Agenda restrictions in multi-issue bargaining

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Abstract

We study a bilateral multi-issue bargaining procedure with complete information and endogenous agenda. In the procedure, proposals must be made on only one issue at a time, although the proposer can choose which issue to bring to the table. When bargaining frictions are small, there is a large multiplicity of equilibrium agreements, including ones with delay. However, equilibrium payoffs cannot be made arbitrarily small—perpetual disagreement cannot be supported in equilibrium. This multiplicity contrasts with the uniqueness found in the literature for a procedure where offers can be made in any subset of remaining issues.

Introduction

Two parties are engaged in negotiations that involve several issues. The informal literature on bargaining (for example, Raiffa (1982)) advises real bargainers on how they should proceed in these matters. In particular, it is claimed there that it is always more efficient to bundle issues in the negotiation procedure to be able to exploit the tradeoffs among different issues. On the other hand, it is often argued, invoking bounded rationality and complexity considerations, that bargaining should proceed issue-by-issue.

Therefore, the following two questions are of interest. First, to what extent the unique and efficient prediction of Rubinstein’s (1982) extends to a multiple issue setting, and second, what factors, inherent to the multi-issue nature of the problem, may produce inefficiencies and delay in equilibrium. By studying two complete information procedures, extensions of Rubinstein’s to the multi-issue case, one can find an answer to both questions in agenda restrictions. While their presence (in an issue-by-issue procedure) yields multiplicity of equilibria and delay in agreement, their absence (in a fully flexible procedure) allows the extension of Rubinstein’s theory.

In the procedure studied here, which we refer to as restricted agenda bargaining, parties bargain using alternating offers according to the following rules. The proposer makes an offer on one and only one of the remaining issues, but he can choose which issue to propose on. The responder either accepts or rejects. If he accepts, that issue is considered settled and he proposes on any one and only one of the remaining issues, and so on. If he rejects, with some probability negotiations break down, while with the rest of probability the rejector makes a counterproposal on any of the pending issues. The game ends either with breakdown of negotiations or when all issues have been settled. The agenda is therefore fully endogenous because each proposer can choose which issue comes to the negotiation table on each round. The completely endogenous nature of the agenda makes the analysis far from trivial.

In this game and when the breakdown probability is small, we find a large multiplicity of subgame perfect equilibrium (SPE) outcomes. The large multiplicity is a consequence of a strong form of non-concavity of the feasible agreements frontier caused by the restricted agenda. This allows multiple stationary SPE to appear. Following standard arguments, the stationary equilibria serve to support a plethora of non-stationary ones, including equilibria with delay in agreement. Thus, the inefficiency of not being able to exploit the tradeoffs in the marginal rates of substitution (MRS) between issues, a result of the restricted agenda procedure, is accentuated by the presence of delay in equilibrium. However, these inefficiencies cannot be made extreme; for example, perpetual disagreement is not an equilibrium phenomenon thanks to the unique Rubinstein SPE outcome in single-issue subgames. Also, given the large multiplicity of SPE, we cannot provide a clear answer to the question of which issue to bring up first, perhaps confirming formally the variety of opinions found on the matter in the informal literature.

Our procedure, as described, exhibits a clear friction: bargainers are forced to negotiate one issue at a time. This suggests that one should examine a procedure where the issues can be bundled. A basic observation in this respect is that if one studies a procedure where all issues must be bundled in every offer, Rubinstein’s uniqueness and efficiency result extends (indeed, the same proof as in Osborne and Rubinstein (1990) applies). Instead, in a different paper (In and Serrano, 2003), we investigate a procedure suggested by Inderst (2000) and others. We refer to it as unrestricted agenda bargaining. In it, each proposer can make an offer on any subset of the remaining issues. With this only modification, the rules are as described above. Generalizing Inderst’s result to a considerably larger class of utility functions (where separability across issues and concavity are not assumed), In and Serrano find that there is a unique and efficient SPE outcome, which turns out to be described by the same stationary equations as in Rubinstein. By permitting MRS tradeoffs in the bundled offers, this procedure extends Rubinstein’s theory to the multi-issue setting, and despite allowing the offers to be completely flexible, one does not get equilibria other than Rubinstein’s.1

The formal literature on multi-issue bargaining has already produced a handful of papers. Fershtman, 1990, Fershtman, 2000 studies procedures where the agenda (order of the two issues) is exogenous and provides examples to illustrate how each party is treated in each of the procedures. Busch and Horstmann (1997a) also study a procedure for two issues with exogenous agenda and focus on the effects of different bargaining frictions. We learn from these papers the comparative statics of equilibrium when agendas are exogenously fixed. On the other hand, as is the case in this paper, Bac and Raff, 1996, Busch and Horstmann, 1997b, Busch and Horstmann, 1999, Busch and Horstmann, 2002, Inderst, 2000 and Lang and Rosenthal (1999) study the more realistic case of endogenous agendas. As in our restricted agenda procedure, Inderst also finds multiple equilibria and the possibility of delay in the presence of discrete “strictly controversial” issues. This adds a zero-sum component to the problem and, strictly speaking, places the analysis beyond pure bargaining situations. In Bac and Raff (1996) and Busch and Horstmann, 1997b, Busch and Horstmann, 1999, Busch and Horstmann, 2002, the main actor is asymmetric information: their work extends the important insight of using delay as a signaling device, already captured in many studies in single-issue bargaining. Finally, Weinberger (2000) looks at multi-issue bargaining when the responder can accept selectively subsets of proposals and also finds inefficiencies if one issue is indivisible.

The paper follows this plan. Section 2 studies the restricted agenda procedure. Section 3 concludes. The proofs are gathered in Appendix A.

Section snippets

Restricted agenda bargaining

Our purpose in this section is to show that multiple equilibria are possible. For this, it will suffice to concentrate on the case of two issues and linear utility functions.

Two players 1 and 2 are bargaining over how to split two divisible issues or “pies.” Let L={1,2} be the set of issues. The size of each pie has been normalized to 1. Starting at time t=0, bargaining takes place over potentially infinitely many discrete periods according to the following procedure.

For i=1,2 and SL, the

Conclusion

This paper has clarified to what extent Rubinstein’s theory extends to multi-issue settings. Our focus has been on agenda restrictions. Restricting agendas yields multiplicity of equilibrium outcomes because it creates strong forms of non-concave payoff frontiers. Before any offer is made, a proposer is unable to exploit tradeoffs among issues, and after a partial offer has been accepted, efficiency and strategic considerations are quite distinct. These multiplicity forces are overcome when

Acknowledgements

The first version of this paper was circulated with the title “First Things First: Setting the Agenda in Multi-Issue Bargaining”. We thank Leslie Marx, Bob Rosenthal, Rajiv Vohra, Oscar Volij, an anonymous referee, and seminar participants at Brown, Texas Austin, Pompeu Fabra, Carlos III, National University of Singapore, Melbourne, Auckland, and the 2001 North American Summer Meetings of the Econometric Society, for useful comments. In gratefully acknowledges research support from Brown

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