Randomness for Free

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.

💡 Research Summary

The paper investigates two‑player zero‑sum games played on finite graphs and asks a fundamental question: in which settings does randomness—either in the transition function or in the strategies—actually affect the outcome? To answer this, the authors first introduce a systematic taxonomy based on two orthogonal dimensions. The first dimension concerns the information available to the players: (a) partial‑observation (both players see only a projection of the true state), (b) one‑sided complete‑observation (one player has full knowledge while the other does not), and (c) complete‑observation (both players see the exact state). The second dimension captures the mode of interaction: (a) concurrent (both players choose actions simultaneously) and (b) turn‑based (players move alternately). This yields six distinct classes of games.

The central contribution is a complete characterization of those classes where randomness is “for free”, i.e., where it can be eliminated without changing the value of the game or the set of optimal strategies. The authors treat the two sources of randomness separately.

1. Randomness in the transition function.
   They identify structural conditions under which a probabilistic transition can be simulated by a deterministic one. Roughly, if every probabilistic choice can be expressed as a uniform distribution over a set of deterministic successors, and if the underlying graph admits a deterministic “cover” of all possible moves, then the stochastic transition can be replaced by a deterministic gadget that reproduces the same distribution. The construction preserves the payoff structure, so the game’s value remains unchanged. This result applies to all complete‑observation games and to one‑sided complete‑observation games where the informed player can enforce the deterministic cover.

2. Randomness in strategies.
   The paper shows that in many of the six classes, pure (deterministic) strategies are as powerful as mixed (randomized) strategies. For complete‑observation turn‑based games, the classic determinacy theorem already guarantees the existence of optimal pure strategies. The authors extend this to concurrent complete‑observation games by exploiting the fact that the payoff function is measurable and the action spaces are finite; a standard “purification” argument yields an equivalent pure strategy profile. In the one‑sided complete‑observation setting, they use a belief‑state transformation: the player with full observation can maintain a deterministic belief about the opponent’s hidden state, turning the original game into a deterministic belief‑MDP. Optimal pure strategies in the belief‑MDP lift back to pure strategies in the original game. Consequently, randomness in strategies does not increase the achievable value.

By intersecting the two analyses, the authors obtain a precise map: four of the six classes (all complete‑observation cases and the one‑sided complete‑observation cases) admit both deterministic transitions and pure strategies without loss of generality. The remaining two classes—partial‑observation concurrent games—are shown to genuinely require randomness either in the transition function or in the strategies.

The paper leverages this classification to derive new undecidability results. For the problematic partial‑observation concurrent class, the authors reduce the halting problem of a Turing machine to the question “does there exist a strategy that ensures reaching a target set with probability > ½?”. This reduction demonstrates that even when the transition function is deterministic, the presence of partial observation and concurrency makes the decision problem undecidable. Moreover, they prove that for the deterministic‑transition, pure‑strategy subclasses, the quantitative decision problem is PSPACE‑complete, matching known bounds for deterministic turn‑based games.

In summary, the work provides a clean dichotomy: randomness is essential only in the narrow setting of partial‑observation concurrent games; in all other natural graph‑based zero‑sum settings, one can safely work with deterministic transitions and pure strategies. This insight simplifies algorithmic analysis, informs the design of verification tools, and clarifies the true source of computational hardness in stochastic game models.