Deceptively Simple Strategy Game

I've recently become interested in this simple strategy game... I'll lay out the rules and also provide an example of a complete game.

The Rules
2 Players (A and B) are each given 6 cards labeled 1 through 6. The Pot is also given 6 cards labeled 1 through 6.

The game consists of 6 rounds. In each round one of the Pot's cards is randomly selected, and its value is shown to both players. Players A and B now have to decide what they want to bid for the card. So, each player selects a card from his own stack to play. And, both cards are revealed simultaneously. The player that played the higher card wins all 3 cards.

Once a card has been played, it cannot be played again.

If there is a tie, then all the cards remain in the pot. The winner of the next round will then claim all the cards in the pot. If the 6th and final round results in a tie, then no one wins the cards in the pot, and the game ends.

The winner is the player who has accumulated more points (sum of all values of the cards won). In this game, there are 63 points up for grabs (1 + 2 + 3 + 4 + 5 + 6) * 3. So, you automatically win if you get 32 or more points. However, you can also win a game with fewer points due to ties.

Below is an example of a complete game. Note that after Round 4, Player A cannot lose if he plays it correctly. The carryover results in a pot with 15 points for Round 5. Player A has the all-powerful 6, and is guaranteed to win at least 22 points in that round. And, since Player A has 13 at that point, a win is guaranteed.

Anyway, I'm looking for more insight and thoughts on this game and how to play this more correctly. A lossless strategy does not exist, but I do believe that a computer playing perfect against a typical human would hold a large edge.

Example Game:

Round 1
Inventory
P: 1 2 3 4 5 6
A: 1 2 3 4 5 6
B: 1 2 3 4 5 6
Total Up For Grabs: 63

Pot randomly plays: 5
A / B simultaneously play: 5 / 3
Result - A (5 5 3 = 13), B (0)

Round 2
Inventory
P: 1 2 3 4 6
A: 1 2 3 4 6
B: 1 2 4 5 6
Total Up For Grabs: 50

Pot randomly plays: 1
A / B simultaneously play: 1 / 2
Result - A (5 5 3 = 13), B (2 1 1 = 4)

Round 3
Inventory
P: 2 3 4 6
A: 2 3 4 6
B: 1 4 5 6
Total Up For Grabs: 46

Pot randomly plays: 6
A / B simultaneously play: 2 / 6
Result - A (5 5 3 = 13), B (2 1 1 6 6 2 = 18)

Round 4
Inventory
P: 2 3 4
A: 3 4 6
B: 1 4 5
Total Up For Grabs: 32

Pot randomly plays: 4
A / B simultaneously play: 4 / 4
Result - A (5 5 3 = 13), B (2 1 1 6 6 2 = 18)

Round 5
Inventory
P: 2 3
A: 3 6
B: 1 5
Total Up For Grabs: 20

Pot randomly plays: 3 (4 4 4 roll over due to tie in previous round) = 15
A / B simultaneously play: 6 / 1
Result - A (5 5 3 3 4 4 4 6 1 = 35), B (2 1 1 6 6 2 = 18)

Round 6
Inventory
P: 2
A: 3
B: 5
Total Up For Grabs: 10

Pot randomly plays: 2
A / B simultaneously play: 3 / 5
Result - A (5 5 3 3 4 4 4 6 1 = 35), B (2 1 1 6 6 2 2 3 5 = 28)

Final Result: A beats B with a score of 35 - 28.