Is this game solvable? Suppose we have two stacks of cards on a table. On each turn, each of two players draws a card from the top of one of the stacks. The game ends when there are no cards left in either stack. The person with the greatest sum then wins. If this were a chess problem, for example, one could attempt to solve this game by starting from a winning final position, and then inducting backwards. However, in this game, the final card picked up has little bearing, as it is the overall sum that results in a winner. Is it possible to find a game plan for perfect play, assuming both players pursue this game plan?

EDIT: The values of all the cards are known by both players, i.e. the players are omniscient when it comes to information about the cards. The card deck need not be complete, just some random cards in two stacks on a table, and we only care about the values.

EDIT 2: I'm wondering whether this can be solved efficiently with an algorithm that does not have to go through all the potential decisions in a decision tree.
I haven't thought it carefully, but I think You can use dynamic programming. If we have n cards in first deck and m cards in second deck, then there is n*m possible states of stack.  

Best_Move(i,j)  
    return Max(Stack1[i]-Best_Move(i-1,j),Stack2[j]-Best_Move(i,j-1))


You just have to remember all partial results, so they won't be calculated twice. This should give You the bes strategy in O(n*m).