A card problem

I was recently party to a very simple, but nonetheless interesting card game. In the game, four players begin the game holding one suit of cards each from a single deck. Then, for thirteen rounds (until the players have run out of cards) every player discards one of their cards simultaneously into the centre, with the player who discards the highest-value card winning the round. If two or more players discard cards of the same face value, then the round is a draw. The winner of the most rounds at the end then wins the game.

While this game is pretty simple and at first glance largely up to luck, it did make me wonder whether there were any strategies that could increase a player’s chance of winning. It would seem so, especially since the starting hands are constant. Therefore, it would be interesting to find both the optimal strategy assuming everyone else plays randomly (i.e. equal probability to play any remaining card), as well as the optimal strategy assuming everyone else plays optimally.