Mirror Games Against an Open Book Player
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Mirror games were invented by Garg and Schnieder (ITCS 2019). Alice and Bob take turns (with Alice playing first) in declaring numbers from the set 1,2, ...2n. If a player picks a number that was previously played, that player loses and the other player wins. If all numbers are declared without repetition, the result is a draw. Bob has a simple mirror strategy that assures he won't lose and requires no memory. On the other hand, Garg and Schenier showed that every deterministic Alice needs memory of size linear in n in order to secure a draw. Regarding probabilistic strategies, previous work showed that a model where Alice has access to a secret random perfect matching over 1,2, ...2n allows her to achieve a draw in the game w.p. a least 1-1/n and using only polylog bits of memory. We show that the requirement for secret bits is crucial: for an `open book' Alice with no secrets (Bob knows her memory but not future coin flips) and memory of at most n/4c bits for any c>2, there is a Bob that wins w.p. close to 1-2^-c/2.
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