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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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