A Nested Family of $k$-total Effective Rewards for Positional Games

TL;DR

This paper introduces a new family of effective reward functions called $k$-total for positional games, extending known reward concepts and establishing fundamental properties like saddle points and strategy optimality.

Contribution

It defines the $k$-total reward functions for deterministic stochastic games and proves the existence of saddle points with optimal strategies, also showing how these games embed into higher-order reward games.

Findings

Existence of saddle points with pure stationary strategies.

Embedding of $k$-total reward games into $(k+1)$-total reward games.

Extension of reward concepts from mean payoff and total reward to a nested family.

Abstract

We consider Gillette's two-person zero-sum stochastic games with perfect information. For each $k \in \ZZ_+$ we introduce an effective reward function, called $k$-total. For $k = 0$ and $1$ this function is known as {\it mean payoff} and {\it total reward}, respectively. We restrict our attention to the deterministic case. For all $k$, we prove the existence of a saddle point which can be realized by uniformly optimal pure stationary strategies. We also demonstrate that $k$-total reward games can be embedded into $(k+1)$-total reward games.