I want to find a strategy to maximize the probability of winning the following game:
Suppose that we have two players, with each player rolls a six-sided dice. They cannot see their own outcome but they can see the outcome of the other player, and no other information is passed between the two players. They must guess their own dice, and if they are both correct, they win. If either is wrong, they lose.
The maximum strategy I can think of is to either guess exactly the outcome of the other player, or guess an outcome that, summed by the outcome by the other player, will equal to 7. Both of them will guarantee an probability of $\frac{1}{6}$.
I just wonder whether it could be better than $\frac{1}{6}$. If so, what could it be; if not, then how to prove the above strategies are the best.
Let $A$ be the event of the first person guessing correctly and $B$ be the event of the second person guessing correctly. No matter what strategy they use, it seems clear that $P(A)$ and $P(B)$ cannot be greater than $\frac16$, since they have no information that would inform their guess. The only thing that they have control over is coordinating a strategy such that $P(B\mid A)=1$ so that the probability of one being right is the same as the probability of both being right. As either of the strategies you mention achieve that, it seems evident to me that it is ideal.