I created a simple game to explore the nature of information in making a series of decisions to win a game. It's a variant of the game of "war" that I call "Fog".

In Fog, each player is given ten cards of increasing value from 1 to 10. Each player then chooses a card to play from their hand and places it face down. Once both players have made their choices their cards are revealed. The player with the highest card gets the points of lower card. In the event of a tie, neither player scores. The goal of the game is to have the higher score after all cards have been played.

For example, if you play the 7 card and your opponent plays the 1 card, you win 1 point. If your opponent plays the 6 card, you gain 6 points. If your opponent plays the 7 card, you get zero points. If your opponent plays an 8 or higher card they get 7 points.

Question: Is there an optimal strategy or play possible for this simple game?