Dicey Games: Randomness & Strategy

• Dicey games are competitive scenarios with inherent randomness, where dice rolls shape strategies and outcomes.
• They utilize formal metrics like span, entropy, and outlier proportion to quantitatively assess randomness and game dynamics.
• Research in dicey games uncovers nontransitive relations, phase transitions, and computational challenges in strategic decision-making.

A dicey game is any structured competitive or cooperative scenario in which randomization—most canonically via dice—directly influences strategy, outcome, or both. The term “dicey games” also refers to the formal models where shared or structured randomness, intransitive relations, transient dominance, and complex stochastic mechanisms yield behaviors such as cycles, paradoxes, phase transitions, and computational hardness. Recent literature treats dicey games through the lenses of probability theory, combinatorics, distributed systems, and algorithmic game theory, highlighting the subtle interplay between stochasticity, coordination, and skill.

1. Metrics and Formalisms for Randomness in Dicey Games

The quantitative assessment of randomness in dice-driven tabletop games is rigorously operationalized by evaluating how outcome distributions vary under controlled “seeded” stochasticity. Let $G_s$ be the deterministic game tree produced by fixing the sequence of all random events (the “game seed” $s$), such as dice rolls or shuffles. Play $N$ independent matches on $G_s$ with two identically configured MCTS agents and record the first-player win-rate $p_s \in [0,1]$. For a collection of seeds $S$, the distribution $\{p_s : s \in S\}$ admits the following diagnostics:

• Span: $\mathrm{Span} = \max_s p_s - \min_s p_s$, measuring maximal outcome range across random event choices.
• Trimmed Span: Computed after discarding $\alpha$-percentile outliers to mitigate rare-seed effects.
• Entropy: Discretize $p_s$ into $B$ bins, compute $q_i$ as fraction in bin $i$, then $S = -\sum_{i=1}^B q_i \log q_i$.
• Outlier Proportion: Fraction of $p_s$ outside the 99% theoretical binomial CI for $\bar{p}$.

For random variable $X$ denoting first-player win per match, $X|s \sim \mathrm{Bernoulli}(p_s)$, so

$$\mathrm{Var}(X) = \mathbb{E}_s[p_s(1-p_s)] + \mathrm{Var}_s(p_s).$$

However, over random $s$, $X$ is Bernoulli$(\bar{p})$, justifying classical binomial CIs for aggregate win-rates. Mirrored-seed pairing (role-swapping) reduces variance, particularly when $p_s\not\equiv0.5$.

Empirical studies (e.g., Can’t Stop: Span $\approx0.52$, Outliers $60\%$; Catan: Span $0.70$, Outliers $81\%$) demonstrate significant seed-dependent variability (Goodman et al., 4 Mar 2025). Decomposing Catan’s randomness using sub-seeds reveals the board setup (hex layout) is a more dominant randomness source than dice rolls.

As agent skill (measured by MCTS budget) increases, the span and the variance of outcomes also increase, as agents better exploit or suffer random state configurations. Designer-facing “randomness scores” can be formulated as:

$$RS(G) = \alpha\, \mathrm{T\!-\!Span}(G) + \beta\, \mathrm{OutlierPct}(G) + \gamma\, \frac{\mathrm{Entropy}(G)}{\mathrm{Entropy_{max}}}$$

with task-specific weights (Goodman et al., 4 Mar 2025).

2. Intransitive (“Sucker’s”) Dice and Cycles

A “sucker’s bet” is a nontransitive system: a set of $K$ dice or decks, each fair but with nonstandard face-labels, such that $P(D_i \text{ beats } D_j)>1/2$ for each $i$ in a directed cycle. Classical examples, such as Efron’s four dice,

$$A=[0,0,4,4,4,4],\; B=[3,3,3,3,3,3],\; C=[2,2,2,2,6,6],\; D=[1,1,1,5,5,5]$$

exhibit pairwise win-probabilities of $2/3$ in cyclic order: $A\succ B\succ C\succ D\succ A$ (Ekhad et al., 2017). The essential metric is the pairwise winning probability:

$$P(A > B) = \frac{|\{(a,b): a \in A, b\in B, a > b\}|}{36} = \frac{24}{36}$$

Enumeration of $k$-deck systems with given sizes uses multivariate generating functions and recurrence relations, efficiently counting all nontransitive tuples for moderate parameters (e.g., five nontransitive three-deck systems with three cards each). Asymptotic frequencies for random triples remain on the order of $10^{-3}$. The existence of such cycles implies that the “is more likely to win” relation is not transitive; designers or casino operators can exploit this by always responding to a selection with a counter-die guaranteed (in expectation) to beat the player’s choice.

Markov models for board games such as Snakes and Ladders similarly reveal intransitive cycles among initial board positions: e.g., square 69 beats 79, 79 beats 73, and 73 beats 69, with strictly positive pairwise “edge” probabilities (Sorkin, 2022).

3. Paradoxical and Anti-Inductive Dice Phenomena

Dicey games can exhibit anti-inductive phenomena: given dice $A$ and $B$, and $k$ rolls per player, the dominance relation $A[k]\succ B[k]$ (i.e., $P(A[k]>B[k]) > P(A[k]<B[k])$) can invert as $k$ increases. For example, for $\Delta = \{+3, -1, -1, -1\}$, $P(\Delta[1]>0)=0.25$, $P(\Delta[3]>0)\approx0.578$, but $P(\Delta[4]>0)\approx0.262$—winning only at $k=3$ before reverting (Eldridge et al., 20 Mar 2025).

Families parameterized by face values (e.g., 3- or 4-sided dice) display complex “tongues” or fractal-like dominance regions in parameter space as $k$ varies. The limiting sign is eventually determined by normalized skewness, but the threshold $N$ before no further reversals is often astronomically large. Enumeration over parameter space reveals rich transient structure, with tight reversals tightly linked to combinatorial properties of the face-value multiset.

4. Dicey Games with Structured Shared Randomness

“Dicey Games,” in the formal sense, are zero-sum team games where shared sources of randomness—represented as a hypergraph $H=(T,\mathcal E)$ over team members $T$—determine the information structure for strategies (Brice et al., 26 Jan 2026). Each hyperedge $e\in\mathcal E$ corresponds to a “die” shared among its members. Individual strategies map observed dice to actions ($\sigma_i: [0,1]^{\# \text{dice seen}} \to \{0,1\}$).

• Strategy value: For the Boolean win/loss function $u$, the team maximizes $\min_{a_D} \int u(\sigma(r), a_D)\, dr$.
• Key examples: With independent dice, the win rate is $2^{-n-1}$ for $n$ players; with a fully shared die, $1/2$ is achievable. Pairwise-only sharing on cycles (e.g., triangular) yields intermediate achievable rates ($\approx 0.2781$ for $n=3$).
• Main theorems: All optimal strategies can be represented as “grid” strategies with at most $k$ partitions per die (where the adversary has $k$ pure actions). The decision problems (e.g., can the team achieve value $\ge t$?) are computationally hard, lying in EXPSPACE and being NEXP-hard; however, structure (e.g., fixed maximal dice per player) lowers complexity.

Allocation of limited shared dice is itself hard: the allocation-threshold problem lies in EXPSPACE, while coverage-type payoff functions permit constant-factor approximations. The existence of cycles in the sharing hypergraph is required for nontrivial gains over factorized independent randomization (Brice et al., 26 Jan 2026).

5. Outcomes, Skill, and Phase Transitions in Dice-Driven Systems

In paradigmatic games such as RISK, dice outcomes drive high-dimensional state transitions. The single-engagement probabilities (e.g., attacker vs defender dice using order statistics and tie-break rules) can be exactly computed:

• For ordinary RISK ($3$ attacker dice vs $2$ defender dice, 6-sided): probabilities for defender losing two armies $\approx0.372$, each losing one $\approx0.336$, attacker loses two $\approx0.293$ (Hendel et al., 2015).

These exact per-engagement odds, combined via negative-binomial or multinomial models, yield closed-form conquer probabilities over multiple battles. For large-scale engagements, normal approximations apply: the probability of attacker success is sharply concentrated around the “$86\%$+$2$” rule—i.e., attackers need at least $0.86$ times the number of defender armies, plus two, to cross the $50\%$ win-probability threshold. The phase transition in win-probability occurs within $O(\sqrt{D})$ of this critical point.

Skill interacts with randomness: increased agent search budget amplifies the dependency on seed configurations, with Span and other randomness metrics rising as stronger players more adroitly exploit favorable states (Goodman et al., 4 Mar 2025). A small number of configurations exhibit non-monotonic skill dependence (“k-level traps”).

6. Threshold and Phase-Transition Gambles: Risk, Greed, and Path-Dependence

Repeated multiplicative gambles—typified by games in which a stake is repeatedly multiplied by “up” ($u$) or “down” ($d$) factors per round—demonstrate a nontrivial separation between arithmetic expectations and typical outcomes (Göll et al., 2021). For capped systems (maximum payout), the probability of finishing with a profit approaches zero as the number of rounds $n$ increases unless $u^p d^{1-p} > 1$ ($p$ is success probability). At a critical threshold ($u d = 1$ for unbiased coins), the expected profit transitions sharply between “profit” and “ruin” zones. In the uncapped setting, the fair frontier is $pu + qd = 1$.

This leads to practical misperceptions in dicey gambling: positive mean outcome per round (arithmetic average) does not ensure typical net winnings due to pathwise multiplicative effects, non-ergodicity, and rare-event dominance in final distributions. The mathematical model and Python algorithms confirm abrupt phase-switching and show that most participants lose, even when naive expectations are positive (Göll et al., 2021).

7. Broader Implications and Open Directions

Dicey games highlight the diversity of behaviors arising from simple stochastic primitives. They reveal:

• Structural nontransitivity, both in devices (dice, cards) and in state-space transitions (e.g., Markovian board games).
• Subtle dependency on the architecture of shared randomness in multi-agent distributed systems, with computational tractability and optimality contingent on hypergraph cycles and die allocation (Brice et al., 26 Jan 2026).
• Fragility of intuition in multi-stage or multiplicative games, where geometric means and rare-event statistics override arithmetic expectations.

Analysis and formal modeling of dicey games extends to algorithmic game theory, stochastic optimization, and the design of robust, fair, or intentionally paradoxical competitive systems. Further research directions include tightening upper/lower bounds for value computation in the presence of limited shared random sources, dynamic dicey game extensions, and exploration of multi-valued action sets. These investigations illuminate fundamental principles of randomness, coordination, and paradox in interactive systems driven by dice and analogous stochastic mechanisms.

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References: (Goodman et al., 4 Mar 2025, Ekhad et al., 2017, Eldridge et al., 20 Mar 2025, Hendel et al., 2015, Göll et al., 2021, Brice et al., 26 Jan 2026, Sorkin, 2022).