Some Unpleasant Bargaining Arithmetics? - STICERD

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n − δ(n − 1) and vU. 1 = vU. 2 = ... = vU n−1 = 0. ▻ Verify that this is (an) equilibrium: Hülya Eraslan and Antonio Merlo. Some Unpleasant Bargaining Arithmetics ...
Some Unpleasant Bargaining Arithmetics? H¨ ulya Eraslan and Antonio Merlo

April 2010

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Introduction

H¨ ulya 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 have become a staple of political economy and have been used in numerous applications.

H¨ ulya 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 have become a staple of political economy and have been used in numerous applications.

I

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

H¨ ulya 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 have become a staple of political economy and have been used in numerous 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 different voting rules in a variety of bargaining models.

H¨ ulya 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 have become a staple of political economy and have been used in numerous 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 different voting rules in a variety of bargaining models.

I

Emphasis has been primarily on the efficiency of equilibrium outcomes.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

This paper

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

This paper

I

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

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

This paper

I

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

I

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

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

This paper

I

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

I

It is commonly believed that, contrary to majority rule, unanimity rule protects minorities from the possibility of expropriation 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 more unequal (or less equitable) than equilibrium outcomes under majority rule.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment

I

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

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment

I

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

I

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

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment

I

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

I

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

I

At most one project can be implemented.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment

I

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

I

If player i were to provide, the amount of surplus potentially available 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 to implement (if any), and how to distribute the available surplus.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (example)

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (example) I

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

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (example) I

There is a single project that needs to be completed, which entails 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 of production costs) is then available for distribution.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (example) I

There is a single project that needs to be completed, which entails 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 of production costs) is then available for distribution.

I

There are n ≥ 2 players, who are endowed with different technologies for producing x.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (example) I

There is a single project that needs to be completed, which entails 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 of production costs) is then available for distribution.

I

There are n ≥ 2 players, who are endowed with different technologies 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 .

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (example) I

There is a single project that needs to be completed, which entails 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 of production costs) is then available for distribution.

I

There are n ≥ 2 players, who are endowed with different technologies 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 of surplus potentially available for distribution would be yi = 1 − ci , with y1 ≤ y2 ... ≤ yn .

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (example) I

There is a single project that needs to be completed, which entails 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 of production costs) is then available for distribution.

I

There are n ≥ 2 players, who are endowed with different technologies 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 of surplus potentially available for distribution would be yi = 1 − ci , with y1 ≤ y2 ... ≤ yn .

I

The players have to collectively decide who (if anybody) will produce x, and how to distribute the surplus y generated in the event that production takes place. H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (con’d)

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (con’d)

I

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

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (con’d)

I

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

I

In the event that agreement is never reached, all players receive a payoff of zero.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (cont’d)

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (cont’d) I

We model the collective decision-making process as a bargaining problem.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (cont’d) I

I

We model the collective decision-making process as a bargaining problem. The protocol we consider is as follows:

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (cont’d) I

I

We model the collective decision-making process as a bargaining problem. 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 with probability 1/n.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (cont’d) I

I

We model the collective decision-making process as a bargaining problem. The protocol we consider is as follows: I

I

In each period, a player is randomly offered the possibility of submitting a proposal for completing the project with probability 1/n. The selected player j may then make a (binding) proposal specifying the way the surplus generated yj would be distributed among all the players if her project were chosen, or forego the opportunity.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (cont’d) I

I

We model the collective decision-making process as a bargaining problem. The protocol we consider is as follows: I

I

I

In each period, a player is randomly offered the possibility of submitting a proposal for completing the project with probability 1/n. The selected player j may then make a (binding) proposal specifying the way the surplus generated yj would be distributed among all the players if her project were chosen, or forego the opportunity. If a proposal is submitted, all players then vote (sequentially) on whether or not to approve it.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (cont’d) I

I

We model the collective decision-making process as a bargaining problem. The protocol we consider is as follows: I

I

I

I

In each period, a player is randomly offered the possibility of submitting a proposal for completing the project with probability 1/n. The selected player j may then make a (binding) proposal specifying the way the surplus generated yj would be distributed among all the players if her project were chosen, or forego the opportunity. If a proposal is submitted, all players then vote (sequentially) on whether or not to approve it. If q players vote in favor, then the proposal is implemented and the game ends. Otherwise, a new player is selected and the process repeats itself (possibly ad infinitum).

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (cont’d)

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (cont’d)

I

q ∈ {1, ..., n} specifies the voting rule (e.g., q = n denotes unanimity and q = (n + 1)/2 majority rule), and hence the game.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

The environment (cont’d)

I

q ∈ {1, ..., n} specifies the voting rule (e.g., q = n denotes unanimity and q = (n + 1)/2 majority rule), and hence the game.

I

i ∈ {1, ..., n} specifies each player’s ranking in the endowment distribution (with 1 denoting the least productive and n the most productive player).

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Comments on the model environment

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Comments on the model environment

I

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

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

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.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

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 additional dimensions of heterogeneity other than in the players’ technology endowments.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Notation and some useful definitions

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Notation and some useful definitions I

Restrict attention to stationary subgame perfect (SSP) equilibria.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Notation and some useful definitions I

I

Restrict attention to stationary subgame perfect (SSP) equilibria. Let v q = (v1q , ..., vnq ) denote the payoff vector generated by an SSP strategy profile in the q-game.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Notation and some useful definitions I

I

I

Restrict attention to stationary subgame perfect (SSP) equilibria. Let v q = (v1q , ..., vnq ) denote the payoff vector generated by an SSP strategy profile in the q-game. Let P 2 ni=1 iyi n+1 G = Pn − n i=1 yi n denote the Gini coefficient for the endowment distribution.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Notation and some useful definitions I

I

I

I

Restrict attention to stationary subgame perfect (SSP) equilibria. Let v q = (v1q , ..., vnq ) denote the payoff vector generated by an SSP strategy profile in the q-game. Let P 2 ni=1 iyi n+1 G = Pn − n i=1 yi n denote the Gini coefficient for the endowment distribution. Let P 2 ni=1 iviq n+1 q G = Pn q − n n i=1 vi denote the Gini coefficient for the equilibrium payoff distribution in the q-game.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Notation and some useful definitions I

I

I

I

I

Restrict attention to stationary subgame perfect (SSP) equilibria. Let v q = (v1q , ..., vnq ) denote the payoff vector generated by an SSP strategy profile in the q-game. Let P 2 ni=1 iyi n+1 G = Pn − n i=1 yi n denote the Gini coefficient for the endowment distribution. Let P 2 ni=1 iviq n+1 q G = Pn q − n n i=1 vi denote the Gini coefficient for the equilibrium payoff distribution in the q-game. Let G U and G M denote the Gini coefficients for the equilibrium payoff distribution under unanimity and majority rule. H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

A leading example

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

A leading example

I

Suppose c1 = c2 = ... = cn−1 =  and cn = 0, or y1 = y2 = ... = yn−1 = 1 −  and yn = 1.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

A leading example

I

I

Suppose c1 = c2 = ... = cn−1 =  and cn = 0, or y1 = y2 = ... = yn−1 = 1 −  and yn = 1. (1−)n For every  > 0, if δ > (1−)n+ , in the unique SSP equilibrium of the unanimity game

vnU =

1 U and v1U = v2U = ... = vn−1 = 0. n − δ(n − 1)

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

A leading example

I

I

Suppose c1 = c2 = ... = cn−1 =  and cn = 0, or y1 = y2 = ... = yn−1 = 1 −  and yn = 1. (1−)n For every  > 0, if δ > (1−)n+ , in the unique SSP equilibrium of the unanimity game

vnU =

I

1 U and v1U = v2U = ... = vn−1 = 0. n − δ(n − 1)

Verify that this is (an) equilibrium:

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

A leading example

I

I

Suppose c1 = c2 = ... = cn−1 =  and cn = 0, or y1 = y2 = ... = yn−1 = 1 −  and yn = 1. (1−)n For every  > 0, if δ > (1−)n+ , in the unique SSP equilibrium of the unanimity game

vnU =

I

1 U and v1U = v2U = ... = vn−1 = 0. n − δ(n − 1)

Verify that this is (an) equilibrium: I

Player n’s payoff is the solution to vnU =

H¨ ulya Eraslan and Antonio Merlo

1 n

+

n−1 U n δvn .

Some Unpleasant Bargaining Arithmetics?

A leading example

I

I

Suppose c1 = c2 = ... = cn−1 =  and cn = 0, or y1 = y2 = ... = yn−1 = 1 −  and yn = 1. (1−)n For every  > 0, if δ > (1−)n+ , in the unique SSP equilibrium of the unanimity game

vnU =

I

1 U and v1U = v2U = ... = vn−1 = 0. n − δ(n − 1)

Verify that this is (an) equilibrium: I I

U Player n’s payoff is the solution to vnU = n1 + n−1 n δvn . If i ≤ n − 1 is the proposer, he cannot obtain approval of 1 player n because 1 −  < δvnU = δ n−δ(n−1) .

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

A leading example

I

I

Suppose c1 = c2 = ... = cn−1 =  and cn = 0, or y1 = y2 = ... = yn−1 = 1 −  and yn = 1. (1−)n For every  > 0, if δ > (1−)n+ , in the unique SSP equilibrium of the unanimity game

vnU =

I

1 U and v1U = v2U = ... = vn−1 = 0. n − δ(n − 1)

Verify that this is (an) equilibrium: I I

I

U Player n’s payoff is the solution to vnU = n1 + n−1 n δvn . If i ≤ n − 1 is the proposer, he cannot obtain approval of 1 player n because 1 −  < δvnU = δ n−δ(n−1) . So player i’s payoff is 0.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

A leading example (cont’d)

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

A leading example (cont’d) I

For every  < majority game

δ 2+δ n−1 n

, in the unique SSP equilibrium of the

v1M = v2M = ... = vnM =

H¨ ulya Eraslan and Antonio Merlo

1 + (n − 1)(1 − ) = vM. n2

Some Unpleasant Bargaining Arithmetics?

A leading example (cont’d) I

For every  < majority game

δ 2+δ n−1 n

, in the unique SSP equilibrium of the

v1M = v2M = ... = vnM =

I

1 + (n − 1)(1 − ) = vM. n2

Verify that this is (an) equilibrium:

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

A leading example (cont’d) I

For every  < majority game

δ 2+δ n−1 n

, in the unique SSP equilibrium of the

v1M = v2M = ... = vnM =

I

1 + (n − 1)(1 − ) = vM. n2

Verify that this is (an) equilibrium: I

If i is the proposer, he offers δv M to q − 1 other players and keeps yi − δ(q − 1)v M .

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

A leading example (cont’d) I

For every  < majority game

δ 2+δ n−1 n

, in the unique SSP equilibrium of the

v1M = v2M = ... = vnM =

I

1 + (n − 1)(1 − ) = vM. n2

Verify that this is (an) equilibrium: I

I

If i is the proposer, he offers δv M to q − 1 other players and keeps yi − δ(q − 1)v M . Let µi denote the (endogenous) probability that i receives his continuation payoff when someone else is the proposer.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

A leading example (cont’d) I

For every  < majority game

δ 2+δ n−1 n

, in the unique SSP equilibrium of the

v1M = v2M = ... = vnM =

I

1 + (n − 1)(1 − ) = vM. n2

Verify that this is (an) equilibrium: I

I

I

If i is the proposer, he offers δv M to q − 1 other players and keeps yi − δ(q − 1)v M . Let µi denote the (endogenous) probability that i receives his continuation payoff when someone else is the proposer. So player i’s payoff must satisfy v M = n1 (yi − δ(q − 1)v M ) + µi δv M .

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

A leading example (cont’d) I

For every  < majority game

δ 2+δ n−1 n

, in the unique SSP equilibrium of the

v1M = v2M = ... = vnM =

I

1 + (n − 1)(1 − ) = vM. n2

Verify that this is (an) equilibrium: I

I

I

I

If i is the proposer, he offers δv M to q − 1 other players and keeps yi − δ(q − 1)v M . Let µi denote the (endogenous) probability that i receives his continuation payoff when someone else is the proposer. So player i’s payoff must satisfy v M = n1 (yi − δ(q − 1)v M ) + µi δv M . 1 In equilibrium, we must have µP i ≥ 0 and µi ≤ 1 − n for all i; n M M y1 − (q − 1)δv ≥ δv ; and i=1 µi = q − 1.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

A leading example (cont’d) I

For every  < majority game

δ 2+δ n−1 n

, in the unique SSP equilibrium of the

v1M = v2M = ... = vnM =

I

1 + (n − 1)(1 − ) = vM. n2

Verify that this is (an) equilibrium: I

I

I

I

I

If i is the proposer, he offers δv M to q − 1 other players and keeps yi − δ(q − 1)v M . Let µi denote the (endogenous) probability that i receives his continuation payoff when someone else is the proposer. So player i’s payoff must satisfy v M = n1 (yi − δ(q − 1)v M ) + µi δv M . 1 In equilibrium, we must have µP i ≥ 0 and µi ≤ 1 − n for all i; n M M y1 − (q − 1)δv ≥ δv ; and i=1 µi = q − 1. Since  < 2+δδn−1 , we can find µ1 , . . . , µn satisfying these. n

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

A leading example (cont’d)

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

A leading example (cont’d)

I

In this environment 0 = GM < G < GU.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Intuition

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Intuition Under unanimity rule:

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Intuition Under unanimity rule: I

Equilibrium is efficient.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Intuition Under unanimity rule: I

Equilibrium is efficient.

I

If players are patient enough, efficiency requires agreement only when player n proposes.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Intuition Under unanimity rule: I

Equilibrium is efficient.

I

If players are patient enough, efficiency requires agreement only 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 never propose. Hence, their equilibrium payoff must be zero.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Intuition Under unanimity rule: I

Equilibrium is efficient.

I

If players are patient enough, efficiency requires agreement only 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 never propose. Hence, their equilibrium payoff must be zero.

Under majority rule:

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Intuition Under unanimity rule: I

Equilibrium is efficient.

I

If players are patient enough, efficiency requires agreement only 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 never propose. Hence, their equilibrium payoff must be zero.

Under majority rule: I

All proposals are accepted (it only takes a majority in favor to approve a project).

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Intuition Under unanimity rule: I

Equilibrium is efficient.

I

If players are patient enough, efficiency requires agreement only 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 never propose. Hence, their equilibrium payoff must be zero.

Under majority rule: I

All proposals are accepted (it only takes a majority in favor to approve a project).

I

Player n loses his advantage.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Intuition Under unanimity rule: I

Equilibrium is efficient.

I

If players are patient enough, efficiency requires agreement only 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 never propose. Hence, their equilibrium payoff must be zero.

Under majority rule: I

All proposals are accepted (it only takes a majority in favor to approve a project).

I

Player n loses his advantage.

I

This “egalitarian” force generates regression toward the mean that equalizes expected equilibrium payoffs.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results

I

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

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results

I

I

If there is agreement is reached when the cake size is yi , then there is agreement when the cake size is yi+1 . q Payoffs are monotone: viq ≤ vi+1 .

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results

I

I

I

If there is agreement is reached when the cake size is yi , then there is agreement when the cake size is yi+1 . q Payoffs are monotone: viq ≤ vi+1 .

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

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d)

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d)

I

Under unanimity rule, there exists a unique SSP payoff:

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d)

I

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

n X δ 1 (yi − = max{0, yj )}. n(1 − δ) n − δ(κU − 1) U j=κ

where κU = min{i : yi − δ

X

vjU ≥ 0}.

j

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d)

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d) I

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

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d) I

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

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

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d) I

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

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

I

Equilibrium payoffs: 2 1 v1M = max{(0.4 − δvjM ), δv1M } + δv1M 3 3 1 vjM = (1 − δv1M ) + µj δvjM , j = 2, 3 3

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d) I

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

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

I

Equilibrium payoffs: 2 1 v1M = max{(0.4 − δvjM ), δv1M } + δv1M 3 3 1 vjM = (1 − δv1M ) + µj δvjM , j = 2, 3 3

I

Player 1 needs the approval of either player 2 or 3 but not both: µ2 = µ3 =1/6 if 0.4 ≥ δvjM

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d) I

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

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

I

Equilibrium payoffs: 2 1 v1M = max{(0.4 − δvjM ), δv1M } + δv1M 3 3 1 vjM = (1 − δv1M ) + µj δvjM , j = 2, 3 3

I

Player 1 needs the approval of either player 2 or 3 but not both: µ2 = µ3 =1/6 if 0.4 ≥ δvjM

I

If there is no agreement then players 2 and 3 both receive their continuation payoffs with probability 1/3: µ2 = µ3 = 1/3 if 0.4 < δvjM

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d) I

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

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

I

Equilibrium payoffs: 2 1 v1M = max{(0.4 − δvjM ), δv1M } + δv1M 3 3 1 vjM = (1 − δv1M ) + µj δvjM , j = 2, 3 3

I

Player 1 needs the approval of either player 2 or 3 but not both: µ2 = µ3 =1/6 if 0.4 ≥ δvjM

I

If there is no agreement then players 2 and 3 both receive their continuation payoffs with probability 1/3: µ2 = µ3 = 1/3 if 0.4 < δvjM

I

One equilibrium: v1M = 0, v2M = v3M = 0.476

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d) I

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

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

I

Equilibrium payoffs: 2 1 v1M = max{(0.4 − δvjM ), δv1M } + δv1M 3 3 1 vjM = (1 − δv1M ) + µj δvjM , j = 2, 3 3

I

Player 1 needs the approval of either player 2 or 3 but not both: µ2 = µ3 =1/6 if 0.4 ≥ δvjM

I

If there is no agreement then players 2 and 3 both receive their continuation payoffs with probability 1/3: µ2 = µ3 = 1/3 if 0.4 < δvjM

I

One equilibrium: v1M = 0, v2M = v3M = 0.476

I

Another equilibrium: v1M = 0.0533, v2M = v3M = 0.3733 H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d)

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d)

I

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

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d)

I

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

I

Under some conditions, unanimity outcome is strictly more unequal than the fundamentals: If δ > yy¯1 where y¯ is the average surplus, then G U > G .

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d)

I

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

I

Under some conditions, unanimity outcome is strictly more unequal than the fundamentals: If δ > yy¯1 where y¯ is the average surplus, then G U > G .

I

If (yn − y1 ) is “small” then there exists an equilibrium under majority rule such that 0 = G M .

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d)

I

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

I

Under some conditions, unanimity outcome is strictly more unequal than the fundamentals: If δ > yy¯1 where y¯ is the average surplus, then G U > G .

I

If (yn − y1 ) is “small” then there exists an equilibrium under majority rule such that 0 = G M .

I

Under some conditions, G M ≤ G ≤ G U .

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

General results (cont’d)

I

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

I

Under some conditions, unanimity outcome is strictly more unequal than the fundamentals: If δ > yy¯1 where y¯ is the average surplus, then G U > G .

I

If (yn − y1 ) is “small” then there exists an equilibrium under majority rule such that 0 = G M .

I

Under some conditions, G M ≤ G ≤ G U .

I

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

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Conjecture

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Conjecture

I

There exists a q¯ < n such that for any q < q¯ and q 0 > q¯, and for any equilibria of q-game and q 0 -game, we have 0 Gq ≤ G ≤ Gq .

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Concluding remarks

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Concluding remarks

I

In the original Baron and Ferejohn (1989) environment, G M = G = G U . However, ex post, majority leads to more inequality.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?

Concluding remarks

I

In the original Baron and Ferejohn (1989) environment, G M = G = G U . However, ex post, majority leads to more inequality.

I

In our environment, G M ≤ G ≤ G U . Ex post it depends.

H¨ ulya Eraslan and Antonio Merlo

Some Unpleasant Bargaining Arithmetics?