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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.
MSC:
91A10 Noncooperative games
55M99 Classical topics in algebraic topology
55M05 Duality in algebraic topology
55N45 Products and intersections in homology and cohomology
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