Randomness for Free

TL;DR

This paper characterizes classes of two-player zero-sum games on graphs where randomness does not provide additional strategic power, leading to new undecidability results.

Contribution

It provides a complete characterization of game classes where randomness in transitions and strategies is not beneficial, and derives related undecidability results.

Findings

Randomness in transitions can be simulated by deterministic transitions in certain classes.

Pure strategies are as powerful as randomized strategies in these classes.

New undecidability results are established for these game classes.

Abstract

We consider two-player zero-sum games on graphs. These games can be classified on the basis of the information of the players and on the mode of interaction between them. On the basis of information the classification is as follows: (a) partial-observation (both players have partial view of the game); (b) one-sided complete-observation (one player has complete observation); and (c) complete-observation (both players have complete view of the game). On the basis of mode of interaction we have the following classification: (a) concurrent (both players interact simultaneously); and (b) turn-based (both players interact in turn). The two sources of randomness in these games are randomness in transition function and randomness in strategies. In general, randomized strategies are more powerful than deterministic strategies, and randomness in transitions gives more general classes of games. In this work we present a complete characterization for the classes of games where randomness is not helpful in: (a) the transition function probabilistic transition can be simulated by deterministic transition); and (b) strategies (pure strategies are as powerful as randomized strategies). As consequence of our characterization we obtain new undecidability results for these games.