One of my favorite problems in NP $\cap$ co-NP is deciding who wins a simple stochastic game. The game is played on a directed graph by two players, call them A and B. This graph contains several types of nodes. There is a source node and two sink nodes, one for each of the players. There are also random nodes (which include the source), "A" nodes, and "B" nodes. At the start of the game, for each "A" or "B" node, the corresponding player chooses one of the edges leading away from it, without seeing the other player's choices.
Each player's goal is to maximize the probability that the token lands on their sink node. The question in NP $\cap$ co-NP is: does player A have a winning strategy that ensures the token lands on his sink node with probability at least $\frac{1}{2}$?