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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
0.3
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
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 6 12 18 24 30 36 42 48Bid #
Wage
Effort
Session 3
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 7 14 21 28 35 42 49 56Bid #
Wage
Effort
Session 4
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
0.90
1.00
0 8 16 24 32 40 48 56 64Bid #
Wage
Effort
Session 2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 6 12 18 24 30 36 42 48Bid #
Wage
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 + 5e
ui(w,e) = 10 – e + 5w
Riedl & Tyran 04• Tax-side equivalence
j(w,e) = 30 – w + 10e [– 20]
ui(w,e) = w – e [– 20]
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 16 32
w (ToF/ToW)
e (ToF/ToW)
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 Story
1. Time trend => repeated game effects
2. But finite repetition => unraveling
3. 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 1p 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 distribution
2. 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 1p 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 Experiments
1. Replicate FKR93 at Caltech Same instructions, protocol, payoffs
2. Same as (1.), but decisions are public
Wages & effort linked to ID#s
Effort 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 correlatedCorrelation 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
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 16 32
w (ToF/ToW)
e (ToF/ToW)
w (ToW/ToF)
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…