Some Unpleasant Bargaining Arithmetics? › seminarpapers › et24062010.pdf · Huly a Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics? The environment (example)

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 Antonio Merlo Some Unpleasant Bargaining Arithmetics? The environment (example)

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