Insurance Bet and Fluctuations

[MMalmuth]

Hi Everyone:

This is the first time I've posted on this forum.

In my Gambling Theory book I argue that those strategies which increase expectation are usually accompanied by a higher standard deviation. However, this is not a hard and fast rule. Two examples where I know it doesn't hold are the ability to read hands well in poker and the surrender rule in blackjack. Both of these will increase your expectation and lower your fluctuations at the same time (which is nice).

I'm now wondering about the insurance rule in blackjack. Assuming you're taking insurance at the appropriate times (according to the count) it will increase your expectation. But what does it do to the standard deviation?

I have my opinion, but would like to hear what others think,

Mason

[aceside]

About the insurance rule in blackjack, you will increase your expectation and decrease your deviation for shoe games.
For infinite deck games, neither.

[bejammin075]

Hi Everyone:

This is the first time I've posted on this forum.

In my Gambling Theory book I argue that those strategies which increase expectation are usually accompanied by a higher standard deviation. However, this is not a hard and fast rule. Two examples where I know it doesn't hold are the ability to read hands well in poker and the surrender rule in blackjack. Both of these will increase your expectation and lower your fluctuations at the same time (which is nice).

I'm now wondering about the insurance rule in blackjack. Assuming you're taking insurance at the appropriate times (according to the count) it will increase your expectation. But what does it do to the standard deviation?

I have my opinion, but would like to hear what others think,

Mason

Hi Mason,

I enjoyed reading your book Blackjack Essays. I don't have any intuition about your question, but I can run a simulation in CVData. This simulation used 6 decks, 1 deck pen, Pennsylvania rules. Two players using the same Wong Halves (doubled tags) strategy. The "seat effect" is removed. Player 1 takes insurance at +7, whereas Player 2 never takes insurance.

At the TCs below 7, they have the same standard deviation, but as the TC rises above TC 7, the player 1 who takes insurance has a relatively lower standard deviation than the player 2 who does not take insurance. Now what I don't get is that there is another standard deviation at the bottom summary which is larger for player 1.

Mason Player 1.jpg


Mason Player 2.jpg

[Mr. Ed]

Hmmm. I need to think about this. “Even Money” will reduce your sd, since now your payout is always 1 (I.e. 1 bet). Insuring a 20 reduces sd, since most of the time your payout +1/2 (instead of 1), or 0 instead of -1.

But if you have a stiff hand, say 14, your payout will be zero instead of -1 about 1/3 of the time if he has BJ, but if he does not, your payout is most likely -1.5 instead of -1 if you lose the hand, and +0.5 instead of +1 if in the off chance you win. Sd is increased.

So you have to look at insurance for all player hands to see if sd is increased or decreased overall. I don’t have the patience to do this, but I am a bit curious if insurance really reduces Sd, since it sure feels like it increases it.

P.s. This might be nit-picking, but please don’t say, “opinion” when you are talking about math. You might be right, you might be wrong, but your opinion does not matter.

[MMalmuth]

Hmmm. I need to think about this. “Even Money” will reduce your sd, since now your payout is always 1 (I.e. 1 bet). Insuring a 20 reduces sd, since most of the time your payout +1/2 (instead of 1), or 0 instead of -1.

But if you have a stiff hand, say 14, your payout will be zero instead of -1 about 1/3 of the time if he has BJ, but if he does not, your payout is most likely -1.5 instead of -1 if you lose the hand, and +0.5 instead of +1 if in the off chance you win. Sd is increased.

So you have to look at insurance for all player hands to see if sd is increased or decreased overall. I don’t have the patience to do this, but I am a bit curious if insurance really reduces Sd, since it sure feels like it increases it.

P.s. This might be nit-picking, but please don’t say, “opinion” when you are talking about math. You might be right, you might be wrong, but your opinion does not matter.

Hi Mr. Ed:

My inclination, when you put everything together, is that properly taking insurance is that properly taking insurance will lower the standard deviation.

Best wishes,
Mason

[Norm]

Unlike the summary, the SD per TC is not in dollars. The insurance player has a better EV. Therefore, the optimal bets are increased. This increases overall SD. Of course, it's SCORE that matters as it takes into account both EV and SD.

Oh, and welcome to the site Mason.

[bejammin075]

Unlike the summary, the SD per TC is not in dollars.

So if those standard deviations by true count are not calculated using dollars, what is being plugged into the formula?

[Norm]

In answer to the original question, assuming you bet with the same ramp instead of Kelly optimal, the SD is lower for the insurance player. BUT, the difference is tiny. Running 25 billion hands each:

No insurance SD/hour: $193.91
Correct insurance SD/hour: $193.29

This difference is so small, it might be reversed for another count.

[Secretariat]

In answer to the original question, assuming you bet with the same ramp instead of Kelly optimal, the SD is lower for the insurance player. BUT, the difference is tiny. Running 25 billion hands each:

No insurance SD/hour: $193.91
Correct insurance SD/hour: $193.29

This difference is so small, it might be reversed for another count.

Norm, is that for perfect insurance?
Nice to see you here, Mr, Malmuth. I've also enjoyed your book.

[MMalmuth]

In answer to the original question, assuming you bet with the same ramp instead of Kelly optimal, the SD is lower for the insurance player. BUT, the difference is tiny. Running 25 billion hands each:

No insurance SD/hour: $193.91
Correct insurance SD/hour: $193.29

This difference is so small, it might be reversed for another count.

This is an interesting result. If the SD would have been much smaller, it would imply that for players on a small bankroll that for survival purposes it would be correct to also take insurance in situations where the bet was still slightly negative. This result says no to that idea.

Best wishes,
Mason

[MMalmuth]

In answer to the original question, assuming you bet with the same ramp instead of Kelly optimal, the SD is lower for the insurance player. BUT, the difference is tiny. Running 25 billion hands each:

No insurance SD/hour: $193.91
Correct insurance SD/hour: $193.29

This difference is so small, it might be reversed for another count.

Hi Norm:

I've been away from blackjack for some time. So let me clarify something you wrote. When you say "you bet with the same ramp instead of Kelly optimal" what kind of bet spread are you using, and how is this different from Kelly Optimal?

Mason

[DSchles]

Hi Norm:

I've been away from blackjack for some time. So let me clarify something you wrote. When you say "you bet with the same ramp instead of Kelly optimal" what kind of bet spread are you using, and how is this different from Kelly Optimal?

Mason

Whatever the spread, it's the same for both players in Norm's sim, whereas, if the insurance player were betting optimally, he'd be betting more than the non-insurance player, because he has a greater overall advantage by virtue of insuring when it's correct to do so. Those larger bets would, in turn, lead to a greater s.d. than the one for the non-insurance player, provided both bet optimally.

Don

[MMalmuth]

In answer to the original question, assuming you bet with the same ramp instead of Kelly optimal, the SD is lower for the insurance player. BUT, the difference is tiny. Running 25 billion hands each:

No insurance SD/hour: $193.91
Correct insurance SD/hour: $193.29

This difference is so small, it might be reversed for another count.

Hi Norm:

I'm coming back to this since I now understand better what exactly is happening.

Having a standard deviation of $193.91 per hand (I think) for the non-insurance player and a standard deviation of $193.29 for the player who is taking insurance is an amazing result, and for practical purposes I think we can say that these two standard deviations are essentially the same. Yet, the player who is taking insurance is putting more money in action, so wouldn't you expect his standard deviation to go up?

What I think is happening is that some of his bets are not independent of each other and, in fact, are inversely correlated. Specifically when the dealer has a blackjack, unless the player also has a blackjack (which should happen only rarely) he must then lose the standard bet. This produces a result of $0 which probably has the effect of lowering the standard deviation. Next, which we can call the second case, when he losses the insurance bet but wins the standard bet he may also be lowering his standard deviation since his result is now half of what it would be if he hadn't taken insurance (and I'm ignoring things like doubling down and splitting to keep this explanation simple). It's only when, the third case, the player losses both bets that he drives his standard deviation up since he's actually losing 1-and-a-half bets instead of 1 bet. My guess is, based on the numbers that Norm has provided, that all of this approximately washes out.

But based on the way I understand things, which comes from mostly poker (see my Gambling Theory book), if taking insurance increases your win rate (which is sometimes different from advantage) but you keep your standard deviation the same, your bankroll requirements go down. So, here is an example that will appear counterintuitive to many people.

Following up on this a little more, if you play with the equations in my Gambling Theory book you'll end up with this equation (which appears in my book Poker Essays):

BR = [(9)(SD)^2]/[(4)WR] where BR is bankroll, (SD)^2 is the standard deviation squared (or the variance), and WR is the win rate. Now, this equation is not perfect, but in my experience it does a very good job of estimating the required bankroll at three standard deviations. And in this problem which we're discussing, since according to Norm's simulation the standard deviation is essentially the same, but the win rate will obviously go up for the player taking insurance, the required bankroll for the player taking insurance will be less even though he's putting more money into action.

To finish, DSchles points out that the player taking insurance should be using a more aggressive betting scheme than the one not taking insurance. I think it would be interesting to see what this does to the standard deviation. Perhaps Norm can run another simulation.

Best wishes,
Mason