Fair Bargains: Distributive Justice and Nash Bargaining Theory

The Suppes–Sen dominance relation is a weak criterion of impartiality in distributive justice. I propose its application to Nash bargaining theory. The Nash Bargaining Solution (NBS) is characterized by replacing the controversial Independence of Irrelevant Alternatives axiom with an axiom embodying the Suppes–Sen principle. This allows a clear interpretation of the NBS as a fair arbitration scheme. In addition, the proposed characterization is more robust than the standard one with respect to variations in the domain of bargaining problems.     

Bargaining with Imperfect Commitment

We study the effect of imperfect commitment in noncooperative two-person bargaining games. By establishing the reputation for being stubborn, a player sometimes commits to her initial demand, becoming unable to change her demands or to accept an inferior offer from her opponent. When the probability of being stubborn is small, the set of equilibria is shown to be small and agreement may be reached immediately despite the possibility of stubbornness. A player has greater bargaining power when she is more patient and/or is more likely to be stubborn. Journal of Economic Literature Classification Number: C78.     

Identifying Human-Capital Externalities: Theory with Applications

The identification of aggregate human-capital externalities is still not fully understood. The existing ( Mincerian ) approach confounds positive externalities with wage changes due to a downward sloping demand curve for human capital. As a result, the Mincerian approach yields positive externalities even when wages equal marginal social products. We propose an approach that identifies human-capital externalities, whether or not aggregate demand for human capital slopes downward. Another advantage of our approach is that it does not require estimates of the individual return to human capital. Applications to U.S. cities and states between 1970 and 1990 yield no evidence of significant average-schooling externalities.     

Bargaining with subjective mixtures

This paper reconsiders the Bargaining Problem of Nash (Econometrica 28:155–162, 1950). I develop a new approach, Conditional Bargaining Problems, as a framework for measuring cardinal utility. A Conditional Bargaining Problem is the conjoint extension of a Bargaining Problem, conditional on the fact that the individuals have agreed on a “measurement event”. Within this context, Subjective Mixture methods are especially powerful. These techniques are used to characterise versions of the Nash and the Kalai–Smorodinsky solutions. This approach identifies solutions based only on the individuals’ tastes for the outcomes. It is therefore possible to do Bargaining theory in almost complete generality. The results apply to Biseparable preferences, so are valid for almost all non-expected utility models currently used in economics.     

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).     

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.     

Bargaining with Rent Seekers

Consider a rent‐seeking game, which has government bargain with firms over dividing the rents. In period 1, each firm can invest to increase the probability that the rent will appear. In period 2, the parties bargain. In equilibrium, though firms will invest more than the socially optimal level, rent‐seeking expenditures may be low. Firms that collude to restrict investment maximize joint profits by investing at a positive, non‐infinitesimal level, and restrict investment even if the cost of rent‐seeking effort is zero.     

Bargaining with endogenous information

This paper analyses information acquisition in ultimatum bargaining with common values. Because of an endogenous lemons problem the equilibrium payoffs of the agents are non-monotonic in the information cost. The mere possibility of information acquisition can cause no trade although the agents maintain symmetric information in equilibrium and the gain from trade is common knowledge. The agent responding to a take-it-or-leave-it offer may capture some or even the full trading surplus in a perfect Bayesian equilibrium. The implications for sequential bargaining are discussed.     

Multi-issue Bargaining with Endogenous Agenda

The first part of this paper shows that in a noncooperative bargaining model with alternating offers and time preferences the timing of issues (the agenda) matters even if players become arbitrarily patient. This result raises the question of which agenda should come up endogenously when agents bargain over a set of unrelated issues. It is found that simultaneous bargaining over “packages” should be a prevailing phenomenon, but we also point to the possibility of multiple equilibria involving even considerable delay. Journal of Economic Literature Classification Number: C78.     

Bargaining Outcomes with Double-Offer Arbitration

Increasingly, arbitration is becoming used to resolve bargaining disputes in a variety of settings. Reducing dispute rates is often listed as a main goal in designing arbitration mechanisms. Conventional arbitration and final-offer arbitration are two commonly used procedures, but theoretical examinations of these arbitration procedures show that disputants’ final bargaining positions do not converge and disagreement is likely. This article contains results from a set of experiments designed to compare bargaining outcomes under the two commonly used arbitration procedures with outcomes under an innovative procedure called “double-offer” arbitration (Zeng et al., 1996). This procedure requires that disputants make two final offers at impasse: a primary and a secondary offer. The arbitrator evaluates the pairs of offers using a linear criterion function, and theory suggests the secondary offers converge to the median of the arbitrator’s preferred settlement distribution. Because the procedure’s rules are that convergence of offers generates a settlement at those offers, this theoretical convergence result implies that arbitration is not needed in the end. Experimental results indicate that dispute rates in double-offer arbitration are, on average, about the same as dispute rates in conventional arbitration. However, other results show reason to favor double-offer arbitration. Specifically, in repeated bargaining, there is concern over whether use of an arbitration procedure becomes addictive and makes bargainers more likely to use the procedure in the future-a “narcotic effect.” The data show that double-offer arbitration is non-addictive, whereas both conventional and final-offer arbitration are.     

Alternating-Offer Bargaining with Two-Sided Incomplete Information

I study alternating-offer bargaining games with two-sided incomplete information about the players' discount rates. For both perfect Bayesian equilibrium and a rationalizability-style notion, I characterize the set of expected payoffs which may arise in the game. I also construct bounds on agreements that may be made. The set of expected payoffs is easy to compute and incorporate into applied models. My main result is a full characterization of the set of perfect Bayesian equilibrium payoffs for games in which the distribution over the players' discount rates is of wide support, yet is in a weak sense close to a point mass distribution. I prove a lopsided convergence result: each player cannot gain from a slight chance that she is a strong type, but the player can suffer greatly if there is a slight chance that she is a weak type.     

Bargaining with a possibly committed seller

We consider negotiations with an open time horizon where a buyer has private information about his valuation and does not know whether the seller is committed to the advertised price. This setting combines two common specifications made in the non-cooperative bargaining literature: one side is privately informed about its valuation, which is drawn from a continuum, and the other side is possibly committed to a fixed offer. We analyze the game both in discrete and in continuous time and show convergence of the two settings, which extends results from Abreu and Gul [2000. Bargaining and reputation. Econometrica 68, 85–117]. One interesting result is that as time proceeds, the non-committed seller becomes less likely to concede in a given period, i.e., it appears as if he becomes more “stubborn.” We further show that a seller may prefer to negotiate with a “worse” buyer as this enhances the value of his possible commitment.     

Repeated Bargaining with Persistent Private Information

The paper analyses repeated contract negotiations involving the same buyer and seller where the contracts are linked because the buyer has persistent (but not full y permanent) private information. The size of the surplus being divided is specified as a two-state Markov chain with transitions that are synchronized with contract negotiation dates. Equilibrium involves information cycles triggered by the success or failure of aggressive demands made by the seller. Because there is persistence in the Markov chain generating the surplus, a successful demand induces the seller to make another aggressive demand in the next negotiation, since the buyer's acceptance reveals that the current surplus is large. Rejection of an aggressive demand, on the other hand, leads the seller to be pessimistic about the size of the surplus in the next contract, so the seller makes a "soft" offer that is sure to be accepted. Then, after several such offers have been accepted, the seller is optimistic enough to again make an aggressive demand, creating an information cycle. An interesting feature of this cycle is that the soft price is not constant, but declines as the cycle continues, so as to offset the buyer's option value of re-starting the cycle when the current state is bad. An explicit mapping is given for the relationship between the basic parameters and the equilibrium prices and quantities; in particular, there is a closed-form solution for the threshold belief that makes the seller indifferent between hard and soft offers.     

Bargaining over loan contracts with signaling

This paper combines a sequential bargaining game between an enterprise and a fixed number of banks with a signaling game through which the enterprise reveals her project quality as well as her market-speed on the lending market. We characterize subgame-perfect Nash equilibrium loan contracts that are supported by separating perfect Bayesian equilibria in the signaling game. In contrast to existing models of lending markets, low-quality investment projects might be rewarded with more favorable equilibrium loan contracts than high-quality projects. Also in contrast to existing models, an increase in the competitive pressure between banks reduces the aggregate welfare in our model. The reason is that more favorable loan conditions come with a greater incentive for the ‘strong’ entrepreneur to distinguish herself from her ‘weak’ counterpart through socially wasteful signaling costs.     

Coalitional Bargaining with Competition to Make Offers

The paper analyzes a simple discrete-time noncooperative coalitional bargaining game in which, at each stage, there is a contest for the right to make a proposal. The pure stationary subgame-perfect equilibrium payoff set is equal to the core.Journal of Economic LiteratureClassification Numbers: 026.     

Bargaining with asymmetric information in non-stationary markets

Summary. The Rubinstein and Wolinsky bargaining-in-markets framework is modified by the introduction of asymmetric information and non-stationarity. Non-stationarity is introduced in the form of an arbitrary stochastic Markov process which captures the dynamics of market entry and pairwise matching. A new technique is used for establishing existence and characterizing the unique outcome of a non-stationary market equilibrium. The impact of market supply and demand on bilateral bargaining outcomes and matching probabilities is explored. The results are useful for examining such questions as why coordination failures and macroeconomic output fluctuations are correlated with real and monetary shocks. Received: July 22, 1994; revised version: January 21, 1998     

Introduction to Search Theory and Applications

This introduces the Symposium on Search Theory and Applications.     

Bargaining over effort

Employment contracts generally specify both the reward for labour and the conditions of employment. Although it is clear how unions affect the payment received by workers, the effect of unions upon working conditions, particularly the contracted effort level that must be supplied, is not so well-understood. This note analyses the generalised Nash bargain between a union and a firm over employment level and effort level. Under standard assumptions, it is shown that an increase in union power always leads to a reduction in the agreed effort level. The effect upon employment is dependent upon whether the firm has the ‘right-to-manage’.     

Bargaining and Boldness

We study a multiperson bargaining problem with general risk preferences through the use of Shaked's game of cycling offers with exogenous breakdown. If preferences are “smooth,” then as the risk of breakdown vanishes, the limiting outcome is one in which bargainers are equally marginally bold; where a bargainer's marginal boldness measures his willingness to risk disagreement in return for a marginal improvement in his position. Under smoothness, any (ordinal-)Nash solution is an equally marginally bold outcome. However, unlike the concept of the (ordinal-)Nash solution, a unique equally marginally bold outcome exists in natural cases—in particular, if all bargainers have risk-averse preferences of the rank-dependent expected utility type. For these preferences, the equally marginally bold outcome maximizes a “bargaining power”-adjusted (asymmetric) Nash product where the degree of asymmetry is determined by the disparity in the marginal valuation of certainty among bargainers. Journal of Economic Literature Classification Numbers: C72, C78, D81.