Card guessing after an asymmetric riffle shuffle

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.

💡 Research Summary

The paper studies a single‑riffle‑shuffle card‑guessing game with full feedback under an asymmetric version of the Gilbert‑Shannon‑Reeds (GSR) model. In the classical GSR model the deck is cut near the middle (p = ½) and the two piles are interleaved with probabilities proportional to their current sizes. Here the authors replace the symmetric cut by a Binomial(n, p) cut, where p∈(0,1) and p≠½, so that the expected cut position is p·n. The shuffled permutation can be described equivalently in three ways: (i) a probability measure Rifₚ on the symmetric group Sₙ, (ii) a binomial cut followed by size‑biased interleaving, and (iii) a random word of length n over the alphabet {a,b} with P(a)=p, P(b)=1‑p.

The first technical step is Lemma 3, which gives the exact distribution of the first revealed card. For m=1 the probability is p+(1‑p)ⁿ; for m≥2 it equals (1‑p)·BinomialPMF(n‑1, p; m‑1). This simple formula already shows a qualitative change at p=½: when p>½ the number 1 is most likely to appear first, while for p<½ the mode of the binomial distribution (≈⌊np⌋+1) can dominate.

Using this distribution the authors derive the optimal guessing policy. The policy depends on whether p≥½ or p<½.

If p≥½: Guess “1” for the first card. If the guess is correct, continue guessing the next integer (2,3,…). If a guess fails and the revealed card is m>j (where j is the current guess), the deck is instantly split into two increasing subsequences: {1,…,m‑1} and {m+1,…,n}. The remaining cards are then guessed proportionally to the lengths of these subsequences. This “keep the natural order as long as possible” rule is proved optimal.

If p<½: Define n₀ = ⌈ln(½‑p)/ln(1‑p)⌉. For n≤n₀ the probability p+(1‑p)ⁿ exceeds ½, so the strategy coincides with the p≥½ case (guess 1). For n₀<n≤n₁ (where n₁ is the smallest n≥n₀ such that P(first = 1)≥max_{m≥2}P(first = m)), the optimal first guess is κₙ = 1+⌊np⌋, i.e., the mode of the binomial B(n‑1, p) plus one. After observing κₙ, the same subsequence‑splitting rule applies. For n>n₁ the exponential term (1‑p)ⁿ becomes negligible and again “guess 1” is optimal. Table 2 and Proposition 1 summarize these regimes.

Having fixed the optimal policy, the authors obtain a recursive distributional equation for Xₙ, the total number of correct guesses (Theorem 4). Solving this recursion yields the limit laws summarized in Theorem 1:

Boundary cases p=0 or p=1: Xₙ converges to the deterministic value n·p* (p* = max{p,1‑p}).

Asymmetric interior (p∈(0,1){½}): Xₙ = n·p* + G + o(1) in distribution, where G follows a geometric distribution with success probability ρ = (1‑p*)/p*. Thus the number of correct guesses fluctuates around its linear mean by a discrete “overshoot” with exponential tails.

Symmetric case p=½: Xₙ = n/2 + √n·GG + o(√n), where GG has the Maxwell‑Boltzmann (generalized gamma) density f(x)= (q/2π)·8x² e^{‑2x²}, x≥0. This continuous limit reflects the higher variability when the cut is truly balanced.

The paper further investigates phase transitions when p approaches the critical values ½, 0, or 1 at rates depending on n. Setting p = ½ + αₙ with αₙ→0, the authors distinguish regimes based on the speed of αₙ:

– If αₙ ≫ n^{‑½}, the geometric overshoot dominates and the limit law remains the geometric one.

– If αₙ ∼ c·n^{‑½}, a non‑trivial mixture of geometric and generalized‑gamma components appears.

– If αₙ ≪ n^{‑1}, the generalized‑gamma (Maxwell‑Boltzmann) law re‑emerges.

Analogous analyses are performed for p→1 (or 0), where the exponential decay of (1‑p)ⁿ forces the first‑card probability to concentrate on 1, and Xₙ becomes essentially deterministic. These transition results are formalized in Theorems 9 and 10.

Overall, the paper contributes three major advances: (1) a precise characterization of the optimal guessing strategy under an asymmetric riffle shuffle, (2) an exact recursive description of the distribution of correct guesses, and (3) a comprehensive limit‑theoretic picture that captures how the distribution morphs from geometric to generalized‑gamma as the bias parameter p varies. The work bridges combinatorial probability, optimal stopping theory, and the probabilistic analysis of shuffling, and it opens avenues for applications in areas such as randomized algorithms, statistical experiment design, and even psychophysical studies of human prediction under biased random processes.