The space complexity of mirror games

Abstract

We consider a simple streaming game between two players Alice and Bob, which we call the mirror game. In this game, Alice and Bob take turns saying numbers belonging to the set $\{1, 2, \dots,2N\}$. A player loses if they repeat a number that has already been said. Bob, who goes second, has a very simple (and memoryless) strategy to avoid losing: whenever Alice says $x$, respond with $2N+1-x$. The question is: does Alice have a similarly simple strategy to win that avoids remembering all the numbers said by Bob? The answer is no. We prove a linear lower bound on the space complexity of any deterministic winning strategy of Alice. Interestingly, this follows as a consequence of the Eventown-Oddtown theorem from extremal combinatorics. We additionally demonstrate a randomized strategy for Alice that wins with high probability that requires only $\tilde{O}(\sqrt N)$ space (provided that Alice has access to a random matching on $K_{2N}$). We also investigate lower bounds for a generalized mirror game where Alice and Bob alternate saying $1$ number and $b$ numbers each turn (respectively). When $1+b$ is a prime, our linear lower bounds continue to hold, but when $1+b$ is composite, we show that the existence of a $o(N)$ space strategy for Bob implies the existence of exponential-sized matching vector families over $\mathbb{Z}^N_{1+b}$.