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