Quantum-over-classical Advantage in Solving Multiplayer Games

TL;DR

This paper explores the advantages of quantum algorithms over classical ones in solving multiplayer subtraction games, extending known results to multi-player scenarios and demonstrating quantum speedups in computational game theory.

Contribution

It generalizes quantum complexity results from two-player to multiplayer subtraction games, maintaining the same complexity bounds and highlighting quantum advantages.

Findings

Quantum algorithms outperform classical algorithms in multiplayer subtraction games.

Complexity bounds for multiplayer games are similar to the two-player case.

Quantum solutions achieve sublinear time complexity in solving these games.

Abstract

We study the applicability of quantum algorithms in computational game theory and generalize some results related to Subtraction games, which are sometimes referred to as one-heap Nim games. In quantum game theory, a subset of Subtraction games became the first explicitly defined class of zero-sum combinatorial games with provable separation between quantum and classical complexity of solving them. For a narrower subset of Subtraction games, an exact quantum sublinear algorithm is known that surpasses all deterministic algorithms for finding solutions with probability $1$. Typically, both Nim and Subtraction games are defined for only two players. We extend some known results to games for three or more players, while maintaining the same classical and quantum complexities: $\Theta\left(n^2\right)$ and $\tilde{O}\left(n^{1.5}\right)$ respectively.