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Social Choice Welfare (1985) 2:95-117 Social Choice dWelfare © Springer-Verlag 1985 Multi-Agent Equilibria with Market Share and Ranking Objectives* A. Denzau 1, A. Kats 2, S. Slutsky a Washington University, Department of Economics, St. Louis, MO 63130, USA 2 Virginia Polytechnic Institute & State University, Department of Economics, Blacksburg, VA 24061, USA 3 University of Florida, Department of Economics, Gainsville, FL 32601 Received July 18, 1984/Accepted December 27, 1984 Abstract. A model of nonprice unidimentional spatial competition between firms or between political candidates is studied. Typically in such models agents are assumed to maximize their shares of customers (as a proxy for profits) by firms, or of voters by candidates. This paper derives conditions for the existence, and characterization, of equilibrium outcomes when there are more than two agents, all of whom are assumed to minimize their rank as well as maximize their shares. The rank of an agent is defined as the sum of the number of agents with greater shares and a fraction of the number of agents with equal shares. It is shown that rank minimization tends to require more symmetry in location than share maximi- zation, that the greater the weight on agents with equal shares in defining rank, the more extreme are the requirements for equilibrium until, for weights greater than 1/3, no equilibrium exists for more than three competing agents, and that when agents care about both rank and shares, the relative weights put on each is irrelevant in determining the equilibrium locations. 1. Introduction A classic model illustrating the operation of competition is the spatial model first proposed by Hotelling [8] with elaborations by Smithies [10] and Chamberlin [4]. The model has yielded insights into such things as locational choices and decisions on product quality. A major conclusion of the early literature was the "principle of minimum differentiation" which asserted that there would be bunching of firms at a single location. Only recently have the equilibrium patterns in this model been com- pletely characterized (see Eaton and Lipsey [7] for example) and has price adjustment been fully integrated (see, for example, Salop [9], Spence [11], d'Aspremont et al. [1], and Stiglitz [13]). While extreme minimum differentiation does not seem to hold as a general principle, a weaker conclusion, that there will be some inefficient bunching of firms instead of optimal dispersion, does seem to be valid. * We acknowledge very fruitful discussions with Ted Bergstrom on the original conception of this paper