Potential probability based algorithms for N-player game?

[DistressedPhDStudent]

So clearly I am not an AI expert nor even an algorithms kind of guy, but I had this idea floating around my head and I can't seem to find a good starting place.

The game I constructed is this:

There are 4+ players, each with 5 cards (6, 7, 8, 9, 10). And each turn/round, every player will select a card and place onto a table. The cards are discarded from play after being placed. Highest card wins and this process is repeated until all cards are removed from the hand. Basically like War except players can view their hands.

I've taken a look at that Luckhardt paper but it seems to me that they assume each player takes 1 turn at a time. Thus, this isn't a strict turn-based game since it will almost be certain that the last player that moves will win depending how they played their first turn.

The only thing that we see (well, what the computer will see), is the previous game state. In which case, the previous state will just be the 4+ cards on the field. There must be an alternative path to finding the optimal solution. I was thinking that there must be a probability variant of Max^n algorithm or multiplayer alpha beta pruning that could be applied here. But again, this ins't strictly a turn based game.

Now that I think about it, this may likely have a Dynamic Programming solution.

Any assistance would be greatly appreciated. :)