=1-.5[(i,t)=1-.5[kk ( S ( S-i-ik k (t) - (t) - =1=1

t t SS-i-ikk(()/t ) )/t ) 2 2 ]]

φφ close to 1 when stable, dips to 0 when close to 1 when stable, dips to 0 when unstableunstable

Example: Price matching with loyalty Example: Price matching with loyalty rewards rewards (Capra, Goeree, Gomez, Holt AER ‘99)(Capra, Goeree, Gomez, Holt AER ‘99)

Players 1, 2 pick prices [80,200] ¢Players 1, 2 pick prices [80,200] ¢

Price is P=min(PPrice is P=min(P1,1,,P,P22))

Low price firm earns P+RLow price firm earns P+R

High price firm earns P-RHigh price firm earns P-R

What happens? (e.g., R=50)What happens? (e.g., R=50)

Ultimatum offers across societies Ultimatum offers across societies (mean shaded, mode is largest circle…)(mean shaded, mode is largest circle…)

1

3

5

7

9

80

81~

90

91~

100

101~

110

111~

120

121~

130

131~

140

141~

150

151~

160

161~

170

171~

180

181~

190

191~

200

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Prob

Period

Strategy

Empirical Frequency

1

3

5

7

9

80

81~

90

91~

100

101~

110

111~

120

121~

130

131~

140

141~

150

151~

160

161~

170

171~

180

181~

190

191~

200

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Prob

Period

Strategy

Thinking fEWA

1

3

5

7

9

80

81~

90

91~

100

101~

110

111~

120

121~

130

131~

140

141~

150

151~

160

161~

170

171~

180

181~

190

191~

200

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

Prob

Period

Strategy

Empirical Frequency

A decade of empirical studies of learning: A decade of empirical studies of learning: Taking stockTaking stock

Early studies show models can track basic Early studies show models can track basic features of learning pathsfeatures of learning paths– McAllister, ’91 Annals OR; Cheung-Friedman ’94 GEB; McAllister, ’91 Annals OR; Cheung-Friedman ’94 GEB;

Roth-Erev ’95 GEB,’98 AERRoth-Erev ’95 GEB,’98 AER

Is one model generally better?: “Horse races”Is one model generally better?: “Horse races”– Speeds up process of single-model explorationSpeeds up process of single-model exploration– Fair tests: Common games & empirical methodsFair tests: Common games & empirical methods

““match races” in horse racing: Champions forced to competematch races” in horse racing: Champions forced to compete

Development of hybrids which are robust Development of hybrids which are robust (improve on failures of specific models) (improve on failures of specific models) – EWA (Camerer-Ho ’99, Anderson-Camerer ’00 Ec Thy)EWA (Camerer-Ho ’99, Anderson-Camerer ’00 Ec Thy)– fEWA (Camerer-Ho, ’0?)fEWA (Camerer-Ho, ’0?)– Rule learning (Stahl, ’01 GEB)Rule learning (Stahl, ’01 GEB)

5. Automatic reduction of belief in noncredible threats 5. Automatic reduction of belief in noncredible threats (subgame perfection)(subgame perfection)

row levelrow level 0 1 2 3+0 1 2 3+

TT 4,4 4,4 .5 1 0 0.5 1 0 0 LL R R

BB 6,36,3 0,10,1 .5 0 1 1.5 0 1 1

(T,R) Nash, (B,L) subgame perfect(T,R) Nash, (B,L) subgame perfectCH Prediction: (CH Prediction: (=1.5) =1.5)

89% play L89% play L56% play B56% play B (Level 1) players do not have (Level 1) players do not have

enough faith in enough faith in rationality of others rationality of others (Beard & Beil, 90 Mgt Sci; Weiszacker (Beard & Beil, 90 Mgt Sci; Weiszacker

’03 GEB)’03 GEB)