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arxiv: 2601.18303 · v2 · pith:JJ7C4AL5new · submitted 2026-01-26 · 💻 cs.GT · cs.LO· cs.MA

Dicey Games: Shared Sources of Randomness in Distributed Systems

L\'eonard Brice , Thomas A. Henzinger , K. S. Thejaswini This is my paper

Pith reviewed 2026-05-16 11:19 UTC · model grok-4.3

classification 💻 cs.GT cs.LOcs.MA
keywords dicey gamesshared randomnessdistributed systemsmatching penniesgame theoryoptimal strategiesrandomness allocationcoordination
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0 comments X

The pith

In the four-player Matching Pennies game, a team using only pairwise shared randomness sources can win with probability strictly above one quarter.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

A team of three players faces the Devil in a game where everyone chooses heads or tails simultaneously; the team wins only if all four choices match. Independent randomization gives the team a 1/8 win chance, while full shared randomness raises it to 1/2. The paper shows that restricting shared randomness to each pair of teammates still lets the team exceed a 1/4 win probability. This observation motivates the Dicey Games framework, which models how limited shared randomness sources can be allocated to improve coordination in distributed settings. The framework then characterizes when optimal strategies exist, how they can be represented, and how to allocate scarce randomness sources most effectively.

Core claim

Dicey Games formalize team-versus-adversary games in which coordination occurs solely through an allocation of shared randomness sources; for the concrete 4-player Matching Pennies instance, pairwise sharing already yields a win probability strictly greater than 1/4, and the framework supplies existence, representation, and complexity results for optimal strategies under arbitrary allocations.

What carries the argument

Dicey Games, the model in which a team’s coordination graph determines which subsets of players share independent randomness sources, and optimal mixed strategies are computed over the resulting joint distributions.

If this is right

  • Optimal strategies exist for any finite Dicey Game and any allocation of shared sources.
  • Optimal strategies admit a finite-support representation whose size is bounded by the number of players and sources.
  • Finding an optimal strategy is computationally hard in general, but polynomial for fixed numbers of players.
  • Given a budget of randomness sources, the problem of choosing the best allocation can be solved by enumeration over graphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same pairwise-sharing pattern may improve coordination in other simultaneous-move distributed protocols that already use limited shared randomness.
  • Extending the model to sequential games or noisy channels would test whether the >1/4 threshold generalizes.
  • Concrete small instances could be implemented on hardware with limited entanglement or shared seeds to measure empirical gains.

Load-bearing premise

The team’s only coordination mechanism is the pre-assigned shared randomness sources; all choices are made simultaneously with no further communication.

What would settle it

An explicit calculation or linear-program solution for the 4-player Matching Pennies instance that reports a maximum win probability of at most 1/4 when each pair of teammates shares one private random bit.

Figures

Figures reproduced from arXiv: 2601.18303 by K. S. Thejaswini, L\'eonard Brice, Thomas A. Henzinger.

Figure 1
Figure 1. Figure 1: A strategy to win with probability 1/6 matching-pennies case (Ada vs. Devil) and the max min value be￾comes 1/2. We now pose an illustrative question that motivates the broader class of games that we call dicey games, addressed in this work. Triangular matching pennies with the Devil What is the optimal strategy—and corresponding max min value—for achieving an all-Heads or all-Tails outcome in the matching… view at source ↗
Figure 2
Figure 2. Figure 2: An optimal collective strategy 1.2 Motivation In the context of zero-sum games, dicey games represent a natu￾ral intermediate notion between the independent randomisation studied by Cournot [19], von Neumann and Morgenstern [39], and Nash [33], and the shared randomness inherent to correlated equi￾libria [3, 5]. In correlated equilibria, all players can coordinate using one common source of randomness, whi… view at source ↗
Figure 3
Figure 3. Figure 3: The collective strategy 𝜎¯ 0 . . . • else, we have 𝜇(𝑎¯) = 𝑥𝑛+1. For example, in a matching pennies game in which all players choose between the actions Heads (written H) and Tails (written T), the fact that the team gets payoff 1 if all actions match, and 0 other￾wise, can be described by the list (H)𝑝∈Π∪{𝔇} , 1  , [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Applying Carathéodory’s theorem Geometrically, if HHH denotes the part of the cube of rolls [0, 1] 3 in which the three players choose H, then 𝑓 (𝑥) is the area of the slice HHH ∩ ({𝑥} × [0, 1] 2 ), and similarly for 𝑔. Let us note that the probability of all players choosing H is then ∫ 𝑓 , and that the probability of all choosing T is ∫ 𝑔 (without more precision, integrals are considered on the whole dom… view at source ↗
Figure 5
Figure 5. Figure 5: The collective strategy 𝜑1 (𝜎¯ 0 ) Proposition 3.6. The collective strategy 𝜎¯ ′ has a value at least as large as the collective strategy 𝜎¯. Proof. We prove this proposition by showing the inequalities: P [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: The collective strategy 𝜑2 ◦ 𝜑1 (𝜎¯ 0 ) 𝑥 𝑦 H T (a) Ada’s strategy 𝑧 𝑦 T H (b) Bertrand’s strategy 𝑥 𝑧 H T (c) Claude’s strategy 𝑥 𝑦 𝑧 (d) The sets HHH and TTT [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The collective strategy 𝜎¯ ★ The second one is analogous. Then, the value of the collective strategy 𝜎¯ ′ is: min  P [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The set of Fritz John points 4.1 A system of polynomial inequalities To formally define a system of inequalities analogous to the one presented at the end of Section 3.1, we first define some notation. Throughout this section, we assume, up to renaming them, that our dice form the set Δ = {1, . . . , 𝑛}. We let 𝑘 = |𝐴𝔇|. Definition 6 (Strategy scheme, 𝑎¯¯-strategy). Let D be a dicey game. A strategy scheme… view at source ↗
Figure 9
Figure 9. Figure 9: A few more examples Here, one might expect that the optimal strategy is similar to the one that was found for triangular matching pennies: one threshold 𝜆 obtained by some intricate polynomial equation, and each player choosing Heads if all the rolls they see are above that threshold, and Tails otherwise. The reality is quite different. The max min value of this game is 1/4, which is obtained by the follow… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces Dicey Games, a formal framework for distributed systems in which agents coordinate via allocated shared randomness sources. It analyzes a 4-player Matching Pennies instance in which a team of three players faces the Devil, showing that pairwise shared randomness sources allow the team to win with probability strictly greater than 1/4 (compared with 1/8 for independent randomness and 1/2 for a single shared source). The authors characterize existence, representation, and computational complexity of optimal strategies within the framework and study the problem of optimally allocating a limited number of randomness sources among team members.

Significance. If the characterizations hold, the work supplies a new, self-contained model for reasoning about partial coordination in distributed systems, with direct relevance to multi-agent game theory and distributed computing. The concrete demonstration that pairwise sharing exceeds the 1/4 threshold is nontrivial and falsifiable. The paper provides explicit characterizations of optimal strategies together with complexity results, which constitute a reusable technical contribution.

major comments (2)
  1. [§3.2] §3.2, the explicit strategy for the pairwise-sharing case: the claimed winning probability >1/4 is stated but the derivation of the exact value and the verification that it is optimal are not cross-referenced to the general representation theorem in §4.1; without this link the central claim rests on an unverified calculation.
  2. [§5] §5, the complexity result for optimal allocation: the reduction establishing NP-completeness assumes that the number of randomness sources is part of the input, yet the paper does not discuss whether the hardness persists when the number of sources is fixed (a regime relevant to the motivating example).
minor comments (2)
  1. [Abstract] The abstract asserts a 'surprising (and nontrivial) answer is yes' but does not state the concrete probability achieved by the pairwise strategy; adding this value would improve readability.
  2. [§2] Notation for the shared-randomness graph (vertices = players, edges = shared sources) is introduced without a small illustrative diagram; a figure would clarify the pairwise-sharing configuration used in the main example.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the precise comments. We address each major point below and will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3.2] §3.2, the explicit strategy for the pairwise-sharing case: the claimed winning probability >1/4 is stated but the derivation of the exact value and the verification that it is optimal are not cross-referenced to the general representation theorem in §4.1; without this link the central claim rests on an unverified calculation.

    Authors: We agree that an explicit link is needed. In the revised version we will add a direct cross-reference from §3.2 to the representation theorem in §4.1, together with a short derivation showing how the pairwise strategy yields a winning probability strictly greater than 1/4 and confirming optimality via the general result. revision: yes

  2. Referee: [§5] §5, the complexity result for optimal allocation: the reduction establishing NP-completeness assumes that the number of randomness sources is part of the input, yet the paper does not discuss whether the hardness persists when the number of sources is fixed (a regime relevant to the motivating example).

    Authors: The referee is correct that the reduction treats the number of sources as part of the input. We will add a short discussion in §5 noting that, for any fixed constant number of sources, all possible allocations can be enumerated in polynomial time (though evaluating the value of each allocation remains hard); this regime is directly relevant to the 4-player motivating example. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper introduces Dicey Games as a new formal framework for modeling distributed systems with shared randomness sources, using the 4-player Matching Pennies example to illustrate that pairwise shared randomness yields win probability strictly above 1/4. It then characterizes existence, representation, and computational complexity of optimal strategies within this framework. No equations, fitted parameters, or self-citations are visible that reduce any claimed result to its own inputs by construction; the central claims rest on the newly defined model and its internal analysis rather than self-definitional loops, renamed empirical patterns, or load-bearing prior self-citations. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The framework rests on standard simultaneous-move game assumptions and introduces the new concept of Dicey Games to capture restricted shared randomness; no free parameters or invented physical entities are mentioned.

axioms (1)
  • standard math Players choose simultaneously and payoffs depend only on the tuple of choices.
    Implicit in the definition of the matching-pennies payoff structure.
invented entities (1)
  • Dicey Games no independent evidence
    purpose: Formal model of distributed games with arbitrary patterns of shared randomness sources.
    Newly defined to study the allocation and strategy problems described.

pith-pipeline@v0.9.0 · 5474 in / 1146 out tokens · 40849 ms · 2026-05-16T11:19:30.183099+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

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