Group Reputations, Stereotypes, and Cooperationin a Repeated Labor Market

Paul J. Healy

U. Pitt. Feb. 2006

Carnegie Mellon University

The Plan

• Introduce the basic game

• Review of past experiments– What is really going on?

• Develop a model of group reputations

• New experiments & fit to model

The “Gift Exchange” Game• n workers, m firms, n > m• Stage 1

– Each firm j posts a wage wj {5,10,15,…}

– Workers can each accept one wage offer– Each firm hires only one worker unemployment

• Stage 2– Each hired worker i selects an effort ei {1,…,10}

• Payoffs– Firm: j(w,e) increasing in e, decreasing in w– Worker: ui(w,e) increasing in w, decreasing in e– Prediction: e*=1 and w* = min{w : ui(w,e*) 0}

Fehr Kirchsteiger & Riedl ‘93• Two rooms: buyers & sellers• Period: 3 min. or all firms are matched• Wage offers: open out-cry• Wage revisions: must beat outstanding offers• Worker acceptance: open out-cry (hectic!)• Effort choice:

– Private decisions at end of period

FKR93 Payoff Functions

j(w,e) = (126 – w)(e/10) “your” buyer’s conversion rate depends on you

ui(w,e) = w – 26 – c(e)

Unmatched agents receive zero payoff

e 1 2 3 4 5 6 7 8 9 10

c(e) 0 1 2 4 6 8 10 12 15 18

w 0, 5, 10, 15, 20,

e 1 w 30

FKR93 Isoprofit Lines

FKR93 Results

FKR93 Results

• Wage/effort correlation => reciprocity

Charness 98• Treatment: Wage chosen or randomly assigned• Exogenous matching design• Results:

– Reciprocity (wage/effort correlation)– Less reciprocity with random wages

Charness 98

0.1

0.2

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0.4

1 2 3 4 5 6 7 8 9 10Period

Average Wage

Average Effort

Fehr & Falk 99• Wages: Double auction• Treatments: Effort chosen or given

Hannan, Kagel & Moser ‘02• One-sided posted wage market• Motivation: USA vs. Europe• Result: MBAs different from undergrads

FKR98

• Same design + exogenous effort treatment

• Results– Significant reciprocity in 13/16 workers– High wages are generally profitable– Group reputation: does et affect wt+1? No

– Exogenous effort => wages drop toward equil.

Session 1

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Session 3

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Session 4

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Session 2

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Effort

Charness, Frechette & Kagel 01

• Less cooperation when payoff table is given• More significant time trends (“humpback”)

Rigdon ‘02• (Slightly) Increase marginal cost of effort• Computerized => less experimenter effect

j(w,e) = v(e) – w ui(w,e) = w – 2(e –1) e from 1 to 6 w from 10 to 35

Brandts & Charness 03• Excess supply of workers vs. firms• Minimum wage

j(w,e) = 10 – w + 5eui(w,e) = 10 – e + 5w

Riedl & Tyran 04• Tax-side equivalence j(w,e) = 30 – w + 10e [– 20]

ui(w,e) = w – e [– 20]

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w (ToW/ToF)e (ToW/ToF)

Lynch Miller Plott Porter #21• Wage: Double auction• Effort: Binary (e = 0 or 1)• ID# and effort choices are publicly observed• Subjects experienced in >2 other treatments• Multiple units, Quasi-linear payoffs

j(w,e) = v(q,e) – w– High cost:

ui(w,e) = w – 20 – 100e– Low cost:

ui(w,e) = w – 20 – 25e

LMPP #21 Results

Other LMPP Results

• Some end-game effects (low effort at high price)

• Price received by a seller depends on:– own history of effort (usually)– market history of effort (sometimes)

• Also get “lemons” with high costs when IDs & choices are private

Summary of Previous Data• Reciprocity is very strong• Individual-level heterogeneity• Subject pool effects• Wage & effort need to be choice variables• Matching mechanism not important• Surplus of labor not important• Cooperation sensitive to payoffs

– Quasi-linearity not necessarily important• Often time trends, end-game effects

Developing a Story1. Time trend => repeated game effects2. But finite repetition => unraveling3. Kreps, Milgrom, Roberts & Wilson (1982)

“Reputation building” in repeated P.D.We do observe heterogeneity…

4. But anonymity => no reputation building

• Recall LMPP: “market reputation”• Need a modified KMRW story

The Model I: 2 players, 2 actions

• 1 worker, 1 firm, 1 period

w w,w e e , e

1 -1 2 0

1 2 -1 0

The Model I: 2 players, 2 actions

• Unconditionally reciprocal worker

(Generally, e is a particular increasing function of w)

1 -1 2 0

1 2 -1 0

ew e ew e

X X

The Model I: 2 players, 2 actions

• Firm uncertainty:

1 0 1 -1 2 0

1 0 1 2 -1 0

E w 2p 1E w 0

PrReciprocal p

w, e w, e w, e w, e

p w, e 1 p w, e w, e

p w , e w , e w , e w , e

: p

The Model I: Full Reputation Equil.• T-Period “Full Reputation Equilibrium”:

Common knowledge prior: p0 0, 1p t p0 if e was "reciprocal" tp t 0 otherwise

w t w tet e t TRational: eT eReciprocal: eTw e

eTw e

The Model I: Full Reputation Equil.• Period T :

• Period T-1 :Assume wT 1 w

e pT p0 uw, e uw, e e pT 0 uw, e uw, e

e e uw , e uw , e uw , e uw , e

:

1, and we assume 1Assume wT 1 w

Playing e reduces payoff now, sets pT 0.

Worker behavior is sequentially rational.Firm: assume p0 p . So pT p0 w

The Model I: Full Reputation Equil.• Period T-1 Firm:

Proposition: With 2 players, a full reputation equilibrium exists iff

Note: there are many other sequential equilibria!

p0 p and

w w, e Period T with pT pT 1w w, e Period T with pT pT 1

The Model II: Multiple Agents

• m firms, n workers, m > n• “Random” matching

– Unmatched players earn zero• Identities, partners, & actions are public info

• Same argument, except1. Each worker has their own reputation (pj)

2. Worker discount factor in T-1:

Proposition: FRE exists iff

m/n

p0 p and m/n

The Model III: Anonymous Matching

1. All actions are publicly observed• At least: all wages are publicly observed

2. Firms don’t know ID of other firms’ workers

3. Firms don’t know ID of their own worker• Minimizes reputation-building• Beliefs are symmetric, tractable• Matches experimental environments

• Firms carry one belief about all workers: pt

• One worker defects in period t :

• Suppose If one worker defects,

Proposition: A full reputation equilibrium exists iff

The Model III: Anonymous Matching

p t 1 p t n 1n

pT 1 p nn 1

pT p wT w

p0 p , nn 1 p

and m/n

The Model III: Anonymous Matching

• FKR93:

• Need something else…

w, e 30, 1 w, e 100, 10m/n 2/3 0. 257p 0. 299p 0. 299, 0. 336

The Model IV: Type Correlation

“Stereotyping” parameter

1. Rational: uncertainty about type distribution2. Irrational: stereotyping

PrRecip i p0

Pr Recip i |Selfishj 1 p0

0, 1

Stereotyping in the Lab• Economics Experiments:

– McEvily, Weber et al.: Trustworthiness in trust games is inferred from (irrelevant) group membership (MGP)

• Social Psychology Experiments:– Acknowledging heterogeneity no stereotyping– Stereotype formation when the group affects you– Stereotype formation in group competitions– Under cognitive load:

Stereotype formation Stereotypes change more dramatically w/ new info

The Model IV: Type Correlation

• One worker defects in period t :

• Proposition: A full reputation equilibrium exists iff

• Gamma large => back to public matching:

p t 1 1 p t n 1n

p0 p , 11

nn 1 p

and m/n

p0 p and m/n

Summary of Previous Data (Again)

• Reciprocity is very strong• Individual-level heterogeneity• Subject pool effects• Wage & effort need to be choice variables• Matching mechanism not important• Surplus of labor not important• Cooperation sensitive to payoffs

– Quasi-linearity not necessarily important• Often time trends, end-game effects

The Model IV: Type Correlation• Get and from experiment parameters

– Depend on and

• Don’t know and

• Assume:

p

p0

p0 U0, 1 U0, 1

w, e w, e

Prexists p 1 ln p n

n 1nn 1 1 if m/n

0 if m/n

maxPrexists e n 1n

New Experiments1. Replicate FKR93 at Caltech

Same instructions, protocol, payoffs

2. Same as (1.), but decisions are publicWages & effort linked to ID#sEffort chosen immediately

3. Same as (2.), but payoffs make F.R.E. unlikely

Predictions: 1• Treatment 1: p 0. 299 m/n

Existence if p0 0. 3, 0. 68 and 1/2

Results

Results

Results• Effort and wage are positively correlated

Correlation coefficients > 0.446, significant

• Cooperation “pops” completely in final periodSignificant

• Wage increasing in periods 1 through 11

• Suggests: subject pool differences

Predictions: 2• Treatment 2:

Hypothesis: individual reputations strengthen reciprocity

p 0. 299 m/n

Results

Results

Results• Wages and efforts are significantly higher

• End-game effects

• Heterogeneity in end-game play

Predictions: 3• Treatment 3: w, e 126ve w

uw, e w 26 3ce

e c(e) v(e)

1 0 .35

2 1 .42

3 2 .49

4 4 .57

5 6 .64

6 8 .71

7 10 .78

8 12 .86

9 15 .93

10 18 1.0

Predictions: 3• Don’t need stereotyping to get reputation effect• But…

F.R.E. rarely exists & bound on beliefs is high

m/n 6/9 0. 771p 0. 855

Results

Results

Results

Results

Results• Min. effort is modal choice

• Effort converges to stage game equilibrium

• Reservation wage is modal choice

Switching: FKR New

Switching: FKR New

Riedl & Tyran

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e (ToW/ToF)

Rigdon

Lynch Miller Plott Porter

Conclusions• Past data suggestive of repeated game effects• Group reputation-building story is plausible

– Can turn reciprocity on and off!– Stereotyping can increase social welfare

• Applications & directions– Any repeated moral hazard setting with many agents

• Insurance, IMF loans, unemployment, lemons markets…

– Group reputation effects in other domains• Asset bubbles, public goods, coordination problems…