# pr.probability – Guess the parts of each other

I recently thought about the next game (has it been considered before?).

Alice and Bob collaborate. Alice observes a sequence of independent unbiased random bits $$(A_n)$$, then choose an integer $$a$$. Similarly, Bob observes a sequence of independent unbiased random bits $$(B_n)$$, independent of $$(A_n)$$, then choose an integer $$b$$. Alice and Bob are not allowed to communicate. They win the game if $$A_b = B_a = 1$$.

What is the optimal probability of winning $$p_ {opt}$$? A strategy for each player is a function (Borel) $$f: {0,1 } ^ { mathbf {N}} to mathbf {N}$$, and we want to maximize the probability of winning on pairs of strategies $$(f_A, f_B)$$.

Constant strategies win with probability $$1/4$$and it may be counterintuitive that you can do better. To choose $$f$$ to be the index of the first $$1$$ earn with probability $$1/3$$. This is not optimal however, by running a small program trying randomly modified strategies on a finished window, I could find that $$p_ {opt} geq 358/1023 approx 0.3499$$, with a pair (with $$f_A = f_B$$) devoid of any apparent motive.

But a more interesting question is: can you prove an upper limit on $$p_ {opt}$$, besides the trivial $$p_ {opt} leq 1/2$$?