Bargaining with history-dependent preferences

Two impatient players bargain over a pie of size one according to the infinite alternating-offers procedure. Players’ payoffs depend not only on the outcome but also on the process of the bargaining. Specifically, they prefer impasse to any agreement that gives them lower discounted utility than would have been derived from accepting earlier offers. We characterize the essentially unique subgame perfect equilibrium path, which consists of gradual concessions. The more patient players are, the longer it takes them to reach an agreement. When players become infinitely patient, the efficiency loss is substantial, yet the equilibrium division converges to the Nash solution.     

Coase theorem, complexity and transaction costs

This paper, by introducing complexity considerations, provides a dynamic foundation for the Coase theorem and highlights the role of transaction costs in generating inefficient bargaining/negotiation outcomes. We show, when the players have a preference for less complex strategies, the Coase theorem holds in negotiation models with repeated surplus and endogenous disagreement payoffs if and only if there are no transaction costs. Specifically, complexity considerations select only efficient equilibria in these models without transaction costs while every equilibrium induces perpetual disagreement and inefficiency with transaction costs. We also show the latter is true in the Rubinstein bargaining model with transaction costs.     

INEQUALITY AVERSION CAUSES EQUAL OR UNEQUAL DIVISION IN ALTERNATING‐OFFER BARGAINING

A solution to Rubinstein (1982)'s open‐ended, alternating‐offer bargaining problem for two equally patient bargainers who exhibit similar degrees of inequality aversion is presented. Inequality‐averse bargainers may experience envy if they are worse off, and guilt if they are better off, but they still reach agreement in the first period under complete information. If the guilt felt is strong, then the inequality‐averse bargainers split a pie of size one equally regardless of their degree of envy. If the guilt experienced is weak, then the agreed split is tilted away from the Rubinstein division towards a more unequal split whenever the degree of envy is smaller than the discounted degree of guilt. Envy and weak guilt have opposite effects on the equilibrium division of the pie, and envy has a greater marginal impact than weak guilt. Equally inequality‐averse bargainers agree on the Rubinstein division if the degree of envy equals the discounted degree of guilt. As both bargainers' sensation of inequality aversion diminishes, the bargaining outcome converges to the Rubinstein division.     

Delays and partial agreements in multi-issue bargaining

We model a situation in which two players bargain over two pies, one of which can only be consumed starting at a future date. Suppose the players value the pies asymmetrically: one player values the existing pie more than the future one, while the other player has the opposite valuation. We show that players may consume only a fraction of the existing pie in the first period, and then consume the remainder of it, along with the second pie, at the date at which the second pie becomes available. Thus, our model features a special form of bargaining delay, in which agreements take place in multiple stages. Such partial agreements arise when players are patient enough, when they expect the second pie to become available soon, and when the asymmetry in their valuations is large enough.     

Bargaining under a deadline: evidence from the reverse ultimatum game

We study a “reverse” ultimatum game, in which proposers have multiple chances to offer responders a division of some fixed pie. The game ends if the responder accepts an offer, or if, following a rejection, the proposer decides not to make a better offer. The unique subgame perfect equilibrium gives the proposer the minimum possible payoff. Nevertheless, the experimental results are not too different from those of the standard ultimatum game, although proposers generally receive slightly less than half of the surplus.We use the reverse ultimatum game to study deadlines experimentally. With a deadline, the subgame perfect equilibrium prediction is that the proposer gets the entire surplus.Deadlines are used strategically to influence the outcome, and agreements are reached near the deadline. Strategic considerations are evident in the differences in observed behavior between the deadline and no deadline conditions, even though agreements are substantially less extreme than predicted by perfect equilibrium.     

The Formateurs’ Role in Government Formation

Following an election, Heads of State typically choose a formateur to report back a prospective government by some deadline. If this formateur fails then another party leader is chosen on the same terms. We model this mechanism as a bargaining game, in which a single party leader has exclusive access to the Head of State, but does not have a monopoly on making proposals. We characterize the unique equilibrium of the game in which all parties are symmetric and parties split a fixed-size pie. We also show that supra-minimal governments may form when parties are asymmetrically sized: a result consistent with the evidence on government formation in countries which use a formateur mechanism. We are grateful to Maria Montero, Rick Thomas and Paul Warwick for valuable conversations, and to seminar participants in UBC and Warwick.     

Bargaining with revoking costs

A simple two stage bilateral bargaining game is analyzed. The players simultaneously demand shares of a unit size pie. If the demands add up to more than one, the players simultaneously choose whether to stick to their demand or accept the other?s offer. While both parties sticking to their offers leads to an impasse, accepting a lower share than the original demand is costly for each party. The set of pure strategy subgame perfect equilibria of the game is characterized for continuously differentiable payoff and cost functions, strictly increasing in the pie share and the amount conceded, respectively. Higher cost functions are shown to improve bargaining power. The limit equilibrium prediction of the model, as the cost functions are made arbitrarily high, selects a unique equilibrium in the Nash Demand Game that corresponds to a Proportional Bargaining Solution of Kalai (1977).     

Disapproval aversion or inflated inequity acceptance? The impact of expressing emotions in ultimatum bargaining

Past experimental research has shown that when rating systems are available, buyers are more generous in accepting unfair offers in ultimatum bargaining. However, it also suggests that, under these conditions, sellers behave more fairly to avoid receiving negative feedback. This paper experimentally investigates which effect is stronger with the use of a rating system: buyers’ inflated inequity acceptance or sellers’ disapproval aversion. We explore this question by varying the information condition on the buyers’ side. Our experiment shows that in a setup where the size of the pie is common knowledge for both buyers and sellers, when a rating system is present, the sellers exhibit disapproval aversion but the buyers do not display greater acceptance of inequity. By contrast, when only sellers are aware of the size of the pie, sellers behave aggressively to exploit buyers and their behavior does not change in the presence of a rating system; however, buyers display greater acceptance of inequity when a rating system is present. We discuss how these results can be explained by a theoretical model that includes sellers’ social disapproval aversion and buyers’ disappointment aversion in addition to the players’ inequality aversion.     

Delayed agreements and nonexpected utility

The unexplained occurrence of inefficient delays in reaching agreement is known in the economics literature as the “Hicks paradox.” This paper describes a strategic situation in which players may play a simultaneous move game either before or after a move of Nature. The structure is such that if the players were expected utility maximizers, they would be indifferent over the order of play. However, if at least one of the players is a nonexpected utility maximizer, for example, if player one has preferences over lotteries which exhibit betweenness and fanning out, such a player may strictly prefer to wait before playing the game. If both players exhibit fanning out and betweenness, then there exist games in which both prefer to delay agreement.     

N-PERSON NASH BNARGAINING WITH VARIABEL THREATS*

We consider two models of n-person bargaining problems with the endogenous determination of disagreement points. In the first model, which is a direct extension of Nash's variable threat bargaining model, the disagreement point is determined as an equilibrium threat point. In the second model, the disagreement point is given as a Nash equilibrium of the underlying noncooperative game. These models are formulated as extensive games, and axiomatizations of solutions are given for both models. It is argued that for games with more than two players, the first bargaining model does not preserve some important properties valid for two-person games, e.g., the uniqueness of equilibrium payoff vector. We also show that when the number of players is large, any equilibrium threat point becomes approximately a Nash equilibrium in the underlying noncooperative game, and vice versa. This result suggests that the difference between the two models becomes less significant when the number of players is large.     

Learning by matching patterns

We analyze a model of repeated play between two boundedly rational agents. In each stage each player recalls the outcomes from the most recent few rounds, calculates the distribution of its opponent's past reactions to this outcome pattern, and then optimizes myopically against this distribution. If both players use the same pattern length then the limit points of pattern matching are in the convex hull of the limit points of fictitious play. Thus if fictitious play converges into the set of Nash equilibria then pattern matching converges into the convex hull of Nash equilibria. If the players use different pattern lengths, the more sophisticated player may, but does not generally, succeed in playing as if it could perfectly predict its opponent's play.     

Multi-player contests with asymmetric information

We consider imperfectly discriminating, common-value, all-pay auctions (or contests) in which some players know the value of the prize, others do not. We show that if the prize is always of positive value, then all players are active in equilibrium. If the prize is of value zero with positive probability, then there is some threshold number of informed players such that if there are less, all uninformed players are active, and otherwise all uninformed players are inactive.     

Undominated Nash Implementation with Collusion and Renegotiation

This article looks at implementation in economic environments when agents have perfect information about the state of the world, but cannot commit not to renegotiate bad outcomes or to collude against each other. If renegotiation satisfies a weak condition of disagreement point monotonicity, then any Pareto-efficient social choice function can be implemented if there are at least three agents who play undominated Nash equilibria. The mechanism does not use modulo or integer games, has no bad mixed strategy equilibria, and is “bounded.”Journal of Economic LiteratureClassification Number: D71.     

Multi-activity contests

In many contests, players can influence their chances of winning through multiple activities or “arms”. We develop a model of multi-armed contests and axiomatize its contest success function. We then analyze the outcomes of the multi-armed contest and the effects of allowing or restricting arms. Restricting an arm increases total effort directed to other arms if and only if restricting the arm balances the contest. Restricting an arm tends to reduce rent dissipation because it reduces the discriminatory power of the contest. But it also tends to increase rent dissipation if it balances the contest. Less rent is dissipated if an arm is restricted as long as no player is excessively stronger than the other with that arm. If players are sufficiently symmetric in an arm, both players are better off if that arm is restricted. Nevertheless, players cannot agree to restrict the arm if their costs of using the arm are sufficiently low.     

Folk Theorem For Bilateral Bargaining with Vector Endowments

We analyse a bargaining game of two players on the division of the sum of their vector endowments, with alternating proposals and discounting of single period utilities. The pair of endowments is not weakly Pareto‐efficient. Until they reach an agreement, each of the parties can withdraw from bargaining and keep their endowments. Any strictly individually rational division of the sum of endowments can emerge in some subgame perfect equilibrium if discount factors are close enough to one. Each subgame perfect equilibrium, in which the parties’ decisions do not depend on past rejected proposals, leads to a weakly Pareto‐efficient agreement in the first period.     

Strategic step-by-step negotiation

We analyze a specific type of negotiation process where parties proceed in stages taking into consideration that negotiation may end prematurely with an inefficient agreement. Parties negotiate only one increasing pie, thereby avoiding inefficiencies which are typical for issue-by-issue negotiations. For ann-stage game, we prove the existence of a unique subgame-perfect equilibrium. We then show that step-by-step negotiation can only improve players' expected payoffs if negotiation in stages reduces the difference between their equilibrium offers. For this to occur, however, the risk of negotiation must be affected by the agenda of the bargaining problem.     

Size inequality,coordination externalities and international trade agreements

Developing countries now account for a significant fraction of world trade and two-thirds of the membership of the World Trade Organization (WTO). However, many are still individually small and thus have a limited ability to bilaterally extract and enforce trade concessions from larger developed economies even though as a group they would be able to do so. We show that this coordination externality generates asymmetric outcomes under agreements that rely on bilateral threats of trade retaliation – such as the WTO – but not under agreements extended to include certain financial instruments. In particular, we find that an extended agreement generates improvements in global efficiency and equity if it includes the exchange of bonds prior to trading but not if it relies solely on ex post fines. Moreover, a combination of bonds and fines generates similar improvements even if small countries are subject to financial constraints that prevent them from posting bonds.     

Bargaining and efficiency in networks

We study an infinite horizon game in which pairs of players connected in a network are randomly matched to bargain over a unit surplus. Players who reach agreement are removed from the network without replacement. The global logic of efficient matchings and the local nature of bargaining, in combination with the irreversible exit of player pairs following agreements, create severe hurdles to the attainment of efficiency in equilibrium. For many networks all Markov perfect equilibria of the bargaining game are inefficient, even as players become patient. We investigate how incentives need to be structured in order to achieve efficiency via subgame perfect, but non-Markovian, equilibria. The analysis extends to an alternative model in which individual players are selected according to some probability distribution, and a chosen player can select a neighbor with whom to bargain.     

Challenging the enforcement of environmental regulation

If a firm can contest the enforcement of an environmental regulation, neither increasing the probability nor severity of the fine will guarantee a reduction in a firm's illegally dumped waste. A policy that can unambiguously decrease illegal dumping is lowering the cost of legal disposal. This result occurs because the use of monitoring and fines to increase the probability or severity of enforcement triggers investment to evade enforcement, while a decrease in the costs of legal disposal does not. Investment in the resources to evade enforcement decreases the attractiveness of monitoring by significantly increasing the costs of environmental audits, administrative hearings, and judicial procedures. This occurs even with a high degree of regulator information about the firm's cost structure and no monitoring errors. In addition, if the regulator can only imperfectly monitor a firm's behavior so the firm can be accused of another firm's behavior, observable commitment to challenge enforcement will lead to overinvestment in resources to evade enforcement, an increased level of illegal dumping, and an overall increase in total costs relative to the unobservable case.