Entropic Risk for Turn-Based Stochastic Games

TL;DR

This paper introduces entropic risk as a measure in turn-based stochastic games, establishing its properties, computational complexity, and providing algorithms for decision and approximation, thus extending risk-averse strategies beyond traditional models.

Contribution

It is the first to analyze entropic risk in turn-based stochastic games, showing existence of optimal strategies and exploring the computational complexity of related decision problems.

Findings

Games are determined and admit optimal memoryless deterministic strategies.

Decidability of the threshold problem depends on Shanuel's conjecture.

An approximation algorithm for the optimal ERisk value is provided.

Abstract

Entropic risk (ERisk) is an established risk measure in finance, quantifying risk by an exponential re-weighting of rewards. We study ERisk for the first time in the context of turn-based stochastic games with the total reward objective. This gives rise to an objective function that demands the control of systems in a risk-averse manner. We show that the resulting games are determined and, in particular, admit optimal memoryless deterministic strategies. This contrasts risk measures that previously have been considered in the special case of Markov decision processes and that require randomization and/or memory. We provide several results on the decidability and the computational complexity of the threshold problem, i.e. whether the optimal value of ERisk exceeds a given threshold. In the most general case, the problem is decidable subject to Shanuel's conjecture. If all inputs are rational, the resulting threshold problem can be solved using algebraic numbers, leading to decidability via a polynomial-time reduction to the existential theory of the reals. Further restrictions on the encoding of the input allow the solution of the threshold problem in NP$\cap$coNP. Finally, an approximation algorithm for the optimal value of ERisk is provided.