Léonard Brice

Léonard Brice, Thomas A. Henzinger, K. S. Thejaswini

Consider a 4-player version of Matching Pennies where a team of three players competes against the Devil. Each player simultaneously says "Heads" or "Tails". The team wins if all four choices match; otherwise the Devil wins. If all team players randomise independently, they win with probability 1/8; if all players share a common source of randomness, they win with probability 1/2. What happens when each pair of team players shares a source of randomness? Can the team do better than win with probability 1/4? The surprising (and nontrivial) answer is yes! We introduce Dicey Games, a formal framework motivated by the study of distributed systems with shared sources of randomness (of which the above example is a specific instance). We characterise the existence, representation and computational complexity of optimal strategies in Dicey Games, and we study the problem of allocating limited sources of randomness optimally within a team.

Léonard Brice

To verify the robustness of a program or protocol, it is common in the computer science community to rely on the theoretical framework of game theory. In particular, if one seeks to enforce a desired property, or specification, despite an unpredictable environment, a useful abstraction is to model the situation as a two-player zero-sum game. The goal is then to find a strategy for the system that guarantees the specification against any strategy of the environment. However, to model more complex situations, such as multiple systems with different objectives or an environment composed of various agents, the richer framework of multiplayer games must be considered. In this setting, a natural question is to identify equilibria, i.e., strategy profiles that are robust in the sense that no player has an incentive to deviate. The most well-known equilibrium concept is the Nash equilibrium, but several alternatives exist. We study five such notions and, for each of them, we provide complexity results for the constrained existence problem, which consists of deciding whether a given game contains an equilibrium that ensures each player a payoff within a specified interval.

Léonard Brice, Filip Cano, Krishnendu Chatterjee et al.

Partially Observable Markov Decision Processes (POMDPs) are systems in which one agent interacts with a stochastic environment, and receives only partial information about the current state. In a multi-environment POMDP (MEPOMDP), the initial state is unknown, and assumed to be adversarially chosen. In this work we focus on computing the optimal value and policy in MEPOMDPs with finite-horizon objectives. That problem is known to be PSPACE-complete in POMDPs. Our main results are as follows: (1) we establish that it is also PSPACE-complete in the more general setting of MEPOMDPs; (2) we present a practical algorithm and evaluate it on classical benchmarks, significantly outperforming the only previously known algorithm.

Léonard Brice, Thomas A. Henzinger, Alipasha Montaseri et al.

We study concurrent graph games where n players cooperate against an opponent to reach a set of target states. Unlike traditional settings, we study distributed randomisation: team players do not share a source of randomness, and their private random sources are hidden from the opponent and from each other. We show that memoryless strategies are sufficient for the threshold problem (deciding whether there is a strategy for the team that ensures winning with probability that exceeds a threshold), a result that not only places the problem in the Existential Theory of the Reals (\exists\mathbb{R}) but also enables the construction of value iteration algorithms. We additionally show that the threshold problem is NP-hard. For the almost-sure reachability problem, we prove NP-completeness. We introduce Individually Randomised Alternating-time Temporal Logic (IRATL). This logic extends the standard ATL framework to reason about probability thresholds, with semantics explicitly designed for coalitions that lack a shared source of randomness. On the practical side, we implement and evaluate a solver for the threshold and almost-sure problem based on the algorithms that we develop.