Coin Hopping Strategy in Bayesian & Game Theory

• Coin Hopping Strategy is an adaptive protocol for sequential decision-making that optimizes coin selection using Bayesian updates and dynamic reward assessments.
• It enables efficient bias identification and optimal plays in combinatorial games through techniques like likelihood ratio updates and explicit Grundy number calculations.
• In applied markets, coin hopping guides miner strategies by switching coins based on revenue-per-hash incentives, ensuring convergence to Nash equilibria.

A coin hopping strategy refers to an adaptive dynamic protocol or rule for sequential decision-making involving the movement, allocation, or selection of coins (or analogous discrete units) under uncertainty, incentive, or positional constraints. Across probability, game theory, combinatorics, and cryptocurrency mining, the term captures a class of strategies whose hallmark is rapid alternation—“hopping”—between coins based on observed information, position, or changing rewards. Theoretical work has elucidated precise optimality properties, computational bounds, and Nash equilibria for major coin hopping paradigms in Bayesian identification, impartial games, and strategic markets.

1. Bayesian Coin Hopping in Bias Identification

In the sequential problem of identifying the most biased coin among a (possibly infinite) set $\{i=1,2,\dots\}$, where each $\theta_i \in \{p_+,p_-\}$ and prior $P(\theta_i=p_+)=\alpha$ independently, the coin hopping strategy designates that at each step, one samples the coin $i^*$ maximizing the current likelihood ratio $L_i$. This ratio, $$L_i = \frac{P(\theta_i=p_+\,|\,D_i)}{P(\theta_i=p_-\,|\,D_i)}$$, is efficiently updated after each toss using the recursive rule: $L_i \leftarrow L_i \times \begin{cases} \frac{p_+}{p_-} & \text{if result is head,}\[6pt] \frac{1-p_-}{1-p_+} & \text{if result is tail.} \end{cases}$ When all $L_i<1$, an untested coin (with $L=1$) is preferable. One terminates when some $L_i$ exceeds a threshold $B=\log\frac{(1-\alpha)(1-\delta)}{\alpha\delta}$ for a given confidence parameter $\delta$, ensuring posterior $P(\theta_i=p_+|D_i) \ge 1-\delta$.

This policy is Bayes-optimal: it minimizes the expected number of tosses for identification, as established via a Markov-game argument. Each coin induces a 1D Markov process with absorbing barriers at $0$ and $B$, and the globally optimal action is always to continue with the coin whose log-likelihood ratio is maximal. The approach achieves an expected sample complexity

$$\mathbb{E}[\#\text{tosses}] \le \frac{16}{\varepsilon^2} \left( \frac{1-\alpha}{\alpha} + \log\frac{(1-\alpha)(1-\delta)}{\alpha\delta} \right)$$

for $\varepsilon=p_+-p_-$, which is strictly superior to nonadaptive strategies for large $n$ (Chandrasekaran et al., 2012).

2. Combinatorial Coin Hopping in Strategic Games

A rich family of impartial games interprets coin hopping as a positional rule-set for two or more tokens (“coins”) on graphs or boards, with the following paradigms:

2.1. Slide-and-Hop on Semi-Infinite Boards

Given two indistinguishable coins at positions $(a,b)$ ($a<b$) on $\mathbb{Z}_{\geq 0}$, a move consists of either sliding a coin left to a lower unoccupied cell or executing a hop-and-push: simultaneously shifting the right coin left by $d$ sites and pushing the left coin left by $d$. The Grundy number $G(a,b)$, crucial for classifying combinatorial game-theory strategies, is determined by a piecewise family decomposition:

• Even-gap: $G(a,b)=3\cdot \frac{b-a}{2}-1$ when $d$ even, $n=3k-1\equiv 2\pmod{3}$, and $a\geq 2(n-2)/3$.
• Slide-up/slide-down families: Explicit formulas connect $(a,b)$ to unique Grundy indices, allowing calculation of optimal plays and last-move winning strategies in disjunctive sums (Miyadera et al., 25 Apr 2025).

2.2. Two-Dimensional Coin Hopping

For two coins on an infinite quadrant $(w,x,y,z)$, the rule-set (allowing sliding left/up, jumping over but not landing on the other coin, and no pushing) yields a complete classification of previous-player win ($\mathcal{P}$) positions. The set of all $\mathcal{P}$-positions is: $\mathcal{P} = (P_0 \cup P_1) \setminus N_0,$ where $P_0$ is given by the nim-sum being zero among $(w-1),(x-1),(y-1),(z-1)$. Exceptional sets $P_1$ (certain same-row/column, offset-by-one formations with nim-sum $1$) and $N_0$ (parity-misaligned near-alignments) are explicitly described. The classification supports efficient algorithmic determination of winning strategies and is tightly connected to classical Nim theory (Miyadera et al., 24 May 2025).

3. Coin Hopping in Markov Decision Processes and Bandit Problems

Hopping strategies in bandit-like settings are formulated in terms of multitoken Markov systems, in which each token represents a candidate (coin, arm, etc.) and transitions correspond to evidence accrual. The Derman–Ono–Ross theorem states that the unique optimal pure strategy is to act on the token in the lowest grade (where grade is a monotonic function of the log-likelihood or Gittins index), which coincides with the Bayesian coin hopping selection. This framework, leveraging monotonicity, guarantees that hopping to the most promising candidate at every step is globally and locally optimal (Chandrasekaran et al., 2012).

4. Strategic Coin Hopping in Applied Markets

In multi-cryptocurrency ecosystems, coin hopping describes a miner’s adaptive switch to the coin with maximal instantaneous revenue-per-unit (RPU) as $RPU_c = F(c)/M_c(s)$, where $F(c)$ is per-block reward and $M_c(s)$ is total hash rate on coin $c$. Each miner’s best response is to hop from current coin $c$ to $c'$ if

$\frac{F(c')}{M_{c'}(s)+m_p} > \frac{F(c)}{M_c(s)},$

leading to a potential-game dynamic with guaranteed convergence to a Nash equilibrium. Manipulation via temporary reward changes—forceful coin hopping—can steer the entire miner population to a new equilibrium that is more favorable to the attacker. Reward design techniques can be explicitly constructed to realize desired system states by orchestrating staged hopping, provided the system’s ordinal potential is maintained (Spiegelman et al., 2018).

5. Coin Hopping in Sequential Decision with Constraints

In the stochastic “set aside at least one coin per round” game, the coin hopping strategy chooses, on each round of tossing $n$ coins, whether to set aside exactly one coin or (in special cases) more, with the goal of maximizing the total expected heads collected. There is a sharp threshold $p_0\approx0.5495021777642$ for the coin’s head probability:

• For $p > p_0$ and large $n$, it is optimal to always set aside exactly one coin per round unless all coins are heads.
• For $1/2 \leq p \leq p_0$, set aside one coin unless at most one coin is tails. A recursion for optimal expected payoff $v_{n,p}$ is established, and a fine-grained regime classification results in precise asymptotics for gains achievable by the coin hopping rule (Doorn, 20 Jun 2024).

6. Algorithmic and Analytical Properties

Coin hopping strategies often enable efficient (O(1) per step) adaptive algorithms with provable optimality guarantees. In Bayesian search, they minimize sample complexity to tight $O(\varepsilon^{-2}\log(1/\delta))$ bounds. In combinatorial games, explicit Grundy-number computations yield algorithmic determination of optimal strategies. In multi-agent systems, ordinal potential guarantees global convergence, while in Markov models, monotonicity of grade underpins myopic optimality with respect to expected costs.

7. Open Problems and Extensions

Several fundamental challenges persist:

• The extension of Bayesian coin hopping optimality to correlated or non-binary priors remains unsolved, as current results crucially exploit independence and the two-point distribution (Chandrasekaran et al., 2012).
• In combinatorial frameworks, variants incorporating additional operations (e.g., pushing or blocking constraints) yield complex parity exceptions and require new classification theorems (Miyadera et al., 24 May 2025).
• In economic contexts, designing robust, manipulation-resistant mining reward schemes that thwart adversarial coin hopping without violating potential-game convergence presents an ongoing systems-design problem (Spiegelman et al., 2018).

Coin hopping strategies thus unify a spectrum of adaptive, position-reactive policies across stochastic search, strategic response, and combinatorial optimization, with deep connections to Bayesian inference, game theory, and applied market design.