Let A = {1, 2, ..., n}, and let X and Y be two random variables on the same space, taking values in A, and distributions given by: P(X = i) = $p_i$, and P(Y = i) = $q_i$, for any i in {1,2,...,n}

What is the maximum possible value of P(X = Y)?

Basically, we have to choose a good joint distribution of X and Y, such that the sum of diagonal entries in the joint distribution table/matrix is maximized. For n = 2, this is easy to do, and I get the answer (1 - |$p_1$ - $q_1$|). But even for n = 3, I am not able to do much. A solution/strategy for general n would be appreciated.