Is there an optimal strategy for this game of card?

Question

This game is based on the concept of minimizing resource wastage.

The Game

1. Cards which do not have numbers are not used e.g. ace, king, queen, joker. For the remaining cards, regardless of the color and shape, the value of a card is equal to the number on it. The cards are shuffled and distributed between the two players. They get equal number of cards.
2. In each round, both player pick a card a place it upside down on the table without showing the opponent to complete their moves. After this they show their cards to each other.
3. The player with the bigger number on the card win the round. The points won by the round winner for this round is equal to the number on card played by the losing players card in this round. E.g if $A$ plays $6$ and $B$ plays $1$ in this round then $A$ wins the round and gets $1$ point.
4. If the number on both cards are equal than the round is a draw and the game continues until someone wins the round or all the cards are used. The round winner get the points equal to the number on the last card played by the round loser.
5. After a round is completed, the played cards are discarded and the game continues with the unused cards remaining with each player until all the cards are used
6. At the end, the player with the greater sum of points win the game.

Minimizing wastage: Clearly we want to win a round by using the smallest card. E.g. if $B$ plays $1$ then $A$ will win the round and get $1$ point if he/she puts any card from $2$ to $10$. But playing a $10$ to win $1$ point is a waste as $10$ can be played to win against bigger numbers later in the match. However before showing, neither player knows what card was played by the opponent.

One may argue that if a player uses up the big cards in the beginning to make small win, it may leave him/her weaker in the later rounds. But using bigger cards in the beginning, leaves the player with smaller cards which leaves the opponent less scope to win more points.

Question: Is there a mathematically optimal strategy that maximizes the chances of winning?

Note: One of the reason why poker is considered a sport as opposed to gambling is because is proven to be a game of strategy and chance and not chance alone.

Answer 1

This answer models the game as a simultaneous game, i.e. one where the game is played in one turn where players announce the orders of playing their cards. This may or may not be what you want.

Assuming the players seek to maximize their end-of-game score, there is no optimal pure strategy, since player one can always play the "right-shift" strategy to counter any strategy played by player two. That is, the strategy to completely maximize payoff for player two is to play a $1$ whenever player one plays a $10$, play a $2$ whenever player one plays a $1$, etc.

There are definitely mixed-strategy equilibria, i.e. one where players randomize between the orders they choose. It is clear that there is an equilibrium whenever both players completely randomize the order in which you play the cards (and the expected end result is a draw), as no player has the incentive to resort to another strategy (the expected end result will still be a draw). From this we can conclude that the expected result in all equilibria will be a draw; otherwise, one player will lose in expectation and that player will have the incentive to completely randomize. There may also be other mixed-strategy equilibria, but they must have same result in expectation. Note that they may be difficult to identify, requiring an ad-hoc method to find all of them.