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Public goods provision and endogenous coalitions
(experimental approach)
Background
• Social dilemmas
– Underprovision of public goods
– Overexploitation of common pool resources
• Experiments on voluntary contributions
– High levels of contribution in early periods
– Decline of contributions over time
– Terminal contribution above equilibrium
What can improve cooperation ?
• Punishments
• Face-to-Face communication
• Commitments through binding agreements
Background
• Some facts about public goods experiments
• Binding agreements
• Theoretical predictions
• Experimental design
• Results
• Discussion
Plan
• Two goods : private and public• N players, : endowment • ci = Contribution of player i to the public good (C = total)
• ui(xi ,y) = xi + y : marginal payoff of the private good : marginal payoff of the public good• y = g( C ) = C • MPCR = • Normalization : • If < 1, ci = 0 is a dominant strategy and ui = • Finitely repeated game : unique subgame-perfect equilibrium ci
= 0 each period• If 1 > 1/N social optimum is ci = adn ui =
The linear public goods game
1 2 3 4 5 6 7 8 9 10 meanall 51,1 47,2 44,1 47,4 46,7 38,1 40,6 35,2 35,8 37,3 42,35M = 0,3 43 35 28 32 26 25 20 17 20 17 26,3M = 0,75 60 59 60 63 67 51 61 53 52 57 58,3unexperiment 53 53 45 50 55 43 50 41 39 44 47,3Experimented 49 41 43 45 38 33 31 30 33 30 37,3N = 4 50 50 38 40 38 30 36 32 38 30 38,2N = 10 56 50 40 41 41 34 32 33 37 35 39,9
Experiment by Isaac, Walker et Thomas (1984)
wi = w = 100
Isaac, Walker, Thomas (1984)
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10
Mean contribution
Isaac, Walker, Thomas (1984)
01020304050607080
1 2 3 4 5 6 7 8 9 10
m = 0,3 m = 0,75
MPCR
Isaac, Walker, Thomas (1984)
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10
n = 4 n = 10
Group Size
Isaac, Walker, Thomas (1984)
0
10
20
30
40
50
60
1 2 3 4 5 6 7 8 9 10
novices expérimentés
Experience
• PUR ALTRUISM• IMPURE ALTRUISM ("war glow giving"); (Andreoni,
1990)• CORRELATED ERRORS (Anderson, Goeree, Holt, 1998)• REPEATED GAME EFFECTS (Kreps, Milgrom, Roberts,
Wilson, 1982)• LEARNING (Andreoni, 1988)• CONDITIONAL COOPERATION (Keser & Van Winden,
2000)• STRENGTH OF THE SOCIAL DILEMMA (Willinger &
Ziegelmeyer, 2001)• FRAMING (Andreoni 1992, Willinger & Ziegelmeyer,
1999)
Possible explanations for overcontribution
Punishment opportunity(Fehr & Gächter, AER 2000)
• Idea : contributions that do not conform to a given « contribution norm » might be punished
• The punishment threat increases cooperation
• Punishments induce losses
• Punishing others is costly for the punisher
Experimental design
• 2 stages :
stage 1 : standard linear public goods game
stage 2 : punishment game
• After stage 1 individual contributions are publicly announced
4
1
1 20j
jiic)c(
Stage 1
Stage 2
ijp Punishment points chosen by j for i
Each punishment point reduces i’s profit by 10%:
ij
ji)p(c
Cost of punishment points for the punishers
ij
jiii)p(c)p(
10111
),pmin(pij
ij
10
Individual profit (per period)
0 1 2 3 4 5 6 7 8 9 10
0 1 2 4 6 9 12 16 20 25 30)p(c ji
jip
Design partners/strangers with/without punishment
(= 4 treatments)
• Agents have the opportunity to make binding agreements– Commitment to a contribution (public good)
– Quota on harversting (common pool)
• An agreement is defined as a coalition
• The size of the coalition determines the level of the members' contribution
• The total amount of public good provided depends on the structure of coalitions
Binding agreements
Why can agreements solve the social dilemma ?
• Positive side :– Agents who belong to the same coalition
maximise the utility of the coalition– Taking into account the group interest reduces
the free rider problem
• Negative side– An agreement covers only its members– Coalitions play a noncooperative game Free
riding occurs across coalitions
Procedures for agreement formation
Sequential procedure
• Veto• Dictator
• One agent is selected to make an agreement proposal
(e.g. choosing a group size)
• Potential members are randomly selected in the
population of potential members
• Selected members decide : accept or reject
• If all accept the proposal becomes binding
• If one potential members rejects the proposal he makes
a new proposal
• The process ends when all agents belong to an
agreement
Procedure with veto
• One agent is selected to make an agreement
proposal (e.g.. choosing a group size)
• Potential members are randomly selected in
the population of potential members
• Selected members cannot reject the proposal
• The process ends when all agents belong to an
agreement
Procedure with a dictator
Questions
• Which coalitions are more likely to emerge in the lab ?
• What is the sequence of coalition formation ?
• Do the realized coalitions come closer to the socially optimum outcome ?
• Does it matter whether potential members have veto power ?
A simplification of the coalition game
Result 1 (Bloch. 1996) : identical players coalition game is equivalent to choosing a coalition size
Result 2 (Ray & Vohra. 1999) : if only size matters the endogenous sharing rule is the egalitarian rule (in each coalition)
An example of pollution control(Ray & Vohra. 2001)
n regions involved in pollution control (pure public good)Stage 1 : binding agreementsState 2 : choice of the level of control in each agreement
Z = total amount of pollution control (pure public good)c(z) = cost of pollution controlProfit for region i :
n
jijni zczzzu
11 )(),...,(
• partition of the n regions into m binding agreements : π = (S1.....Sm)
• Each coalition (agreement) decides about a level of contribution :
j
m
jjiiii
zzszczssMax
ij
i 1
)(
2
2
1)( ii szc
ii sz
2
1
2
2
1i
k
jji ssu
221 2
1,.. i
m
jjmi ssssu
2 players remaining :
Stand alone : ui (B,1,1) = f(B) + 2 – ½ = f(B) + 1.5
Group of 2 : ui (B,2) = f(B) + 4 – ( ½) 4 = f(B) + 23 players remaining :
Stand alone : ui (B,1,2) = f(B) + 5 – ½ = f(B) + 4.5
Group of 3 : ui (B,3) = f(B) + 9 – ( ½) 9 = f(B) + 4.5
f(B) = benefit generated by the existing coalition structure
2221i
jji
ssu
4 players remaining :
Stand alone : ui (B,1,3) = f(B) + 9.5
Group of 2 : ui (B,2,2) = f(B) + 6
Group of 4 : ui (B,4) = f(B) + 8
5 players remaining :
Stand alone : ui (B,1,1,3) = f(B) + 10.5
Group of 2 : ui (B,2,3) = f(B) + 11
Group of 4 : ui (B,4,1) = f(B) + 9
Group of 5 : ui (B,5) = f(B) + 12.5
• N = 2 (2)• N = 3 (3)• N = 4 (4)• N = 5 (5)• N = 6 (1,5)• N = 7 (2,5)
Equilibrium prediction according to population size
2
1
2
2
1i
m
jji ssu
1. The social optimum is the grand coalition
2. The equilibrium coalitional structure is (2, 5)
3. The smaller coalition is formed before the larger one. and freerides on the larger coalition
3 predictions for the 7 players case
# Structure s1 s2 s3 s4 s5 s6 s7 Total 1 (7) 24.5 172.0
(1,6) 36.5 19.0 151.0
(2,5) 27.0 16.5 137.0 2
(4,3) 20.5 17.0 130.0
(1,1,5) 26.5 26.5 14.5 126.0
(1,2,4) 2.50 19.0 13.0 110.5
(1,3,3) 18.5 14.5 14.5 106.0 3
(2,2,3) 15.0 15.0 12.5 97.5
(1,1,1,4) 18.5 18.5 18.5 11.0 99.5
(1,1,2,3) 14.5 14.5 13.0 10.5 86.5 4
(1,2,2,2) 12.5 11.0 11.0 11.0 78.5
(1,1,1,1,3) 12.5 12.5 12.5 12.5 8.5 75.5 5
(1,1,1,2,2) 10.5 10.5 10.5 9.0 9.0 67.5
6 (1,1,1,1,1,2) 8.5 8.5 8.5 8.5 8.5 7.0 56.5
7 (1,1,1,1,1,1,1) 6.5 6.5 6.5 6.5 6.5 6.5 6.5 45.5
Experimental design
N = 7
2 treatments : Veto and Dictator
Same prediction for both treatments : (2 ,5)
Experimental design
• Step 1 : at the beginning of each round each subject receives an ID (letter A, B, C...)
• Step 2 : one ID is randomly chosen to make the first proposal (choose a group size)– If s1 = 1 , a singleton is formed– If 7 > s1 > 1 , the s1 proposed members are randomly
selected• Step 3 : each proposed member has to decide whether
to "accept" or to "reject"– If all proposed members accept the coalition is formed– If at least one proposed member rejects no coalition if
formed• Step 4 : One of the rejectors is selected to make a new
proposal
Experimental design
• The process ends after all subjects are assigned to a coalition
• Individual payoffs are announced after each round for each coalition size that has been formed
• 10-14 subjects per session (random / fixed), 4 veto sessions, 3 dictator sessions
• Random ending
• 92 coalition structures observed in the veto treatment and 60 in the dictator treatment
Results for the Veto treatment
Session 5 group 1 (random) 1 2 3 4 5 6 7 8 9 10 11 12 13 14 5 2 2 3 1 7 7 1 7 1 7 7 7 7 3 2 6 2 5 7 2 6 1 1 1 1 1 1 3 1 7 7 1 2 1 5 5 6 6 6 1 4 2 7 3 4 1 5 1 5 1 6 1 1 2 2 2 5 3 5 1 5 1 5 1 2 2 6 5 1 5 1 2 1 1 1 2 1 1 3 1 2 1 1 6 1 3 3 4 1 2 1 1 4 2 2 4 2 1 1 2 2
Fixed
00,10,20,30,40,50,60,70,8
1 2 3 4 5 6 7
size frequency (f ixed)
Random
00,1
0,20,30,4
0,50,6
0,70,8
1 2 3 4 5 6 7
size frequency (random)
Result 1 : The most frequently realized "agreement" is the singleton.
Result 2 : We observe a large heterogeneity of coalition structures. The equilibrium structure is never observed. The modal structure is the grand coalition (25 overall). More than 50 of the coalition structures contain 3 or more singletons.
random
0,00
0,05
0,10
0,15
0,20
0,25
0,30
0,35
0,40
0,45
0,50
(7)
(1,6
)
(2,5
)
(4,3
)
(1,1
,5)
(1,2
,4)
(1,3
,3)
(2,2
,3)
(1,1
,1,4
)
(1,1
,2,3
)
(1,2
,2,2
)
(1,1
,1,1
,3)
(1,1
,1,2
,2)
(1,1
,1,1
,1,2
)
(1,1
,1,1
,1,1
,1)
fixed
0,00
0,05
0,10
0,15
0,20
0,25
0,30
0,35
0,40
0,45
0,50
(7)
(1,6
)
(2,5
)
(4,3
)
(1,1
,5)
(1,2
,4)
(1,3
,3)
(2,2
,3)
(1,1
,1,4
)
(1,1
,2,3
)
(1,2
,2,2
)
(1,1
,1,1
,3)
(1,1
,1,2
,2)
(1,1
,1,1
,1,2
)
(1,1
,1,1
,1,1
,1)
• Optimal performance (Grand Coalition): 172.00
• Equilibrium performance (2. 5) : 137.00 (72 of the optimum)
• Average performance : 104.59 (46 of the optimum)
Result 2 (cntd)
Result 3 : Coalition structures with low payoff disparity among members are more likely to emerge.
Coalition structure
Total payoff
Gini index
Frequency
(7) 172.00 0.00 0.25 (1,6) 151.00 0.10 0.05 (2,5) 137.00 0.11 0.00 (4,3) 130.00 0.05 0.00 (1,1,5) 126.00 0.14 0.07 (1,2,4) 110.50 0.10 0.02 (1,3,3) 106.00 0.03 0.01 (2,2,3) 97.50 0.04 0.00 (1,1,1,4) 99.50 0.13 0.12 (1,1,2,3) 86.50 0.07 0.04 (1,2,2,2) 78.50 0.02 0.01 (1,1,1,1,3) 75.50 0.09 0.05 (1,1,1,2,2) 67.50 0.04 0.12 (1,1,1,1,1,2) 56.50 0.04 0.14 (1,1,1,1,1,1,1) 45.50 0.00 0.11
Low Gini High Frequency
High Gini High Frequency
Low Gini Low Frequency
Regression :
Dependent variable : Frequency of the coalition structure
Independent variable : Gini coefficient
Coefficient Std Err t P > t Gini index -3.32 1.36 -2.43 0.031 Gini index2 21.32 9.80 2.17 0.050 constant .15 .04 4.00 0.002 Prob > F = 0.0781 R-squared = 0.3462 Adj R-squared = 0.2372 Root MSE = .0613
Total payoff not significant
Result 4
Over the 3 last periods the frequency of the grand coalition increases. and the frequency of coalitions structures containing three or more singletons decreases.
All periods 3 final periods
grand coalition 25 42
at least 3 single 54 46
others 21 13
Result 5 : For 1/3 of the coalition structures. the groups are formed from the smallest to the largest. For 2/3 of the coalition structures there is no precise ordering
Result 6 : Myopic best reply predicts most of the observed coalition structures
Myopic player :Proposer : acts without anticipating
the possibility that subsequent players make couter-proposals
Responder : does not anticipate any counter-proposal except her own
222
2
1ii
ijji sssu
A myopic player always proposes the largest possible agreement of the remaining players
Myopic player 1 proposes the grand coalition
Mixed populations (Myopic + Farsighted) :
A farsighted player always proposes the singleton
Proposition : If k players are farsighted and n – k are myopic. the equilibrium coalition structure is formed by k singletons which form first followed by a unique coalition of size n – k.
Summary of alternative prediction :
Myopic players propose the grand coalition or the largest possible coalition
Farsighted players propose the singleton
ProposalSize of subgame
7 6 5 4 3 2 1
739.31 4.62 4.62 2.89 6.94 7.51 34.1
6 36.61 2.68 1.79 6.25 9.82 42.86
5 36 5.33 2.67 9.33 46.67
4 37.33 4 20 38.67
3 32.69 15.38 51.92
Proposals in subgames
Number of players in the
subgame
Frequency of consistent Proposals Random
Frequency of consistent Proposals
Data
7 28.57 73.41
6 33.33 79.46
5 40.00 82.67
4 50.00 76.00
3 66.67 84.62
Comparison Veto versus Dictator
Fixed
00,10,20,30,40,50,60,70,8
1 2 3 4 5 6 7
dictator veto
Random
00,10,20,30,40,50,60,70,8
1 2 3 4 5 6 7
dictator veto
Frequency of coalition sizes
0
0,05
0,1
0,15
0,2
0,25
0,3
0,35
0,4
(7)
(1,6
)
(2,5
)
(4,3
)
(1,1
,5)
(1,2
,4)
(1,3
,3)
(2,2
,3)
(1,1
,1,4
)
(1,1
,2,3
)
(1,2
,2,2
)
(1,1
,1,1
,3)
(1,1
,1,2
,2)
(1,1
,1,1
,1,2
)
(1,1
,1,1
,1,1
,1)
dictator veto
Frequency of coalition structures
• Optimal performance (Grand Coalition): 172.00
• Equilibrium performance (2. 5) : 137.00
• Average performance Veto: 104.59
• Average performance Dictator: 142.59
Performance
Summary
• Equilibrium prediction – never observed in the Veto treatment– 14% in the Dictator treatment
• Ordering : Smaller groups emerge earlier but only in 1/3 of the cases (veto treatment)
• Performance : below equilibrium in the veto treatment and above equilibrium in the dictator treatment
• High frequency of extreme coalitions : grand coalition and singletons
• Two explanations : – Mixed population equilibrium (myopic + farsighted players)– Inequality aversion
Questions for future research
• Individual behaviour/player types• Negative externality (large groups
emerge earlier in the sequence)• Coalition formation rule :
dictatorial. renegociation