some unpleasant bargaining arithmetics? › seminarpapers › et24062010.pdf · huly a eraslan and...
Post on 02-Feb-2021
12 Views
Preview:
TRANSCRIPT
-
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.
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.
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.
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.
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.
Hülya 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 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.
Hülya 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 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.
Hülya 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 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.
Hülya 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 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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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).
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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).
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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 ofsubmitting 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).
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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).
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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).
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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).
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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).
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
The environment (cont’d)
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).
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
The environment (cont’d)
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).
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
The environment (cont’d)
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).
Hülya 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 additionaldimensions of heterogeneity other than in the players’technology endowments.
Hülya 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 additionaldimensions of heterogeneity other than in the players’technology endowments.
Hülya 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 additionaldimensions of heterogeneity other than in the players’technology endowments.
Hülya 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 additionaldimensions of heterogeneity other than in the players’technology endowments.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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 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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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 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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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 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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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
∑ni=1 µi = q − 1.
I Since � < δ2+δ n−1n
, we can find µ1, . . . , µn satisfying these.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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 � < δ2+δ n−1n
, we can find µ1, . . . , µn satisfying these.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
A leading example (cont’d)
I In this environment
0 = GM < G < GU .
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
A leading example (cont’d)
I In this environment
0 = GM < G < GU .
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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}.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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}.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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}.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
General results (cont’d)
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
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
General results (cont’d)
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
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
General results (cont’d)
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
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
General results (cont’d)
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
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
General results (cont’d)
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
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
General results (cont’d)
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
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
General results (cont’d)
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
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
General results (cont’d)
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
Hülya 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 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 ).
Hülya 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 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 ).
Hülya 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 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 ).
Hülya 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 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 ).
Hülya 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 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 ).
Hülya 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 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 ).
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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′ .
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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′ .
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
-
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.
Hülya Eraslan and Antonio Merlo Some Unpleasant Bargaining Arithmetics?
top related