zbMATH — the first resource for mathematics

Nash equilibria via duality and homological selection. (English) Zbl 1318.91009
Let \(f: \mathbb{D}^m \multimap \mathbb{D}^n\) be a continuous multimap of finite-dimensional discs with compact values. Let \(\operatorname{gr}(f)\) denote the graph of \(f\) and \(\operatorname{gr}(\partial f)\) the graph of the restriction of \(f\) to \(\partial \mathbb{D}^m\). A non-zero \(m\)-dimensional chain \(c_m\) supported in \(\operatorname{gr}(f)\) is called the homological selection of \(f\) if its boundary \(\partial^m c_m\) is supported in \(\operatorname{gr}(\partial f)\) and the projection \(H_m(\operatorname{gr}(f)\), \(\operatorname{gr}(\partial f)) \to H_m(\mathbb{D}^m, \partial \mathbb{D}^m)\) maps \(c_m\) to a non-zero class.
The authors present a homological selection theorem and consider its application to the existence of a Nash equilibrium.
91A10 Noncooperative games
55M99 Classical topics in algebraic topology
55M05 Duality in algebraic topology
55N45 Products and intersections in homology and cohomology
Full Text: DOI arXiv
[1] Basar T and Oldser G J, Dynamic Noncooperative Game Theory (1982) (New York: Academic Press)
[2] Gale D, The game of Hex and the Brouwer fixed-point theorem, Amer. Math. Monthly86 (1979) 818-827 · Zbl 0448.90097
[3] Hart S and Schmeidler D, Existence of correlated equilibria, Math. Oper. Res.14(1) (1989) 18-25 · Zbl 0674.90103
[4] Hatcher A, Algebraic Topology (2002) (Cambridge University Press)
[5] Kohlberg E and Mertens J, On the strategic stability of equilibria, Econometrica54(5) (1986) 1003-1037 · Zbl 0616.90103
[6] Lemke C E and Howson J T, Equilibrium points in bimatrix games, SIAM J. App. Math.12(1964) 413-423 · Zbl 0128.14804
[7] Maehara R, The Jordan curve theorem via the Brouwer fixed point theorem, Amer. Math. Monthly91(10) (1984) 641-643 · Zbl 0556.54025
[8] McKelvey R and McLennan A, The maximal number of regular totally mixed Nash Equilibria. Discussion Paper No. 272, July 1994, Center for Economic Research, Department of Economics, University of Minnesota
[9] McLennan A, The expected number of Nash equilibria of a normal form game, Econometrica73(1) (2005) 141-174 · Zbl 1152.91326
[10] Michael E, Continuous selections. I, Ann. Math.63(2) (1956) 361-382
[11] Michael E, Continuous selections. II, Ann. Math.64(3) (1956) 562-580
[12] Michael E, Continuous selections. III, Ann. Math.65(2) (1957) 375-390
[13] Nash J, Non-cooperative games, Ann. Math.54(2) (1951) 286-295
[14] Schick T and Simon R, Spiez S and Torunczyk H, A parametrized version of the Borsuk-Ulam theorem, Bull. Lond. Math. Soc.43(6) (2011) 1035-1047 · Zbl 1233.55002
[15] Simon R, Spiez S and Torunczyk H, The existence of equilibria in certain games, separation for families of convex functions and a theorem of Borsuk-Ulam type. Israel J. Math.92(1-3) (1995) 1-21 · Zbl 0843.90143
[16] Simon R, Spiez S and Torunczyk H, Equilibrium existence and topology in some repeated games with incomplete information, Trans. Amer. Math. Soc.354(12) (2002) 5005-5026 · Zbl 1043.91011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.