some unpleasant bargaining arithmetics? › seminarpapers › et24062010.pdf · huly a eraslan and...

Some Unpleasant Bargaining Arithmetics?

Hülya Eraslan and Antonio Merlo

April 2010

Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?

Introduction

I Starting with the seminal work of Baron and Ferejohn (1989)noncooperative models of multilateral bargaining havebecome a staple of political economy and have been used innumerous applications.

I The agreement rule (e.g., unanimity, majority, super-majority)is a key component of multilateral bargaining environments.

I Large literature on comparing the performance of differentvoting rules in a variety of bargaining models.

I Emphasis has been primarily on the efficiency of equilibriumoutcomes.

Introduction

This paper

I We focus on comparing the distributional consequences(equity properties) of alternative voting rules in a simplebargaining environment.

I It is commonly believed that, contrary to majority rule,unanimity rule protects minorities from the possibility ofexpropriation and is therefore more equitable.

I We show that this is not necessarily the case in bargaining:unanimity rule can induce equilibrium outcomes that are moreunequal (or less equitable) than equilibrium outcomes undermajority rule.

This paper

The environment

I There are n ≥ 2 players, who are endowed with differenttechnologies for providing some perfectly divisible surplus.

I If player i were to provide, the amount of surplus potentiallyavailable for distribution would be yi , with y1 ≤ y2... ≤ yn.

I At most one project can be implemented.

I The players have to collectively decide which project toimplement (if any), and how to distribute the availablesurplus.

The environment

The environment (example)

I There is a single project that needs to be completed, whichentails the production of a unitary level of a perfectly divisible(private) good, x = 1.

I The surplus generated (i.e., the amount of x net ofproduction costs) is then available for distribution.

I There are n ≥ 2 players, who are endowed with differenttechnologies for producing x .

I Player i ∈ {1, ..., n} can produce x = 1 at cost ci ∈ [0, 1].Without loss of generality let c1 ≥ c2... ≥ cn.

I This implies that if i were to produce x , the amount ofsurplus potentially available for distribution would beyi = 1− ci , with y1 ≤ y2... ≤ yn.

I The players have to collectively decide who (if anybody) willproduce x , and how to distribute the surplus y generated inthe event that production takes place.

The environment (con’d)

I Players have an identical single date payoff function which islinear in their share of the surplus, and discount the future ata common discount factor δ ∈ (0, 1).

I In the event that agreement is never reached, all playersreceive a payoff of zero.

The environment (cont’d)

I We model the collective decision-making process as abargaining problem.

I The protocol we consider is as follows:I In each period, a player is randomly offered the possibility of

submitting a proposal for completing the project withprobability 1/n.

I The selected player j may then make a (binding) proposalspecifying the way the surplus generated yj would bedistributed among all the players if her project were chosen, orforego the opportunity.

I If a proposal is submitted, all players then vote (sequentially)on whether or not to approve it.

I If q players vote in favor, then the proposal is implementedand the game ends. Otherwise, a new player is selected andthe process repeats itself (possibly ad infinitum).

I The protocol we consider is as follows:

I In each period, a player is randomly offered the possibility ofsubmitting a proposal for completing the project withprobability 1/n.

I q ∈ {1, ..., n} specifies the voting rule (e.g., q = n denotesunanimity and q = (n + 1)/2 majority rule), and hence thegame.

I i ∈ {1, ..., n} specifies each player’s ranking in the endowmentdistribution (with 1 denoting the least productive and n themost productive player).

Comments on the model environment

I Baron and Ferejohn (1989) with heterogeneous “cakes”.

I Merlo and Wilson (1995, 1998) and Eraslan and Merlo (2002)with perfect correlation between the “cake” and the“proposer” processes.

I Deliberately “egalitarian” protocol, and no additionaldimensions of heterogeneity other than in the players’technology endowments.

Notation and some useful definitions

I Restrict attention to stationary subgame perfect (SSP)equilibria.

I Let vq = (vq1 , ..., vqn ) denote the payoff vector generated by an

SSP strategy profile in the q-game.

I Let

G =2

∑ni=1 iyi

n∑n

i=1 yi− n + 1

n

denote the Gini coefficient for the endowment distribution.

I Let

Gq =2

∑ni=1 iv

qi

n∑n

i=1 vqi

− n + 1n

denote the Gini coefficient for the equilibrium payoffdistribution in the q-game.

I Let GU and GM denote the Gini coefficients for theequilibrium payoff distribution under unanimity and majorityrule.

I Let

G =2

∑ni=1 iyi

n∑n

i=1 yi− n + 1

n

I Let

Gq =2

∑ni=1 iv

qi

n∑n

i=1 vqi

− n + 1n

I Let

G =2

∑ni=1 iyi

n∑n

i=1 yi− n + 1

n

I Let

Gq =2

∑ni=1 iv

qi

n∑n

i=1 vqi

− n + 1n

I Let

G =2

∑ni=1 iyi

n∑n

i=1 yi− n + 1

n

I Let

Gq =2

∑ni=1 iv

qi

n∑n

i=1 vqi

− n + 1n

I Let

G =2

∑ni=1 iyi

n∑n

i=1 yi− n + 1

n

I Let

Gq =2

∑ni=1 iv

qi

n∑n

i=1 vqi

− n + 1n

I Let

G =2

∑ni=1 iyi

n∑n

i=1 yi− n + 1

n

I Let

Gq =2

∑ni=1 iv

qi

n∑n

i=1 vqi

− n + 1n

A leading example

I Suppose c1 = c2 = ... = cn−1 = � and cn = 0, ory1 = y2 = ... = yn−1 = 1− � and yn = 1.

I For every � > 0, if δ > (1−�)n(1−�)n+� , in the unique SSPequilibrium of the unanimity game

vUn =1

n − δ(n − 1)and vU1 = v

U2 = ... = v

Un−1 = 0.

I Verify that this is (an) equilibrium:I Player n’s payoff is the solution to vUn =

1n +

n−1n δv

Un .

I If i ≤ n − 1 is the proposer, he cannot obtain approval ofplayer n because 1− � < δvUn = δ 1n−δ(n−1) .

I So player i ’s payoff is 0.

A leading example

vUn =1

n − δ(n − 1)and vU1 = v

U2 = ... = v

Un−1 = 0.

1n +

n−1n δv

Un .

A leading example

vUn =1

n − δ(n − 1)and vU1 = v

U2 = ... = v

Un−1 = 0.

1n +

n−1n δv

Un .

A leading example

vUn =1

n − δ(n − 1)and vU1 = v

U2 = ... = v

Un−1 = 0.

I Verify that this is (an) equilibrium:

I Player n’s payoff is the solution to vUn =1n +

n−1n δv

Un .

A leading example

vUn =1

n − δ(n − 1)and vU1 = v

U2 = ... = v

Un−1 = 0.

1n +

n−1n δv

Un .

A leading example

vUn =1

n − δ(n − 1)and vU1 = v

U2 = ... = v

Un−1 = 0.

1n +

n−1n δv

Un .

A leading example

vUn =1

n − δ(n − 1)and vU1 = v

U2 = ... = v

Un−1 = 0.

1n +

n−1n δv

Un .

A leading example (cont’d)

I For every � < δ2+δ n−1

n

, in the unique SSP equilibrium of the

majority game

vM1 = vM2 = ... = v

Mn =

1 + (n − 1)(1− �)n2

= vM .

I Verify that this is (an) equilibrium:I If i is the proposer, he offers δvM to q − 1 other players and

keeps yi − δ(q − 1)vM .I Let µi denote the (endogenous) probability that i receives his

continuation payoff when someone else is the proposer.I So player i ’s payoff must satisfy

vM = 1n (yi − δ(q − 1)vM) + µiδv

M .I In equilibrium, we must have µi ≥ 0 and µi ≤ 1− 1n for all i ;

y1 − (q − 1)δvM ≥ δvM ; and∑n

i=1 µi = q − 1.I Since � < δ

2+δ n−1n, we can find µ1, . . . , µn satisfying these.

n

majority game

vM1 = vM2 = ... = v

Mn =

1 + (n − 1)(1− �)n2

= vM .

i=1 µi = q − 1.I Since � < δ

n

majority game

vM1 = vM2 = ... = v

Mn =

1 + (n − 1)(1− �)n2

= vM .

I Verify that this is (an) equilibrium:

I If i is the proposer, he offers δvM to q − 1 other players andkeeps yi − δ(q − 1)vM .

I Let µi denote the (endogenous) probability that i receives hiscontinuation payoff when someone else is the proposer.

I So player i ’s payoff must satisfyvM = 1n (yi − δ(q − 1)v

M) + µiδvM .

I In equilibrium, we must have µi ≥ 0 and µi ≤ 1− 1n for all i ;y1 − (q − 1)δvM ≥ δvM ; and

∑ni=1 µi = q − 1.

I Since � < δ2+δ n−1n

, we can find µ1, . . . , µn satisfying these.

n

majority game

vM1 = vM2 = ... = v

Mn =

1 + (n − 1)(1− �)n2

= vM .

keeps yi − δ(q − 1)vM .

I Let µi denote the (endogenous) probability that i receives hiscontinuation payoff when someone else is the proposer.

M) + µiδvM .

∑ni=1 µi = q − 1.

n

majority game

vM1 = vM2 = ... = v

Mn =

1 + (n − 1)(1− �)n2

= vM .

continuation payoff when someone else is the proposer.

M) + µiδvM .

∑ni=1 µi = q − 1.

n

majority game

vM1 = vM2 = ... = v

Mn =

1 + (n − 1)(1− �)n2

= vM .

M .

∑ni=1 µi = q − 1.

n

majority game

vM1 = vM2 = ... = v

Mn =

1 + (n − 1)(1− �)n2

= vM .

i=1 µi = q − 1.

n

majority game

vM1 = vM2 = ... = v

Mn =

1 + (n − 1)(1− �)n2

= vM .

i=1 µi = q − 1.I Since � < δ

I In this environment

0 = GM < G < GU .

I In this environment

0 = GM < G < GU .

Intuition

Under unanimity rule:

I Equilibrium is efficient.

I If players are patient enough, efficiency requires agreementonly when player n proposes.

I Since all other players can never obtain unanimous approval(player n would never say yes), it is “as if” they neverpropose. Hence, their equilibrium payoff must be zero.

Under majority rule:

I All proposals are accepted (it only takes a majority in favor toapprove a project).

I Player n loses his advantage.

I This “egalitarian” force generates regression toward the meanthat equalizes expected equilibrium payoffs.

Intuition

General results

I If there is agreement is reached when the cake size is yi , thenthere is agreement when the cake size is yi+1.

I Payoffs are monotone: vqi ≤ vqi+1.

I In the q-game, there is always agreement when player qproposes.

General results

General results (cont’d)

I Under unanimity rule, there exists a unique SSP payoff:

vUi = max{0,1

n(1− δ)(yi −

δ

n − δ(κU − 1)

n∑j=κU

yj)}.

whereκU = min{i : yi − δ

∑j

vUj ≥ 0}.

vUi = max{0,1

n(1− δ)(yi −

δ

n − δ(κU − 1)

n∑j=κU

yj)}.

∑j

vUj ≥ 0}.

vUi = max{0,1

n(1− δ)(yi −

δ

n − δ(κU − 1)

n∑j=κU

yj)}.

∑j

vUj ≥ 0}.

I Under majority rule, we have sufficient conditions foruniqueness but the equilibrium need not be unique.

I 3 players, δ = 0.9, y1 = 0.4, y2 = y3 = 1

I Equilibrium payoffs:

vM1 =1

3max{(0.4− δvMj ), δvM1 }+

2

3δvM1

vMj =1

3(1− δvM1 ) + µjδvMj , j = 2, 3

I Player 1 needs the approval of either player 2 or 3 but notboth: µ2 = µ3 =1/6 if 0.4 ≥ δvMj

I If there is no agreement then players 2 and 3 both receive theircontinuation payoffs with probability 1/3: µ2 = µ3 = 1/3 if0.4 < δvMj

I One equilibrium: vM1 = 0, vM2 = v

M3 = 0.476

I Another equilibrium: vM1 = 0.0533, vM2 = v

M3 = 0.3733

I 3 players, δ = 0.9, y1 = 0.4, y2 = y3 = 1

vM1 =1

3max{(0.4− δvMj ), δvM1 }+

2

3δvM1

vMj =1

3(1− δvM1 ) + µjδvMj , j = 2, 3

M3 = 0.476

M3 = 0.3733

I 3 players, δ = 0.9, y1 = 0.4, y2 = y3 = 1

vM1 =1

3max{(0.4− δvMj ), δvM1 }+

2

3δvM1

vMj =1

3(1− δvM1 ) + µjδvMj , j = 2, 3

M3 = 0.476

M3 = 0.3733

I 3 players, δ = 0.9, y1 = 0.4, y2 = y3 = 1

vM1 =1

3max{(0.4− δvMj ), δvM1 }+

2

3δvM1

vMj =1

3(1− δvM1 ) + µjδvMj , j = 2, 3

M3 = 0.476

M3 = 0.3733

I 3 players, δ = 0.9, y1 = 0.4, y2 = y3 = 1

vM1 =1

3max{(0.4− δvMj ), δvM1 }+

2

3δvM1

vMj =1

3(1− δvM1 ) + µjδvMj , j = 2, 3

M3 = 0.476

M3 = 0.3733

I 3 players, δ = 0.9, y1 = 0.4, y2 = y3 = 1

vM1 =1

3max{(0.4− δvMj ), δvM1 }+

2

3δvM1

vMj =1

3(1− δvM1 ) + µjδvMj , j = 2, 3

M3 = 0.476

M3 = 0.3733

I 3 players, δ = 0.9, y1 = 0.4, y2 = y3 = 1

vM1 =1

3max{(0.4− δvMj ), δvM1 }+

2

3δvM1

vMj =1

3(1− δvM1 ) + µjδvMj , j = 2, 3

M3 = 0.476

M3 = 0.3733

I 3 players, δ = 0.9, y1 = 0.4, y2 = y3 = 1

vM1 =1

3max{(0.4− δvMj ), δvM1 }+

2

3δvM1

vMj =1

3(1− δvM1 ) + µjδvMj , j = 2, 3

M3 = 0.476

M3 = 0.3733

I Unanimity outcome is at most as equal as the fundamentals:GU ≥ G .

I Under some conditions, unanimity outcome is strictly moreunequal than the fundamentals: If δ > y1ȳ where ȳ is the

average surplus, then GU > G .

I If (yn − y1) is “small” then there exists an equilibrium undermajority rule such that 0 = GM .

I Under some conditions, GM ≤ G ≤ GU .

I Gq is not monotonic in q (G 1 = G ).

Conjecture

I There exists a q̄ < n such that for any q < q̄ and q′ > q̄, andfor any equilibria of q-game and q′-game, we haveGq ≤ G ≤ Gq′ .

Conjecture

I There exists a q̄ < n such that for any q < q̄ and q′ > q̄, andfor any equilibria of q-game and q′-game, we haveGq ≤ G ≤ Gq′ .

Concluding remarks

I In the original Baron and Ferejohn (1989) environment,GM = G = GU . However, ex post, majority leads to moreinequality.

I In our environment, GM ≤ G ≤ GU . Ex post it depends.

Concluding remarks

some unpleasant bargaining arithmetics? › seminarpapers › et24062010.pdf · huly a eraslan and...

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