What is the optimal winning strategy for this modified blackjack game?

Questions

Is there a best value to stay on so that I win the greatest percentage of games possible? If so, what is it?

Edit: Is there an exact probability of winning that can be calculated for a given limit, independent of whatever the opponent does? (I haven't done probability & statistics since college). I'd be interested in seeing that as an answer to contrast it with my simulated results.

Edit: Fixed bugs in my algorithm, updated result table.

Background

I've been playing a modified blackjack game with some rather annoying rule tweaks from the standard rules. I've italicized the rules that are different from the standard blackjack rules, as well as included the rules of blackjack for those not familiar.

Modified Blackjack Rules

1. Exactly two human players (dealer is irrelevant)
2. Each player is dealt two cards face down
   • Neither player _ever_ knows the value of _any_ of the opponent's cards
   • Neither player knows the value of the opponent's hand until _both_ have finished the hand
3. Goal is to come as close to score of 21 as possible. Outcomes:
   • If player's A & B have identical score, game is a draw
   • If player's A & B both have a score over 21 (a bust), game is a draw
   • If player A's score is <= 21 and player B has busted, player A wins
   • If player A's score is greater than player B's, and neither have busted, player A wins
   • Otherwise, player A has lost (B has won).
4. Cards are worth:
   • Cards 2 through 10 are worth the corresponding amount of points
   • Cards J, Q, K are worth 10 points
   • Card Ace is worth 1 or 11 points
5. Each player may request additional cards one at a time until:
   • The player doesn't want any more (stay)<