I've been thinking about the following game for a while and am curious if anyone has any ideas of how to analyze it.
Say I have two biased coins: coin 1 that shows heads with probability $p$ and coin 2 that shows heads with probability $q$. You and I both know the statistics of the coins.
The game proceeds in multiple rounds as follows:
In the starting round $n=0$:
- I (privately) pick a coin and flip it, we both observe the outcome
- you decide to make a guess of which coin I just flipped, or continue watching
- if you guess correctly, I pay you $\$100$; if you guess incorrectly, you receive no reward and the game is over
At each subsequent round $n\ge 1$:
- I decide to stay with my current coin or reach into my pocket and swap out the current coin for the other coin
- you can see whether I swapped out the coin or not (assume I must switch if I reach into my pocket)
- I flip the coin and we both observe the outcome
- you decide to guess which coin was just flipped, or keep watching
- if you guess correctly, I pay you $\$100\cdot\delta^n$, where $\delta\in(0,1)$; if you guess incorrectly the game ends with you getting nothing
I want to find the "best" switching strategy to minimize the (expected) amount of money I have to pay you.
The probabilities $p$ and $q$ can take on any value, but let's assume that they cannot be equal.
Since you are trying to maximize your reward, the discount factor $\delta$ incentives you to guess correctly as quickly as possible.
Since there are only two coins and you observe when I switch, you are trying to discern between two possible coin sequences, one where the initial coin was coin 1 and the other where the initial coin was coin 2.
My first thoughts are that I would want to keep the empirical averages (of the two sequences) as close as possible to each other. Intuitively this will be easy if $p$ and $q$ are close, but hard if they are far apart.