Optimal use of a mulligan in tournament blackjack

I sometimes play in a small blackjack tournament. To make the tournament more fun (and tough to analyze), the hands are positive EV (notably, Blackjacks pay 2:1) However, you have the opportunity to make one hand for each round (as long as its one of the first 13 of 14) SUPER positive EV by use of a "mulligan" card.

With the mulligan, you may take one of your first two cards and replace it with the next card from the shoe. If you don't use your mulligan on the first two cards, you may use it on a hit card instead, but once you've hit, you may only replace the most recent hit card. And just to clarify, you cannot "request" a particular card. You're stuck with whatever you're dealt on the first attempt to use the mulligan. However, you may continue to play your hand as normal (hit after using the mulligan unless it was used to replace a double down card). If you receive a blackjack as a result of using the mulligan, it pays the usual 2:1.

I don't have the knowledge or resources to develop a strategy engine, but here's where logic takes me:

1) Only use the card when you're bet is large.
2) The cards future value declines with every hand played (i.e. when all 14 hands remain to be played, you are much more likely to find a hand in which to use it later on, whereas on the 13th hand you might as well use it on just about anything that can be improved)
3) Generally it is better to use it to replace one of the first two cards, and not a hit card, except for double downs.
4) The card is best used on hands where one of the cards is particularly strong (A or T), the other is weak (5 or 6), and the dealer is showing a decent, but beatable card (7 or 8)

Other than that, I'm curious to know a more precise mulligan strategy. Presumably this would be determined by calculating and comparing the EV of a hand versus that same hand with one of the cards replaced with a random card.

So I just did a rough analysis in Excel.

I started by using the composition dependent expected returns for the most similar game per the Wizard's other site: https://wizardofodds.com/blackjack/appendix9-6dh17r4.html. I then modified the returns slightly to reflect important rule changes: I switched the 10,A payout from 1.5 to 2.0, and I added a surrender column, which is always -0.5.

I then used the =MAX function to get the best possible outcome of each situation. I created columns where the high card was replaced with each type of card. I then took the average of these results (correctly weighting 10's as 4x as likely to occur). NOTE: This assumes an infinite deck! That said I don't think the errors involved in approximating 6D with an infinite deck will affect the analysis that much (certainly better than guessing!). I compared the average result with a replaced high card with that of the initial hand to determine the "expected benefit of replacement". I then did the same for replacing the low card.

I once again took the max of high-card replacement benefit and low-card replacement benefit to determine the overall benefit (as we would obviously replace the more beneficial card).

Given that this tournament has 13 hands in a round and you can only use the mulligan on one, it seems that the top 43 (1/13 of the possible blackjack hand combinations) hands to benefit from replacement would be of interest. As it turns out, 36 of the 43 are soft hands--the best of which being medium soft hands against high dealer cards. The ideal candidate being a soft 17 v A. This is undoubtedly due to the amazing 2:1 payout of blackjack. Before I changed the BJ payout to 2:1 (yes, I actually did that step last), the best hand to replace as a 10,6 v 7. This was fairly obvious as well, as a 16 is a terrible hand, and a dealer 7 is quite beatable when starting with a 10. (Given the 2:1 rule it's the 20th best possible hand to use a mulligan on).

Another interesting piece of trivia is that 2,2 v 3 (use), and a 5,3 v A (don't use) are the closest calls.

That said, my analysis has some limitations. First, the fact that the mulligan can also be used on hit cards (maybe we could double our 12's if we knew we had 2 chances at not busting?) Or what about hitting a hard 17 when you know it'll only bust twice 48% of the time? Second the infinite deck assumption. And lastly, the amount of the wager and time remaining in the tournament are far more important than the actual hand to use the mulligan on.

I'd love to see someone smarter and more resourceful than me analyze this fully, but I think I've learned the general rule: Use the mulligan on soft hands with big bets.

Mulligan tournament? Interesting concept. And interesting analysis of your own question.


What casino is this?

Because tourneys sometimes have other weird rules, and considering the 2:1 BJ payout, does a BJ with a dealer BJ push or pay? Does insurance still pay 2:1?

BJ v BJ pushes per normal rules. Insurance is a 2:1. Even money is offered, and is a miserable bet, unless specific game conditions make it worthwhile (for instance if no one else takes insurance and you can take the lead going into the final round with even money).

The casino is the Red Dragon in Mountlake Terrace, WA. Non-tribal WA casinos are "mini casinos" or "cardrooms", limited to 15 tables by state law. The tournament buy in is $30 ($20 rebuy), and the prize pool is guaranteed at $2,000 ($2,500 if over a certain number of players show). Max of 63 players.

I also just calculated the player edge when the mulligan option is still available to you. It's an astounding 32.53%!