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Card guessing after an asymmetric riffle shuffle

Markus Kuba

TL;DR

This paper analyzes card guessing after a single asymmetric riffle shuffle with cut near $np$, under complete feedback. It derives the optimal guessing strategy, provides a distributional decomposition of the total number of correct guesses $X_n$, and proves limit laws: for $p\in(0,1)\setminus\{\tfrac12\}$, $X_n-n p^{*}$ converges to a geometric limit, while at $p=\tfrac12$ the fluctuations follow a Maxwell–Boltzmann-type distribution; near critical boundaries and near $p=\tfrac12$ it identifies several phase transitions with explicit scaling regimes and limiting laws. A central technical component is the link to the two-color guessing problem $C_{m_1,m_2}$ and its limit behavior, which feeds into the distributional equation for $X_n$. The results illuminate how asymmetry in the shuffle interacts with complete feedback to shape optimal play and the probabilistic fluctuations of success, with potential applications in randomized algorithms and statistical testing contexts that involve sequential guessing under partial information. The findings contribute to a richer understanding of phase transitions in combinatorial stochastic processes arising from riffle-shuffle models.

Abstract

We consider a card guessing game with complete feedback. An ordered deck of $n$ cards labeled $1$ up to $n$ is riffle-shuffled exactly one time. Given a value $p\in(0{,}1)\setminus\{\frac12\}$, the riffle shuffle is assumed to be unbalanced, such that the cut is expected to happen at position $p\cdot n$. The goal of the game is to maximize the number of correct guesses of the cards: one after another a single card is drawn from the top, and shown to the guesser until no cards remain. We provide a detailed analysis of the optimal guessing strategy and study the distribution of the number of correct guesses.

Card guessing after an asymmetric riffle shuffle

TL;DR

This paper analyzes card guessing after a single asymmetric riffle shuffle with cut near , under complete feedback. It derives the optimal guessing strategy, provides a distributional decomposition of the total number of correct guesses , and proves limit laws: for , converges to a geometric limit, while at the fluctuations follow a Maxwell–Boltzmann-type distribution; near critical boundaries and near it identifies several phase transitions with explicit scaling regimes and limiting laws. A central technical component is the link to the two-color guessing problem and its limit behavior, which feeds into the distributional equation for . The results illuminate how asymmetry in the shuffle interacts with complete feedback to shape optimal play and the probabilistic fluctuations of success, with potential applications in randomized algorithms and statistical testing contexts that involve sequential guessing under partial information. The findings contribute to a richer understanding of phase transitions in combinatorial stochastic processes arising from riffle-shuffle models.

Abstract

We consider a card guessing game with complete feedback. An ordered deck of cards labeled up to is riffle-shuffled exactly one time. Given a value , the riffle shuffle is assumed to be unbalanced, such that the cut is expected to happen at position . The goal of the game is to maximize the number of correct guesses of the cards: one after another a single card is drawn from the top, and shown to the guesser until no cards remain. We provide a detailed analysis of the optimal guessing strategy and study the distribution of the number of correct guesses.
Paper Structure (15 sections, 12 theorems, 85 equations, 3 figures, 1 table)

This paper contains 15 sections, 12 theorems, 85 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Riffle shuffle and card guessing
  3. Card guessing with colored cards
  4. Main results
  5. Notation and plan of the paper
  6. Asymmetric Gilbert-Shannon-Read model
  7. Riffle shuffle model
  8. First drawn card
  9. Optimal strategy:
  10. Distributional equation
  11. Limit law
  12. More limit laws and phase transitions
  13. Phase transition at the value one-half
  14. Phase transition at the boundary
  15. Summary and Outlook

Key Result

Theorem 1

Given $p\in[0,1]$, let $p^{\ast}=\max\{p,1-p\}$. The random variable $X_n$, counting the number of correct guessing with full feedback after an (a)symmetric riffle shuffle with parameter $p$, satisfies: with $D$ denoting a degenerated random variable, such that ${\mathbb {P}}\{D=0\}=1$, $G$ denoting a geometric distribution with parameter $\rho=(1-p^{\ast})/p^{\ast}$, ${\mathbb {P}}\{G=k\}=\rho^k

Figures (3)

  • Figure 1: Example of a one-time riffle shuffle: a deck of seven cards is split after the number two with probability $\binom72\cdot p^2(1-p)^5$ and then interleaved.
  • Figure 2: Comparison of the values ${\mathbb {P}}\{\mathop{\mathrm{FC}}\nolimits_n=1\}=0.15+0.85^n$ and $\max_{2\le m\le n}{\mathbb {P}}\{\mathop{\mathrm{FC}}\nolimits_n=m\}$, $1\le n\le 40$.
  • Figure 3: Plot of the density functions $f_{\chi}(x)$ of $\frac{1}{2}\chi(3,\sqrt{2}\cdot |b|)$ and the generalized Gamma distribution $f_{GG}(x)$ (case $b=0$ of $f_{\chi}(x)$), for $b\in\{0,0.25,0.5,1,1.5,2\}$ occurring in Proposition \ref{['Prop:twoColorBinLimit']} and Theorem \ref{['the:Half']}.

Theorems & Definitions (33)

  • Theorem 1
  • Definition 1
  • Lemma 2
  • proof
  • Lemma 3: Distribution of the first drawn card
  • Remark 1
  • proof
  • Example 1
  • Example 2
  • Proposition 1: Optimal strategy with an asymmetric Gilbert-Shannon-Read model
  • ...and 23 more