Dicey Games: Shared Sources of Randomness in Distributed Systems

TL;DR

Dicey games study how restricted shared randomness affects cooperative strategies against a single adversary in concurrent games. The authors show that optimal team strategies can be restricted to grid forms and that there exists a finite, exponentially bounded description of such strategies, enabling a systematic complexity analysis. They develop a general framework, prove the key grid-optimality theorem, and provide existential and computational results (including EXPSPACE membership and NEXP-hardness) for threshold, value, and allocation problems, with detailed analysis of small, illustrative examples like triangular and clique matching pennies. The work highlights foundational connections between distributed randomness structures, algebraic-geometry methods, and complexity, offering a rigorous basis for understanding and designing strategies under restricted randomness in distributed systems, economics, and social choice.

Abstract

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.

Figures (5)

• Figure 1: A strategy to win with probability 1/6
• Figure 2: An optimal collective strategy
• Figure 3: The collective strategy ${\bar{\sigma}}^0$
• Figure 8: The set of Fritz John points
• Figure 9: A few more examples

Theorems & Definitions (35)

• Definition 1: Game
• Definition 2: Dice structure, dicey game
• Definition 3: Strategy, collective strategy, strategy profile
• Definition 4: Piecewise constant strategy, grid collective strategy
• Definition 5: The dicey game $\mathcal{D}^\triangledown$
• Theorem 3.1
• Lemma 3.2
• Lemma 3.3
• Theorem 3.7: App. \ref{['app:itsOKtobestraight']}
• Definition 6: Strategy scheme, ${\bar{{\bar{a}}}}$-strategy