Abstract
As a result of some important works [4–6, 10, 15], the complexity of 2-player Nash equilibrium is by now well understood, even when equilibria with special properties are desired and when the game is symmetric. However, for multi-player games, when equilibria with special properties are desired, the only result known is due to Schaefer and Štefankovič [18]: that checking whether a 3-player NE (3-Nash) instance has an equilibrium in a ball of radius half in \(l_{\infty }\)-norm is ETR-complete, where ETR is the class Existential Theory of Reals.
Building on their work, we show that the following decision versions of 3-Nash are also ETR-complete: checking whether (i) there are two or more equilibria, (ii) there exists an equilibrium in which each player gets at least h payoff, where h is a rational number, (iii) a given set of strategies are played with non-zero probability, and (iv) all the played strategies belong to a given set.
Next, we give a reduction from 3-Nash to symmetric 3-Nash, hence resolving an open problem of Papadimitriou [14]. This yields ETR-completeness for symmetric 3-Nash for the last two problems stated above as well as completeness for the class \(\rm {FIXP_a}\), a variant of FIXP for strong approximation. All our results extend to k-Nash, for any constant \(k \ge 3\).
Supported by NSF Grants CCF-0914732 and CCF-1216019.
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Garg, J., Mehta, R., Vazirani, V.V., Yazdanbod, S. (2015). ETR-Completeness for Decision Versions of Multi-player (Symmetric) Nash Equilibria. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_45
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DOI: https://doi.org/10.1007/978-3-662-47672-7_45
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