Price of Locality in Permutation Mastermind: Are TikTok influencers Chaotic Enough?

TL;DR

This paper investigates how local strategies, where consecutive guesses in permutation Mastermind are similar, affect the number of guesses needed, revealing they are less efficient than non-local strategies and analyzing their computational complexity.

Contribution

It introduces and analyzes the impact of local strategies in permutation Mastermind, showing their quadratic guess complexity and computational hardness results.

Findings

Optimal local strategies require quadratic guesses, unlike non-local strategies with O(n log n) guesses.

NP-hardness is proven for satisfiability in $\\ell_3$-local strategies.

A randomized polynomial algorithm exists for the $\\ell_2$-local variant.

Abstract

In the permutation Mastermind game, the goal is to uncover a secret permutation $\sigma^\star \colon [n] \to [n]$ by making a series of guesses $\pi_1, \ldots, \pi_T$ which must also be permutations of $[n]$, and receiving as feedback after guess $\pi_t$ the number of positions $i$ for which $\sigma^\star(i) = \pi_t(i)$. While the existing literature on permutation Mastermind suggests strategies in which $\pi_t$ and $\pi_{t+1}$ might be widely different permutations, a resurgence in popularity of this game as a TikTok trend shows that humans (or at least TikTok influencers) use strategies in which consecutive guesses are very similar. For example, it is common to see players attempt one transposition at a time and slowly see their score increase. Motivated by these observations, we study the theoretical impact of two forms of "locality" in permutation Mastermind strategies: $\ell_k$-local strategies, in which any two consecutive guesses differ in at most $k$ positions, and the even more restrictive class of $w_k$-local strategies, in which consecutive guesses differ in a window of length at most $k$. We show that, in broad terms, the optimal number of guesses for local strategies is quadratic, and thus much worse than the $O(n \lg n)$ guesses that suffice for non-local strategies. We also show NP-hardness of the satisfiability version for $\ell_3$-local strategies, whereas in the $\ell_2$-local variant the problem admits a randomized polynomial algorithm.