Ambitropical geometry, hyperconvexity and zero-sum games

TL;DR

This paper characterizes the fixed point sets of Shapley operators in finite zero-sum games using hyperconvexity, lattice theory, and tropical geometry, revealing their structure as special convex and geometric objects.

Contribution

It provides a novel geometric and lattice-theoretic characterization of fixed points of Shapley operators in finite games, linking game theory with hyperconvex and tropical geometry.

Findings

Fixed point sets are hyperconvex and lattice-structured.

They are retracts of ^n with convex-like properties.

In deterministic cases, fixed points form polyhedral complexes with explicit representations.

Abstract

Shapley operators of undiscounted zero-sum two-player games are order-preserving maps that commute with the addition of a constant. We characterize the fixed point sets of Shapley operators, in finite dimension (i.e., for games with a finite state space). Some of these characterizations are of a lattice theoretical nature, whereas some other rely on metric or tropical geometry. More precisely, we show that fixed point sets of Shapley operators are special instances of hyperconvex spaces: they are sup-norm non-expansive retracts of $\R^n$, and also lattices in the induced partial order. Moreover, they retain properties of convex sets, with a notion of ``convex hull'' defined only up to isomorphism. This provides an effective construction of the injective hull or tight span, in the case of additive cones. For deterministic games with finite action spaces, these fixed point sets are supports of polyhedral complexes, with a cell decomposition attached to stationary strategies of the players, in which each cell is an alcoved polyhedron of $A_n$ type. We finally provide an explicit local representation of the latter fixed point sets, as polyhedral fans canonically associated to lattices included in the Boolean hypercube.