  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}.
