Discounted semi-Markov games with incomplete information on one side

TL;DR

This paper establishes the existence of the value function and optimal policies in discounted semi-Markov games with incomplete information on one side, using dual game techniques and iterative algorithms.

Contribution

It introduces a novel framework for analyzing semi-Markov games with incomplete information, including existence proofs and algorithms for optimal policies for both players.

Findings

Existence of the value function under discounting.

Iterative algorithms for optimal policies.

Dual game approach for uninformed player.

Abstract

This work considers two-player zero-sum semi-Markov games with incomplete information on one side and perfect observation. At the beginning, the system selects a game type according to a given probability distribution and informs to Player 1 only. After each stage, the actions chosen are observed by both players before proceeding to the next stage. Firstly, we show the existence of the value function under the expected discount criterion and the optimality equation. Secondly, the existence and iterative algorithm of the optimal policy for Player 1 are introduced through the optimality equation of value function. Moreove, About the optimal policy for the uninformed Player 2, we define the auxiliary dual games and construct a new optimality equation for the value function in the dual games, which implies the existence of the optimal policy for Player 2 in the dual game. Finally, the existence and iterative algorithm of the optimal policy for Player 2 in the original game is given by the results of the dual game.