Sets of Nash Equilibria in Polymatrix Mixed-Strategy Games

Abstract

The method of intersection of best response mapping graphs is applied to determine a Nash equilibrium set in mixed extensions of noncooperative finite games (polymatrix games). The problem of Nash equilibrium set representation in bimatrix mixed strategy game was considered before by Vorob’ev in 1958 (Theory Probab Appl 3:297–309, 1958) [1] and Kuhn in 1961 (Proc Natl Acad Sci USA 47:1657–1662, 1961) [2], but as stressed by different researchers (see e.g. Raghavan (Handbook of game theory with economic applications. Elsevier Science B.V., North-Holland, 2002) [3]) these results have only been of theoretical interest. They where rarely used practically to compute Nash equilibria as well as the results of Mills (J Soc Ind Appl Math 8:397–402, 1960) [4], Mangasarian (J Soc Ind Appl Math 12:778–780, 1964) [5], Winkels (Game theory and related topics. North-Holland, Amsterdam, 1979) [6], Yanovskaya (Lithuanian Mathematical Collection (Litovskii Matematicheskii Sbornik) 8:381–384, 1968) [7], Howson (Manage Sci 18:312–318, 1972) [8], Eaves (SIAM J Appl Math 24:418–423, 1973) [9], Mukhamediev (U.S.S.R. Computational Mathematics and Mathematical Physics 18:60–66, 1978) [10], Savani (Finding Nash Equilibria of Bimatrix Games. London School of Economics, 2006) [11], and Shokrollahi (Palestine J Math 6:301–306, 2017) [12]. The first practical algorithm for Nash equilibrium computing was the algorithm proposed by Lemke and Howson in 1958 (J Soc Ind Appl Math 12:413–423, 1964) [13]. Unfortunately, it doesn’t compute Nash equilibrium sets. There are algorithms for polymatrix mixed strategy games too, Ungureanu in (Comput Sci J Moldova 42: 345–365, 2006) [14], Audet et al. in (J Optim Theory Appl 129:349–372, 2006) [15]. More the more, the number of publications devoted to the problem of finding the Nash equilibrium set is continuously increasing, see, e.g., 2010’s bibliography survey by Avis et al. (Econ Theory 42:9–37, 2010) [16], and the other by Datta (Econ Theory 42:55–96, 2010) [17].