Can you help me find an optimal mixed strategy for this simple 2-person allocation game?

Question

Consider the following simple 2-person game. Players 1 and 2 each have 100 dollar coins, with a barrier between them, hiding each other's moves. Each player must allocates his 100 coins into 3 piles: A, B, and C. For example 40 in A, 30 in B, 30 in C. Then the barrier is removed. Whichever player has more coins in pile A is the winner of A, same with B and C. The player who wins the most of the 3 piles is the overall winner. Obviously, if Player 1's move is known, Player 2 can allocate his coins to win 2 of the 3 piles, so no pure strategy is optimal. But if both players' moves are unknown to each other, is there an optimal mixed strategy for allocating between the 3 piles?

Answer 1

If I use a strategy that puts all coins on a single random pile and leaves the other piles empty, and you use a strategy that has a positve probability $p$ of not putting all coins on a single pile, then I end up with a minimum of $100$ coins and an expected number of $(1+\frac p3)100$ coins (and accordingly, you end with a maximum of $100$ coins and an expected number of $(1-\frac p3)100$ coins. Hence my strategy is strictly better than any strategy with positive $p$. On the other hand, it ties with every strategy that has $p=0$.