This is another iteration of beat the casino. That question did not require a practical, implementable strategy, whereas this one does.
The rules are the same and I list them below. However, the OP answer to that question only had a theoretical result with no concrete strategy, and the accepted answer was not much of an improvement on the naive 2/3 method. I am looking for a practical, implementable result that achieves at least 70% success rate; well within the bounds of the theoretical result.
The rules are the same:
- Each round, $A$, $B$, and the casino simultaneously decide to show a $0$ or a $1$. If all three numbers match, $A$ and $B$ win that round.
- $A$ and $B$ are working cooperatively and can communicate before the game begins.
- $A$ has a method, just before the game starts, to learn the choices the casino will make over all the rounds. However, after learning this information $A$ cannot communicate with $B$ in any way except by her choices in the game.
- $A$ and $B$ are trying to maximize the fraction $p$ of rounds they win in the worst case.
- the game lasts for $n$ rounds.
What is the best possible $p$ that $A$ and $B$ can achieve as $n \to \infty$?
NOTE: I answered the referenced question (very late) and was able to achieve 67.8% with a relatively easy to describe strategy. I provided exact details on the strategies of each player, which got fairly complicated. If your strategy is easy to describe but complicated to implement, that is fine, so long as you can show the implementation is possible.