Mirror Games Against an Open Book Player

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.