There is not, actually! You can show this strategy does as well as the case where prisoners have full information of each other's moves.[0] So any other strateg

There is not, actually! You can show this strategy does as well as the case where prisoners have full information of each other's moves.[0] So any other strategy where each prisoner is blind to others' actions can always be executed in the full-information case above; so this bound is tight and this strategy is optimal: if there was a strategy that did better in the blind case, you could execute it in the full-information case to get a better outcome, but this is impossible. For a nice exposition of this, see Curtin and Warshauer's article "The Locker Puzzle." --- [0] More specifically, a game where all lockers are left open, so every strategy has the same probability of winning.