Zero-Sum Two Person Perfect Information Semi-Markov Games: A Reduction
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Look at the play of a Perfect Information Semi-Markov game (PISMG). As the game has perfect information, at each time point in any play, all but one player is a dummy. Hence on any particular time instant, at most one player has more than one action available to himself. Thus such games lack real conflict throughout its play and no player directly antagonizes another ever (in each state, the reward matrix is a row or column vector). Above intuition helps us to show that any zero-sum two person PISMG can be reduced to an one-player game, i.e., to a semi-Markov decision process (SMDP), which has a value (Sinha et al.,(2017) [14]). In this paper, we use limiting ratio average pay-off (but any standard pay-off function will do) and prove that any PISMG under such an undiscounted pay-off has a value and both the maximiser (player-I) and minimiser (player-II) have pure semi-stationary optimal strategies. To solve such an undiscounted PISMG, we apply Mondal's algorithm (2017, [11]) on the reduced SMDP obtained from the PISMG.
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