Infinite Subgame Perfect Equilibrium in the Hausdorff Difference Hierarchy

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9541)


Subgame perfect equilibria are specific Nash equilibria in perfect information games in extensive form. They are important because they relate to the rationality of the players. They always exist in infinite games with continuous real-valued payoffs, but may fail to exist even in simple games with slightly discontinuous payoffs. This article considers only games whose outcome functions are measurable in the Hausdorff difference hierarchy of the open sets (i.e. \({ {\Delta }}^0_2\) when in the Baire space), and it characterizes the families of linear preferences such that every game using these preferences has a subgame perfect equilibrium: the preferences without infinite ascending chains (of course), and such that for all players a and b and outcomes xyz we have \(\lnot (z <_a y <_a x \,\wedge \, x <_b z <_b y)\). Moreover at each node of the game, the equilibrium constructed for the proof is Pareto-optimal among all the outcomes occurring in the subgame. Additional results for non-linear preferences are presented.


Infinite multi-player games in extensive form Subgame perfection Borel hierarchy Preference characterization Pareto-optimality 



I thank Vassilios Gregoriades and Arno Pauly for useful discussions. The author is supported by the ERC inVEST (279499) project.


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© IFIP International Federation for Information Processing 2016

Authors and Affiliations

  1. 1.Université Libre de BruxellesBrusselsBelgium

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