One-dimensional bargaining with unanimity rule

The paper examines bargaining over a one--dimensional set of social states, with a unanimity acceptance rule. We consider a class of delta-equilibria, i.e. subgame perfect equilibria in stationary strategies that are free of coordination failures in the response stage.We show that along any sequence of delta-equilibria, as delta converges to one, the proposal of each player converges to the same limit. The limit, called the bargaining outcome, is uniquely determined by the set of players, the recognition probabilities, and the utility functions, and it is independent of the choice of the sequence. We characterize the bargaining outcome as a unique solution of a characteristic equation.mathematical economics;

Sequential Share Bargaining

This paper presents a new extension of the Rubinstein-St°ahl bargaining model to the case with n players, called sequential share bargaining. The bargaining protocol is natural and has as its main feature that the players’ shares in the cake are determined sequentially. The bargaining protocol requires unanimous agreement for proposals to be implemented. Unlike all existing bargaining protocols with unanimous agreement, the resulting game has unique subgame perfect equilibrium utilities for any value of the discount factor. In equilibrium, agreement is reached immediately. The results are therefore qualitatively the same as in the two player case. The result builds on an analysis of so-called one-dimensional bargaining problems. We show that also one-dimensional bargaining problems have unique subgame perfect equilibrium utilities for any value of the discount factor, and that also in one-dimensional bargaining problems agreement is reached immediately.microeconomics ;

One-dimensional Bargaining with Markov Recognition Probabilities

We study a process of bargaining over social outcomes represented by points in theunit interval. The identity of the proposer is determined by a general Markov process and the acceptance of a proposal requires the approval of it by all the players. We show that for every value of the discount factor below one the subgame perfect equilibrium in stationary strategies is essentially unique and equal to what we call the bargaining equilibrium. We provide a general characterization of the bargaining equilibrium. We consider next the asymptotic behavior of the equilibrium proposals when the discount factor approaches one. We give a complete characterization of the limit of the equilibrium proposals. We show that the limit equilibrium proposals of all the players are the same if the proposer selection process satisfies an irreducibility condition, or more generally, has a unique absorbing set. In general, the limit equilibrium proposals depend on the partition of the set of players in absorbing sets and transient states of the proposer selection process. We fully characterize the limit equilibrium proposals as the unique generalized fixed point of a particular function.This function depends in a simple way on the stationary distribution related to the proposer selection process. We compare the proposal selected according to our bargaining model to the one corresponding to the median voter theorem.microeconomics ;

One-dimensional bargaining with a general voting rule

We study a model of multilateral bargaining over social outcomes represented by points in the unit interval. An acceptance or rejection of a proposal is determined by a voting rule as represented by a collection of decisive coalitions. The focus of the paper is on the asymptotic behavior of subgame perfect equilibria in stationary strategies as the discount factor goes to one. We show that, along any sequence of stationary subgame perfect equilibria, as the discount factor goes to one, the social acceptance set collapses to a point. This point, called the bargaining outcome, is independent of the sequence of equilibria and is uniquely determined by the set of players, the utility functions, the recognition probabilities, and the voting rule. The central result of the paper is a characterization of the bargaining outcome as a unique zero of the characteristic equation.microeconomics ;

Multilateral negotiations over climate change policy

Negotiations in the real world have many features which tend to be ignored inpolicy modeling. They are often multilateral, involving many negotiating parties with preferences over outcomes that can differ substantially. They are also often multi-dimensional,in the sense that several policies are negotiated over simultaneously. Trade negotiations are a prime example, as are negotiations over environmental policies toabate carbon dioxide. We demonstrate how one can formally model this type of negotiation process. We use a policy-oriented computable general equilibrium model to generate preference functions which are then used in a formal multilateral bargaining game. The case study is to climate change policy, but the main contribution is to demonstrate how one can integrate formal economic models of the impacts of policies with formal bargaining models of the negotiations over those policies.CGE, bilateral bargaining, CO2, Climate Change

A note on the take-it-or-leave-it bargaining procedure with double moral hazard and risk neutrality

In this note we study a take-it-or-leave-it bargaining procedure between two risk neutral individuals engaged in the joint stochastic production of a commodity. Each individual has to exert effort, that is, to provide a one-dimensional input which is unobserved to the other individual. The output-contingent sharing rule is constrained to lead to nonnegative consumption for both individuals, a limited liability constraint. The individuals enter joint production in one of two possible occupations, or tasks, the p-agent and the a-agent, which differ in their incentive intensity. Hence, incentives are asymmetric. The p-agent makes a take-it-or-leave-it offer to the a-agent, and has therefore all the contractual power, modulo providing the a-agent an exogenously given reservation utility.contract theory; bargaining theory

Judicial precedent as a dynamic rationale for axiomatic bargaining theory

Axiomatic bargaining theory (e.g., Nash's theorem) is static. We attempt to provide a dynamic justification for the theory. Suppose a Judge or Arbitrator must allocate utility in an (infinite) sequence of two-person problems; at each date, the Judge is presented with a utility possibility set in the nonnegative orthant in two-dimensional Euclidean space. He/she must choose an allocation in the set, constrained only by Nash's axioms, in the sense that a penalty is paid if and only if a utility allocation is chosen at date T which is inconsistent, according to one of the axioms, with a utility allocation chosen at some earlier date. Penalties are discounted with t, and the Judge chooses any allocation, at a given date, that minimizes the penalty he/she pays at that date. Under what conditions will the Judge's chosen allocations converge to the Nash allocation over time? We answer this question for three canonical axiomatic bargaining solutions: Nash's, Kalai-Smorodinsky's, and the 'egalitarian' solution, and generalize the analysis to a broad class of axiomatic models.Axiomatic bargaining theory, judicial precedent, dynamic foundations, Nash's bargaining solution

Bargaining one-dimensional policies and the efficiency of super majority rules

We consider negotiations selecting one-dimensional policies. Individuals have single-peaked preferences, and they are impatient. Decisions arise from a bargaining game with random proposers and (super) majority approval, ranging from the simple majority up to unanimity. The existence and uniqueness of stationary subgame perfect equilibrium is established, and its explicit characterization provided. We supply an explicit formula to determine the unique alternative that prevails, as impatience vanishes, for each majority. As an application, we examine the efficiency of majority rules. For symmetric distributions of peaks unanimity is the unanimously preferred majority rule. For asymmetric populations rules maximizing social surplus are characterized

Bargaining Foundations of the Median Voter Theorem

We provide game-theoretic foundations for the median voter theorem in a one-dimensional bargaining model based on Baron and Ferejohn’s (1989) model of distributive politics. We prove that, as the agents become arbitrarily patient, the set of proposals that can be passed in any subgame perfect equilibrium collapses to the median voter’s ideal point. While we leave the possibility of some delay, we prove that the agents’ equilibrium continuation payoffs converge to the utility from the median, so that delay, if it occurs, is inconsequential. We do not impose stationarity or any other refinements. Our result counters intuition based on the folk theorem for repeated games, and it contrasts with the known result for the distributive bargaining model that, as agents become patient, any division of the dollar can be supported as a subgame perfect equilibrium outcome.

Regularity of Pure Strategy Equilibrium Points in a Class of Bargaining Games

For a class of n-player (n ? 2) sequential bargaining games with probabilistic recognition and general agreement rules, we characterize pure strategy Stationary Subgame Perfect (PSSP) equilibria via a finite number of equalities and inequalities. We use this characterization and the degree theory of Shannon, 1994, to show that when utility over agreements has negative definite second (contingent) derivative, there is a finite number of PSSP equilibrium points for almost all discount factors. If in addition the space of agreements is one-dimensional, the theorem applies for all SSP equilibria. And for oligarchic voting rules (which include unanimity) with agreement spaces of arbitrary finite dimension, the number of SSP equilibria is odd and the equilibrium correspondence is lower-hemicontinuous for almost all discount factors. Finally, we provide a sufficient condition for uniqueness of SSP equilibrium in oligarchic games.Local Uniqueness of Equilibrium, Regularity, Sequential Bargaining.