A Catalog of ∃R-Complete Decision Problems About Nash Equilibria in Multi-Player Games

[Schaefer and Stefankovic, Theory of Computing Systems, 2015] provided an explicit formulation of there exists R as the class capturing the complexity of deciding the Existential Theory of the Reals, and established that deciding, given a 3-player game, whether or not it has a Nash equilibrium with no probability exceeding a given rational is there exists R-complete. Four more decision problems about Nash equilibria for 3-player games were very recently shown there exists R-complete via a chain of individual, problem-specific reductions in [Garg et al., Proceedings of ICALP 2015]; determining more such there exists R-complete problems was posed there as an open problem. In this work, we deliver an extensive catalog of there exists R-complete decision problems about Nash equilibria in 3-player games, thus resolving completely the open problem from [Garg et al., Proceedings of ICALP 2015]. Towards this end, we present a single and very simple, unifying reduction from the there exists R-complete decision problem from [Schaefer and Stefankovie', Theory of Computing Systems, 2015] to (almost) all the decision problems about Nash equilibria that were before shown there exists R-complete for 2-player games in [Bile and Mavronicolas, Proceedings of SAGT 2012; Conitzer and Sandholm, Games and Economic Behavior, 2008; Gilboa and Zemel, Games and Economic Behavior, 1989]. Encompassed in the catalog are the four decision problems shown there exists R-complete in [Garg et al., Proceedings of ICALP 2015].