A Closed Form Characterization of the Stationary Outcome in Multilateral Bargaining

In this paper we consider infinite horizon multilateral bargaining with alternate offers. We prove that there exists only one stationary subgame perfect equilibrium outcome and it corresponds to the unique invariantmeasure of a column stochastic matrix. We characterize this stationary subgame perfect equilibrium outcome in a closed form, and also extend the approach to the multilateral bargaining with random moves.     

Multilateral subsidy games

This paper examines the rationale for multilateral agreements to limit investment subsidies. The welfare ranking of symmetric multilateral subsidy games is shown to depend on whether or not investment levels are “friendly”, raising rival profits in total, and/or strategic complements, raising rival profits at the margin. In both Cournot and Bertrand competition, when spillovers are low and competition is intense (because goods are close substitutes), national-welfare-maximizing governments over-subsidize investment, and banning subsidies would improve welfare. When spillovers are high, national governments under-subsidize from a global welfare perspective, but the subsidy game is welfare superior to non-intervention. For helpful comments we are grateful to two referees, to Arijit Mukherjee, and to participants in seminars at Prague and UCD, at the EEA Conference in Lausanne and at the GEP Conference on “New Directions in International Trade Theory” at the University of Nottingham, June 2007. Dermot Leahy acknowledges the support of the Science Foundation Ireland Research Frontiers Programme (Grant MAT 017).     

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.     

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’.     

Pattern Bargaining*

Many unions in the United States have for several years engaged in what is known as pattern bargaining. In this article, we show that pattern bargaining is preferred by a union to both simultaneous industry‐wide negotiations and sequential negotiations without a pattern. Allowing for interfirm productivity differentials within an industry, we show that for small differentials, the union most prefers a pattern in wages, but for a sufficiently wide differential, the union prefers a pattern in labor costs. Finally, we demonstrate that pattern bargaining can be a significant entry deterrent. This provides an explanation for why incumbent firms in an industry may support the use of pattern bargaining in labor negotiations.     

The Timing of Multilateral Lending

Bargaining over the terms of multilateral lending can be a prolonged process. We model failure to come to quick agreement as a screening device. Policy makers who find conditionality especially costly delay to signal their type. We show that increased costs to not reaching agreement, whether borne by borrowers or lenders, can increase the chances of delay.     

Bargaining and competition revisited

We show the robustness of the Walrasian result obtained in models of bargaining in pairwise meetings. Restricting trade to take place only in pairs, most of the assumptions made in the literature are dispensed with. These include assumptions on preferences (differentiability, monotonicity, strict concavity, bounded curvature), on the set of agents (dispersed characteristics) or on the consumption set (allowing only divisible goods).     

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.     

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.     

The Choquet Bargaining Solutions

We axiomatically investigate the problem of rationalizing bargaining solutions by social welfare functions that are linear in every rank-ordered subset of Rⁿ₊ . Such functions, the so-called Choquet integrals, have been widely used in the theories of collective and individual choice. We refer to bargaining solutions that can be rationalized by Choquet integrals as Choquet bargaining solutions. Our main result is a complete characterization of Choquet bargaining solutions. As a corollary of our main result, we also obtain a characterization of the generalized Gini bargaining solutions introduced by Blackorby et al. (1994, Econometrica62, 1161–1178). Journal of Economic Literature Classification Numbers: D71, C78.     

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 over Public Goods

In a simple public good economy, we propose a natural bargaining procedure, the equilibria of which converge to Lindahl allocations as the cost of bargaining vanishes. The procedure splits the decision over the allocation in a decision about personalized prices and a decision about output levels for the public good. Since this procedure does not assume price-taking behavior, it provides a strategic foundation for the personalized taxes inherent in the Lindahl solution to the public goods problem.     

Bargaining one-dimensional social choices

We analyze bargaining over the one-dimension characteristic of a public good among n impatient players when decisions require q favorable votes, q?2. Stationary subgame perfect equilibrium strategies are characterized for all games with deterministic protocol. We provide a monotonicity condition (satisfied by all single-peak, strictly quasi-concave and concave utilities) that assures uniqueness for every q whenever player's utilities are symmetric around the peak. Without symmetry, the monotonicity condition assures uniqueness for qualified majorities, q>n/2, provided that agents are sufficiently patient and utilities satisfy an additional regularity condition. Asymptotic uniqueness is assured for qualified majorities by imposing only the monotonicity condition.     

The Biodiversity Bargaining Problem

We employ cooperative bargaining theory and Nash’s ‘rational threats’ idea to cast light on the biodiversity bargaining problem. The problem of global environmental negotiations is argued to be of the nature of a bargaining problem, in which bargainers must agree on the distribution of cooperative surplus in order to move to the bargaining frontier. We discuss the importance of both efficiency (bargaining frontier) and fairness (recognition of characteristics of bargainers) in the choice of the appropriate contract. We show that the incremental cost contract, used to resolve the biodiversity bargaining problem, is of the form of an extreme point contract that fails to recognise the contributions of the South to the production of cooperative surplus. A rational response to such a contract is the use of threats of biodiversity destruction. Contracts must evince both efficiency and fairness in order to represent lasting solutions.