A simplified version of card game Winner
This should probably be a comment, but it's too long. 

A related game was studied by Jeff Kahn, Jeff Lagarias and Hans Witsenhausen, in the series of articles Single-Suit Two-Person Card Play I, II, III, and On Laskar's Card Game. In the game they studied, each player has $n$ cards, dealt from $2n$ cards numbered $1$ $\ldots$ $2n$. Each trick consists of two cards, the higher card wins the trick, and the winner leads. The object is to take the most tricks. 

They proved a number of interesting facts about the optimal strategy, but were unable to find an efficient algorithm for optimal play, and were also unable to prove that it was NP-hard. 

For the misère game, where each person tries to take the fewest number of tricks, they were able to give the optimal strategy.

For the most part, these results were obtained by first looking at the results of a computer program which found the optimal strategy for small instances, then looking for patterns to obtain conjectures, and finally proving these conjectures. I suspect that this would also be a fruitful approach to take for the OP's game.