Coin flipping and waiting times paradoxes: Why fair coins are exceptional

TL;DR

The paper studies first-hit races of independent, potentially biased, symbolic strings and discovers a sharp fairness dichotomy: with a fair $s$-sided die, stochastic dominance induces a total pre-order over all strings ranked by $\mathbb{E}[\tau_T]$, while bias yields reversals and non-transitive cycles. It develops a compact Hadamard–generating-function calculus to compute head-to-head odds and to certify finite algebraic crossover points, yielding precise examples and a comprehensive computational census up to length 8 patterns. The main contributions include (i) a rigorous link between fairness and comparability, (ii) exact structure and endpoint behavior of parity functions $g_{A,B}(p)$, and (iii) extensive demonstrations of paradoxical reversals and cycles under bias, with exact algebraic endpoints and certified results. The findings illuminate how small biases can qualitatively alter race outcomes, with implications for understanding non-transitivity in independent-source string races and for the design of fair comparisons in probabilistic string games.

Abstract

Penney's Ante exhibits non-transitivity when two target strings race to appear in a shared stream of coin tosses. We study instead independent string races, where each player observes their own independent and identically distributed (i.i.d.) coin/die stream (possibly biased), and the winner is the player whose target appears first (under an explicit tie convention). We derive compact generating-function formulas for waiting times and a Hadamard-generating-function calculus for head-to-head odds. Our main theorem shows that for a fair -sided die, stochastic dominance induces a total pre-order on all strings, ordered by expected waiting time. For binary coins, we also prove a converse: total comparability under stochastic dominance characterises the fair coin (), and any bias yields patterns whose waiting times are incomparable under stochastic dominance. In contrast, bias allows both (i) reversals between mean waiting time and win probability and (ii) non-transitive cycles; we give explicit examples and certified computational classifications for short patterns.

Figures (2)

• Figure 1: The non-transitivity region $\mathcal{R}\subset\Delta^\circ$ (axes $u=p_0$, $v=p_1$, so $p_2=1-u-v$) for the cycle $000>020>001>000$ under random tie-break. The boundary curves are the loci where the relevant pairwise win probabilities equal $1/2$. The asterisk indicates the point $(p_0,p_1,p_2)=\Bigl(\frac{624}{1468},\ \frac{399}{1468},\ \frac{445}{1468}\Bigr)$ mentioned above.
• Figure 2: The volume of independent bias triples $(p_A, p_B, p_C)$ for which the patterns $T_A=\mathrm{HHH}$, $T_B=\mathrm{HTH}$, and $T_C=\mathrm{HHT}$ form a non-transitive cycle under random tie-break (axes are $p_A,p_B,p_C\in(0,1)$). On the left, the twin arches are shown, while the right shows in detail the peak of one arch.

Theorems & Definitions (52)

• Theorem 1: Fairness dichotomy
• Definition 2: Borders and prefix--suffix lengths
• Example 3: Quick examples
• Theorem 4
• Remark 5: Rewriting $h_T$ and the expectation
• Remark 6: Provenance and proof sketch
• Corollary 7
• Corollary 8
• proof
• Example 9