Consider the following game between party 1 and party 2. Party 1 is honest; i.e. it follows the game exactly:
Assume that party 1 flips a fair coin $b$, encrypts it to $c$, and sends $c$ to party 2. Party 2 returns a value $b'$. Finally, party 1 reveals b by decrypting c.
Party 2 wins if $b'=b$, and loses otherwise. Assume that the encryption is not perfect; therefore $c$ reveals partial information about $b$, such that party 2 learns that there's a 60% chance that $b=0$. What strategy maximizes the winning chance of party 2?
I think the problem can be symbolically denoted as follows:
- $\Pr[b=0] = \Pr[b=1] = 0.5$
- $\Pr[b=0 \mid c] = 1-\Pr[b=1 \mid c]= 0.6$
and we want to choose $\Pr[b'=0 \mid c]$ such that $\Pr[b'=b \mid c]$ is maximized.
PS: The encryption used above is in fact a commitment scheme, but let's not care about that.