linear programming: optimal strategy to launch three dependent currencies

Suppose I have three correlated currencies. The marginal probability of a currency head $ i $ is denoted by $ p_i $.

The conditional probability of head per coin $ i $ given the results of the currency $ j $ Y $ k $ is denoted by $ p_i | x_j, x_k $, where $ x_j, x_k in {H, T } $. We can similarly build the conditional probability of $ i $ dice $ x_j $.

Each coin can be thrown at most once and you will receive a $ 1 for one side and – $ 1 for one queue. You don't have to throw all the coins, and your goal is to maximize the total reward.

What would be the optimal sequence of flipping coins in this case?

If the currencies are independent of each other, the order would not matter. The optimal strategy should be to "throw a coin if $ p_i> frac {1} {2} $"However, this does not have to be optimal for correlated currencies. I have been thinking about this problem for a long time, but I cannot find a general solution or intuition that can help …