Theory and Methodology
Constraint proposal method for computing Pareto solutions in multi-party negotiations

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Abstract

The constraint proposal method for computing Pareto-optimal solutions is extended to multi-party negotiations. In the method a neutral coordinator assists decision makers in finding Pareto-optimal solutions so that the elicitation of the decision makers' value functions is not required. During the procedure the decision makers have to indicate their most preferred points on different sets of linear constraints. The method can be used to generate either one Pareto-optimal solution dominating the status quo solution of the negotiation or an approximation to the Pareto frontier. In the latter case a distributive negotiation among the efficient agreements can be carried out afterwards.

Introduction

In this paper the constraint proposal method [5] for computing Pareto-optimal solutions is extended to multi-party negotiations. In the method the decision makers' (DMs') value functions do not have to be elicited. During the procedure a neutral coordinator who works as a mediator assists the DMs in finding efficient agreements so that minimal amount of information on the DMs' preferences is required. The constraint proposal method can be used in variety of ways. One way is to use the method to generate several Pareto-optimal solutions or an approximation to the Pareto frontier with the negotiation then becoming distributive along the frontier. Another way is to use the method to find a solution dominating a tentative agreement (see [11, p. 216]).

The development of methods for searching Pareto-optimal solutions in negotiations is important since Pareto-optimality may not be reached in practice without computational aid [11]. This may be due to numerous issues to be negotiated over and the negotiators' limited understanding of the other parties' interests. In addition, negotiators' unwillingness to disclose private information creates a special need for informationally decentralized methods for computing Pareto-optimal solutions.

The constraint proposal method has been previously studied in a two-DM situation. The underlying theory for the situation is presented in [5]. In [16] a negotiation support system RAMONA that is an example of a constraint proposal method is introduced and it is applied to agricultural negotiations between the Finnish Government and the MTK (the Finnish Farmer's Union). The RAMONA method has also been applied in a firm in Finnish paper industry where the production and marketing departments negotiated over the production schedule [9].

In a two-DM situation the idea of the constraint proposal method is to find a hyperplane tangent to the indifference contours of both DMs' value functions at a Pareto-optimal solution. The search for it is done iteratively using hyperplanes going through a given reference point: At every iteration the DMs indicate their most preferred points, i.e., their optimal solutions, on the hyperplane and the mediator adjusts the hyperplane so that the DMs' optimal solutions coincide. With more than two DMs the geometrical condition for Pareto-optimality is different from the two-DM case and therefore the method for two DMs cannot be directly applied. In the general case there is an affine set tangent to the indifference contours of all the DMs' value functions at a Pareto-optimal solution. The dimension of the affine set depends on the number of DMs and the number of decision variables. Thus, in multi-party setting the idea of the method is to find an affine set so that the DMs' most preferred points on that set coincide. A similar idea of iterating with a line going through a fixed reference point was used in [7], [19] for searching Pareto-optimal solutions in multiplayer games. However, the results in those papers are applicable only in the case where the number of decision variables is equal to the number of players.

There are also some other approaches for computing Pareto-optimal solutions in negotiations so that the elicitation of the DMs' value functions is not needed (for references to literature on negotiation research see, e.g., [15], [17]). A popular approach, inspired by Raiffa's procedure [11], involve seeking joint improvements from a tentative agreement. Teich et al. [18] used a heuristic rule for finding joint improvements and proposed heuristics for approximating the whole Pareto-optimal frontier. Ehtamo et al. [8] showed that if the search direction for joint improvements is taken so that it bisects the cone of jointly improving directions at the current solution, then the convergence of the method to a Pareto-optimal solution can be proved under appropriate convexity assumptions. In [6] this idea was extended further to a multi-party setting.

The rest of the paper is organized as follows. Section 2 presents the formulation and the analysis of the problem for generating a Pareto-optimal solution corresponding to a given reference point. First the assumptions used in the analysis are stated and necessary and sufficient conditions for Pareto-optimality are given. Second, the solution of the problem is shown to always dominate the reference point, i.e., each DM is at least as well off at the solution point as at the reference point. In Section 3 the problem is decomposed so that it can be solved iteratively: At every iteration the mediator forms an affine set going through a fixed reference point and each DM indicates his most preferred point in this set. Conditions under which the solution of the decomposed problem solves the original problem are given. In Section 4 the mediator's problem is formulated as a system of nonlinear equations. The function to be equated to 0 is shown to be continuously differentiable under relatively mild assumptions and a heuristics for solving it is given. Section 5 discusses how the reference point should be changed in order to systematically generate several Pareto-optimal solutions. The behavior of the heuristics for solving the mediator's problem is studied by numerical examples in Section 6.

Section snippets

Pareto-optimal solutions

In this paper we consider negotiations where n DMs are negotiating over the values of m continuous issues. The set of the DMs is denoted by I={1,…,n} and the value of the ith decision variable is denoted by xiR. The decision vector is x=(x1,…,xm)TX, where the decision space X is a compact and convex subset of Rm. The DMs' value functions are denoted by ui:RmR,i∈I. However, it should be emphasized that the value functions are assumed not to be explicitly known.

We make the following assumption

Decomposition

Problem (4) can be decomposed so that the only information needed from the DMs is their optimal solutions on different subsets of the decision space. The decomposition leads to an optimization problem where the mediator's problem is the following: For a fixed rRm, find a matrix PRm×(n−1) such thatx*=x1*(P)=⋯=xn*(P),where xi*(P) is a solution to the ith subproblem; that is, it solves the following maximization problem:maxxui(x)s.t.PT(xr)=0.One should note that in the DMi's maximization

Solving the mediator's problem

The mediator' s problem (5) can be formulated in many ways and several numerical iteration schemes can be used to solve it. One way is to define difference vectors di(p̃)=xi*(p̃)−xi+1*(p̃), for every i=1,…,n−1 and formulate the problem (5) as a system of nonlinear equationsd(p̃)=d1(p̃)dn−1(p̃)=0.We normalize the columns of P by setting p1i=1 for every i=1,…,n−1. For each column, we denote the vector of the remaining independent variables by p̃i, i.e., p̃i=(p2i⋯pmi)T. Finally, p̃ denotes the

Generating several Pareto solutions

In the previous sections we have described a procedure for generating one Pareto-optimal solution corresponding to a fixed reference point. In this section we will study the relation between reference points and the corresponding solutions. Especially, we will discuss how the reference point should be changed to systematically generate several Pareto-optimal solutions. After generating several Pareto-optimal solutions one can approximate the Pareto frontier, for example, by connecting the

Numerical examples

The applicability of the constraint proposal method in practice depends greatly on how many optimization problems of the form (6) each DM has to solve when the heuristics presented in Section 4.1 is used. On the other hand the more DMs and the decision variables there are, the larger and more difficult the mediator's problem becomes. Therefore we will study here the performance of the mediator's heuristics in two imaginary negotiation situations with different number of DMs and decision

Conclusion

In this paper a constraint proposal method for computing Pareto-optimal solutions in multi-party negotiations is developed. With it the DMs' value functions do not have to be explicitly revealed and the DMs have to communicate only their tentative decisions during each phase of computation. Another appealing feature of the method is that the solutions are individually rational with respect to the reference point. This is a desirable property for a solution in practical negotiations.

In the paper

Acknowledgements

The authors would like to thank Y. Ermoliev for helpful discussions as well as S. Salo, M. Verkama and the two anonymous referees for their constructive comments. Pirja Heiskanen acknowledges the financial support of the Finnish Cultural Foundation and the Emil Aaltonen Foundation.

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Present address: Helsinki School of Economics and Business Administration, P.O. Box 1210, FIN-00101 Helsinki, Finland.

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