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Proof of a conjecture about Parrondo's paradox for two-armed slot machines

Huaijin Liang, Zengjing Chen

TL;DR

The paper proves Ethier and Lee's conjecture that a two-armed Futurity slot machine with Futurity parameter $J=2$ exhibits Parrondo's paradox for every nonrandom periodic pattern $D$ with $r,s\ge1$: each arm is individually fair, yet alternating according to $D$ yields a positive long-run casino profit. It derives the casino's asymptotic profit per coup as $R_D=2Q\,S$, where $Q$ and $S$ are explicit functions of the arm probabilities and the pattern, and shows $S>0$ and $Q>0$ under $p_A\neq p_B$, ensuring $R_D>0$. The authors provide two proof strategies—a mixture of rearrangement arguments and a direct formula-based verification—along with a concrete example $D=(1,1,1,2,1,3)$ that yields a positive profit, thereby confirming the conjecture. The results demonstrate that a two-armed Futurity machine remains profitable in the long run under any nonrandom periodic strategy, extending the Parrondo framework to a history-dependent, cam-based setting with rigorous combinatorial and probabilistic analysis.

Abstract

The 1936 Mills Futurity slot machine had the feature that, if a player loses 10 times in a row, the 10 lost coins are returned. Ethier and Lee (2010) studied a generalized version of this machine, with 10 replaced by deterministic parameter J. They established the Parrondo effect for a hypothetical two-armed machine with the Futurity award. Specifically, arm A and arm B, played individually, are asymptotically fair, but when alternated ran-domly (the so-called random mixture strategy), the casino makes money in the long run. They also considered the nonrandom periodic pattern strategy for patterns with r As and s Bs (e.g., ABABB if r = 2 and s = 3). They established the Parrondo effect if r + s divides J, and conjectured it in four other situations, including the case J = 2 with r >= 1 and s >= 1. We prove the conjecture in the latter case.

Proof of a conjecture about Parrondo's paradox for two-armed slot machines

TL;DR

The paper proves Ethier and Lee's conjecture that a two-armed Futurity slot machine with Futurity parameter exhibits Parrondo's paradox for every nonrandom periodic pattern with : each arm is individually fair, yet alternating according to yields a positive long-run casino profit. It derives the casino's asymptotic profit per coup as , where and are explicit functions of the arm probabilities and the pattern, and shows and under , ensuring . The authors provide two proof strategies—a mixture of rearrangement arguments and a direct formula-based verification—along with a concrete example that yields a positive profit, thereby confirming the conjecture. The results demonstrate that a two-armed Futurity machine remains profitable in the long run under any nonrandom periodic strategy, extending the Parrondo framework to a history-dependent, cam-based setting with rigorous combinatorial and probabilistic analysis.

Abstract

The 1936 Mills Futurity slot machine had the feature that, if a player loses 10 times in a row, the 10 lost coins are returned. Ethier and Lee (2010) studied a generalized version of this machine, with 10 replaced by deterministic parameter J. They established the Parrondo effect for a hypothetical two-armed machine with the Futurity award. Specifically, arm A and arm B, played individually, are asymptotically fair, but when alternated ran-domly (the so-called random mixture strategy), the casino makes money in the long run. They also considered the nonrandom periodic pattern strategy for patterns with r As and s Bs (e.g., ABABB if r = 2 and s = 3). They established the Parrondo effect if r + s divides J, and conjectured it in four other situations, including the case J = 2 with r >= 1 and s >= 1. We prove the conjecture in the latter case.
Paper Structure (8 sections, 12 theorems, 186 equations)

This paper contains 8 sections, 12 theorems, 186 equations.

Table of Contents

  1. Introduction
  2. Previous conclusions and the conjecture
  3. Main results
  4. Calculation of the asymptotic profit per coup for the casino
  5. Preliminary calculation of casino asymptotic profits per coup
  6. Asymptotic profit per coup for the casino under a simple strategy
  7. Differences between the two strategies
  8. Proof of the conjecture

Key Result

Theorem 1

Under random-mixture strategy $C$, the casino's asymptotic profit per coup $R_C$ attributable to the two-armed feature of the machine is for $J\ge 2$, $0<\gamma<1$, where $f(z)=J(1-z)z^J/(1-z^J)$, $q_A=1-p_A$, $q_B=1-p_B$. Furthermore, if $p_A\neq p_B$, then $R_C>0$ for any random-mixture strategy $C$. The Parrondo effect is observed for any random-mixture strategy $C$, which results from the fac

Theorems & Definitions (35)

  • Theorem 1: Ethier and Lee 2010
  • Theorem 2: Ethier and Lee 2010
  • Conjecture : Ethier and Lee 2010
  • Remark 3
  • Definition 4
  • Remark 5
  • Theorem 6
  • Remark 7
  • Theorem 8
  • Remark 9
  • ...and 25 more