Problem. There's $n$ coins, $n-1$ are fair and one always shows heads, but all coins look the same. You are allowed to flip $m$ coins, where $m < n$ and the choice of the $i$th flip can depend on previous outcomes. After $m$ flips you give a bet on which coin you think is special. What strategy should be used to maximize the expected probability of winning (guessing right)?
Using computer simulations I'm led to believe the answer is greedy: stick with one coin and keep flipping it (until is shows tails and you rule it out and move to another coin), but I haven't been able to prove optimality.
What I did manage to prove is that if I have priors $p_1,\dots,p_n$ ($p_i$ the probability the special coin is the $i$th coin) and only one toss, then tossing the coin with the maximal $p_i$ will give both
i) the highest expected maximal posterior probability (which is the probability I'll win if the game ends after this toss)
ii) the lowest expected posterior entropy (narrowing down the information still unknown)
but I don't know how to generalize this to several tosses.
I also figured this is a special case of the multi armed bandit game but didn't find relevant information in the literature for this case.