Optimal Strategy in "Guess Who?": Beyond Binary Search

TL;DR

This work treats Guess Who? as a zero-sum Simple Stochastic Game with states $(n,m,P_i)$ and bid-driven transitions, yielding explicit optimal strategies for both players. The main result provides closed-form expressions for the optimal bid $b^{\star}(n,m)$ and winning probability $p^{\star}(n,m)$, distinguishing a 'weeds' regime where $b^{\star}=2^{k}$ and a 'upper hand' regime where $b^{\star}=\left\lfloor \tfrac{1}{2}n\right\rfloor$, together with exact formulas for $p^{\star}(n,m)$ in each case. The analysis further reveals log-periodic asymptotics through a Continuous Guess Who? variant, defining $p_{\infty}^{\star}$ with $\,p_{\infty}^{\star}(2x,2y)=p_{\infty}^{\star}(x,y)$ and showing $p^{\star}(n,m)=p_{\infty}^{\star}(n,m)+O(1/(nm))$. A rigorous proof via induction on $n+m$ confirms the piecewise $q(n,m)$ characterization and the optimal bidding rules, linking discrete and continuous models and offering precise insights into risk-taking strategies in competitive racing to a goal.

Abstract

"Guess Who?" is a popular two player game where players ask "Yes"/"No" questions to search for their opponent's secret identity from a pool of possible candidates. This is modeled as a simple stochastic game. Using this model, the optimal strategy is explicitly found. Contrary to popular belief, performing a binary search is \emph{not} always optimal. Instead, the optimal strategy for the player who trails is to make certain bold plays in an attempt catch up. This is discovered by first analyzing a continuous version of the game where players play indefinitely and the winner is never decided after finitely many rounds.

Figures (2)

• Figure 1: The winning probability $p^\star(n,m)$ (left) and the optimal bid $b^\star(n,m)$ (right) of Player 1 "Guess Who?" on Player 1's turn when Player 1 has $n$ candidates in their pool and Player 2 has $m$ candidates in their pool. The region where $n=1$ or $m=1$ and a player immediatly wins is highlited with stars. Values where $p^\star(n,m)$ is close to 0.5 are also shaded. The "upper hand" regions $U_k$ and "in the weeds" regions $W_k$ are also labelled.
• Figure 2: The probability $p_{\infty}^{\star}(2^{k}\alpha,2^{k}\beta)$ of Player 1 winning "Continuous Guess Who?" from the position $x=2^k\alpha$, $y=2^k\beta$ in the region $(\alpha,\beta)\in(2,\infty)\times(1,2]$ and $(\alpha,\beta)\in(1,2]\times(1,\infty)$. This closely approximates "Guess Who?" when $n,m$ are large: $p^{\star}(n,m)=p_{\infty}^{\star}(n,m)+O(\frac{1}{nm})$.

Theorems & Definitions (31)

• Theorem 1.1
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Proposition 2.4
• proof
• Remark 2.5
• Definition 2.6
• Definition 2.7
• Definition 3.1