Abstract

We consider the following game between two players Alice and Bob, which we call the mirror game. Alice and Bob take turns saying numbers belonging to the set {1, 2, ...,N}. A player loses if they repeat a number that has already been said. Otherwise, after N turns, when all the numbers have been spoken, both players win. When N is even, Bob, who goes second, has a very simple (and memoryless) strategy to avoid losing: whenever Alice says x, respond with N+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 O~(sqrt N) space (provided that Alice has access to a random matching on K_N).
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 (when N != 0 mod (1+b)) implies the existence of exponential-sized matching vector families over Z^N_{1+b}.

@InProceedings{garg_et_al:LIPIcs:2018:10129,
  author =	{Sumegha Garg and Jon Schneider},
  title =	{{The Space Complexity of Mirror Games}},
  booktitle =	{10th Innovations in Theoretical Computer Science  Conference (ITCS 2019)},
  pages =	{36:1--36:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-095-8},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{124},
  editor =	{Avrim Blum},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2018/10129},
  URN =		{urn:nbn:de:0030-drops-101295},
  doi =		{10.4230/LIPIcs.ITCS.2019.36},
  annote =	{Keywords: Mirror Games, Space Complexity, Eventown-Oddtown}
}