New, and probably terrible, blackjack idea

All the discussions about inventing new games, or tweaking existing games, made me think of this variation on blackjack:

In televised blackjack tournaments, I've seen players all given a one-time "do-over" button that they can use to replace a dealt card. And the discussion about dealing seconds got me to thinking...

What if the house sold the option to replace a dealt double down card? You would pay X amount (in proportion to your original bet), and for that, you would get the option to exchange your double down card for a second, unknown card (which you would have to keep). You would have to buy this option BEFORE you got dealt your double down card.

Example: you bet $10 and get dealt a hard 11. You double down for $10 more, and buy (let's say) $5 "double down insurance". You get dealt a 4 and decide to use the option you've bought, and get dealt the next card instead, which is....(probably another 4, but hey, you tried). The house takes the $5 whether you actually exercise the option or not.

I think this option might be appealing to the player for two reasons: One, it would give the player a chance to escape the horrible sinking feeling of getting a bad card on a double down, and two, it would be a wonderful outcome to toss away that deuce you got on your hard 10 and replace it with an Ace. Conversely, the house would probably love it because it would get used either too often or not often enough; there would definitely be an optimal strategy for exercising the option, which most players would never know or bother to find out, so in addition to any inherent house edge for the bet, the house would gain from its being used suboptimally. For that reason, the "double down insurance" bet could be offered at a nominally low house edge.

I have absolutely no idea what the fair proportional amount for the DDI would be. Obviously, it would be very nice if the fair amount was something like, say, 45% of the original bet, because then the house could charge 50% (of the original bet) and make a profit very similar to most table game side bets. I wonder if anyone could figure out how much the DDI bet should cost?

I once worked on a blackjack idea similar to this. I don't remember if I was sworn to secrecy or not. In such cases, I always assume I was, so won't get into details.

Here is my analysis (the beginning of it).
The columns are:
dealer upcard
difference between EV of doubling with insurance (without insurance cost), and doubling without
difference between EV of doubling with insurance and best play without it
EV of doubling with insurance (without insurance cost)
EV of regular double
EV of best play.
Last column is the list of cards you'll want to replace if you have insurance.

With hard 9:

2: 0.3267 0.3134 0.3878 0.0611, 0.0744 [2,3,4,5,6,7,8]
3: 0.3159 0.3159 0.4368 0.1208, 0.1208 [2,3,4,5,6,7,8]
4: 0.3052 0.3052 0.4871 0.1819, 0.1819 [2,3,4,5,6,7,8]
5: 0.2921 0.2921 0.5352 0.2431, 0.2431 [2,3,4,5,6,7,8]
6: 0.3108 0.3108 0.6278 0.3171, 0.3171 [2,3,4,5,6,7,8]
7: 0.5114 0.4438 0.6156 0.1043, 0.1719 [2,3,4,5,6,7,8]
8: 0.5158 0.3910 0.4893 -0.0264, 0.0984 [2,3,4,5,6,7,8]
9: 0.4094 0.1606 0.1084 -0.3010, -0.0522 [2,3,4,5,6,7,8,9]
T: 0.3121 -0.0016 -0.1546 -0.4667, -0.1530 [2,3,4,5,6,7,8]
A: 0.4561 0.0889 0.0232 -0.4329, -0.0657 [2,3,4,5,6,7,8]
Total: 0.3609 (average advantage of buying insurance)


With hard 10:
2: 0.4233 0.4233 0.7822 0.3589, 0.3589 [2,3,4,5,6,7,8]
3: 0.4097 0.4097 0.8190 0.4093, 0.4093 [2,3,4,5,6,7,8]
4: 0.3959 0.3959 0.8568 0.4609, 0.4609 [2,3,4,5,6,7,8]
5: 0.3808 0.3808 0.8933 0.5125, 0.5125 [2,3,4,5,6,7,8]
6: 0.3828 0.3828 0.9583 0.5756, 0.5756 [2,3,4,5,6,7,8]
7: ! 0.5632 0.5632 0.9556 0.3924, 0.3924 [2,3,4,5,6,7]
8: ! 0.5895 0.5895 0.8761 0.2866, 0.2866 [2,3,4,5,6,7,8]
9: ! 0.5888 0.5888 0.7331 0.1443, 0.1443 [2,3,4,5,6,7,8]
T: 0.5031 0.4691 0.4944 -0.0087, 0.0253 [2,3,4,5,6,7,8]
A: 0.5944 0.4989 0.5804 -0.0140, 0.0814 [2,3,4,5,6,7,8]
Total: 0.4877

With hard 11:
2: 0.4834 0.4834 0.9541 0.4706, 0.4706 [2,3,4,5,6,7,11]
3: 0.4681 0.4681 0.9859 0.5178, 0.5178 [2,3,4,5,6,7,11]
4: 0.4525 0.4525 1.0185 0.5660, 0.5660 [2,3,4,5,6,7,11]
5: 0.4358 0.4358 1.0505 0.6147, 0.6147 [2,3,4,5,6,7,11]
6: 0.4322 0.4322 1.0996 0.6674, 0.6674 [2,3,4,5,6,7,11]
7: ! 0.5957 0.5957 1.0586 0.4629, 0.4629 [2,3,4,5,6,11]
8: ! 0.6240 0.6240 0.9747 0.3507, 0.3507 [2,3,4,5,6,7,11]
9: ! 0.6337 0.6337 0.8615 0.2278, 0.2278 [2,3,4,5,6,7,11]
T: ! 0.6086 0.6086 0.7883 0.1797, 0.1797 [2,3,4,5,6,7,8,11]
A: ! 0.6607 0.6268 0.7698 0.1091, 0.1430 [2,3,4,5,6,7,11]
Total: 0.5554


The exclamation mark after the first column denotes the situations where you'd want to buy insurance (and double) IF the cost of insurance was 0.5.

This is where I get stuck. Whatever is the cost, if the player knows what he is doing, he'll only pay it if it makes his expectation better. But in that case, the house will never have advantage by offering the bet, they will always lose (or rather win less). Why would they want to ever do that?

Perhaps, it would make more sense to require that the player buys insurance before seeing the cards? The cost would have to be really low then, to make it worthwhile for the player ... maybe, it could offer fair odds at no HE, just to spice up the game.

Quote: weaselman

This is where I get stuck. Whatever is the cost, if the player knows what he is doing, he'll only pay it if it makes his expectation better. But in that case, the house will never have advantage by offering the bet, they will always lose (or rather win less). Why would they want to ever do that?

Perhaps, it would make more sense to require that the player buys insurance before seeing the cards? The cost would have to be really low then, to make it worthwhile for the player ... maybe, it could offer fair odds at no HE, just to spice up the game.

Great analysis!

There are several ways this could work. If the bet turned out to be +EV overall for the player, it could be offered to compensate for some other shortcoming of the game, like 6:5 or 1:1 blackjack payoffs (as in, "Fun 21", and other variations of that nature). Furthermore, if the bet IS +EV, that doesn't mean everyone will use it optimally--either they will buy DDI when they shouldn't (like hard 11 vs. 6), or exercise the option when they shouldn't (like receiving a 9 after doubling on a hard 9). I see it as analogous to offering a beatable single deck game; there was always a way to beat the game by playing optimally, but the house made money anyway.

If the cost of the insurance changes with the value of the hand or the dealer's hand, then it won't work. I would make it a very expensive proposition for the player such as 100 percent of the bet and perhaps make it 50 percent refundable if the player gets a 19 or higher... something like that.

Quote: boymimbo

If the cost of the insurance changes with the value of the hand or the dealer's hand, then it won't work. I would make it a very expensive proposition for the player such as 100 percent of the bet and perhaps make it 50 percent refundable if the player gets a 19 or higher... something like that.

I think it would only work if it was 50 percent of the original bet, i.e., half of the double down amount. Otherwise, people's heads would explode and spatter all over everything.

I just realized something--this is not terribly unlike the "surrender after a double" option discussed elsewhere. I know that that rule doesn't benefit the player hardly at all, because the situations where is is +EV to use that option are fairly rare. Of course, anyone surrendering (in any BJ situation) is, in effect, turning down 3:1 odds that they will win the hand, so the bet amount influencing the decision is greater than in my hypothetical DDI.