Teasers Revisited

[Bullajami]

If you take a 2-team teaser that pays at -110, then you need a minimum .7237 probability of hitting each leg to be break even (long term results thinking).

If you take a 3-team teaser paying +180, then you need a minimum .7095 probability of hitting each leg to break even. (Feel free to check my math.)

How significant is that (.7237 - .7095 = ) .0142 difference? Is it enough to merit pursuing one strategy over another? (2-team vs 3-team)

[Justin7]

It is very significant. This was what I was alluding to in videos, but didn't want to spell out so clearly

[Ganchrow]

Here's a similar take on the same question but asked from a risk/return perspective.

If a full-Kelly bettor (able to bet both 2-team teasers at -110 and 3-team teasers at +180) were faced with 3 teaser legs, each with the same expected win probability, then ignoring betting limits:

1. What would that win probability need to be for the bettor to optimally bet at least 0.01% of bankroll on the 2-team teasers at -110 (assuming he's also betting optimally on the 3-teamers)?
2. Given the above, what % of bankroll would he also be optimally betting on the 3-team teaser at +180?

Same pair of questions for a half-Kelly bettor.

EDIT: To be clear, the player can (and optimally will) bet both 2- and 3-team teasers simultaneously and isn't limited to one or the other.

[Data]

72.38%
3.43%

[Ganchrow]

72.38%
3.43%

No, it's a bit more complicated that (I probably didn't adequately explain the question). The implication of the question was supposed tp be that he can bet BOTH 2 and 3-team teasers simultaneously. He's not limited to one or the other.

At 72.38% he would indeed bet 3.43% of bankroll on the 3-teamer, but doing that he'd bet NOTHING on the the 2-teamers.

[Data]

At 72.38% he would indeed bet 3.43% of bankroll on the 3-teamer, but doing that he'd bet NOTHING on the the 2-teamers.

I still do not see why he would bet anything on two-teamers.

[Ganchrow]

I still do not see why he would bet anything on two-teamers.

The obvious answer is that that's just how the math works out.

The idea is that the total amount wagered on the 2-teamers as a fraction of the amount wagered on the 3- would be vaguely shaped like an upside down truncated parabola.

One intuitive way to see this is by looking at the first two terms of the Taylor expansion of the full-Kelly utility function ln(1+x), from which we get the following approximation:

E[ln(1+x)] ≈ E[x] - E[x²]/2 for |x|< 1

The right side of the above approximation is referred to as "Markowitz utility" in finance.

If the probability is low enough then there won't be any wager on the 2-teamers because the expected value (the first term in the Taylor expansion) on the higher EV 3-teamers will fully dominate.

As probability starts to increase, however, the EV associated with increased bet size increases at a slower rate than the similarly associated variance (which is a close relative of the second term in the Taylor expansion). This makes betting the 2-teamers relatively more attractive and given the payout odds above eventually makes them sufficiently attractive so as to imply a nonzero bet size.

Of course, as the probability continues to increase towards 100%, the outcome become more certain and hence the variance decreases. This reduces the optimal 2-teamer fraction until it eventually hits zero.

[Data]

Well, thanks for trying. I am still unable to see where you account for the high correlation between 2- and 3-teamers. I have not done any calculations but it seems to me that 3.43% wagered on a 3-teamers "includes" whatever you might want to bet on the 2-teamers.

[Ganchrow]

Well, thanks for trying. I am still unable to see where you account for the high correlation between 2- and 3-teamers. I have not done any calculations but it seems to me that 3.43% wagered on a 3-teamers "includes" whatever you might want to bet on the 2-teamers.

Check out the http://forum.sbrforum.com/handicappe...readsheet.html I posted last week.

[Data]

I just tried it and after entering 72.38 into cells C2:C4 get no stakes for 2-teamers which is precisely what I was expecting.

[Ganchrow]

Anyway, I'll give the answer because solving the problem is neither fun nor particularly instructive. It's really just a matter of continuously optimizing across trial values of p until you hit 0.01% on the 2-teamers.

For full-Kelly:

p = 79.1984%
X_2-teamer ≈ 0.010%
X_3-teamer ≈ 21.704%

For half-Kelly:

p = 79.2011%
X_2-teamer ≈ 0.010%
X_3-teamer ≈ 10.618%

For tenth-Kelly:

p = 79.2231%
X_2-teamer ≈ 0.010%
X_3-teamer ≈ 2.062%

The point of all this (and yes, there actually is one), is that unless you find yourself sufficiently constrained by maximum bet sizes, you're going to require a pretty damn high estimate of winning each leg before betting even a de minimis amount on 2-team teasers at -110 when 3-team teasers are available at +180.

[Ganchrow]

I just tried it and after entering 72.38 into cells C2:C4 get no stakes for 2-teamers which is precisely what I was expecting.

Which is indicative of 72.38% not being the correct answer.

[Data]

Which is indicative of 72.38% not being the correct answer.

I think it was the correct answer to the question the way it was asked. I looked at the solution you provided and I see what you meant which, again, I think was not properly worded. The worse part is that all your answers (79.1984%, 79.2011% and 79.2231%) do not check out with your spreadsheet. The bet sizes for the full Kelly are not even close to 0.01% while the bet sizes for 0.5 and 0.1 Kelly multipliers fail at the point where you said "at least 0.01% of bankroll" as both come a tiny bit short of 0.01%.

[Ganchrow]

The worse part is that all your answers (79.1984%, 79.2011% and 79.2231%) do not check out with your spreadsheet. The bet sizes for the full Kelly are not even close to 0.01% while the bet sizes for 0.5 and 0.1 Kelly multipliers fail at the point where you said "at least 0.01% of bankroll" as both come a tiny bit short of 0.01%.

Are you clicking "Read Odds" after making any parameter updates and prior to clicking Solve? Have you set min and max parlay sizes to 2 & 3 respectively? Have you made sure only to include 3 items (each with the same probability)? Are you using fixed parlay odds for 2 & 3 team teasers (-110 and +180, respectively)?

Realize that because we're dealing with very small numbers fairly close to the convergence level if the above proves ineffective you might want to try manually zeroing out the previous bet size solutions before running an optimization (it uses whatever bet sizes are already in there as a "best guess" on how to seed the optimization -- not doing this can drastically slow down cause more complex optimizations). This is part of the reason why I don't recommend Excel Solver for any serious optimization work.

Anyway, these numbers are completely correct and were actually obtained from another source. The Excel generic Kelly spreadsheet only serves to verify this. If you want to compare on your own, then enter the solution set I gave above and verify that my values do indeed produce higher utility for that value of p than yours.

If you believe you have a better solution (a lower probability that yields optimal 2-team bet size ≥ 0.01% after rounding) then by all means do post it here. I'll wager you my 2 dinners to your 1 that you do not.

And actually if you wouldn't mind, and if you've tried all the above to no avail, I'd appreciate if you could post a screenshot of the results you're seeing for full-Kelly (both from the "Main" tab and the "Events" tab).

[Data]

manually zeroing out the previous bet size solutions before running an optimization (it uses whatever bet sizes are already in there as a "best guess" on how to seed the optimization -- not doing this can drastically slow down cause more complex optimizations).

That fixed the "not being close" issue. However, the bet sizes are still short of 0.01%. For the full Kelly, I got 0.00964% on two different computers. What do you get?

[Ganchrow]

However, the bet sizes are still short of 0.01%. For the full Kelly, I got 0.00964% on two different computers. What do you get?

Come on, Man ... you can't really be serious about this.

Round it to the 3 decimal places after the percent (which happens to be the default precision in the spreadsheet) and what do you get? 0.010%. I think that that's what one can rather safely call rounding error.

And considering that the question only asked for an optimal bet of at least 0.01% ... well that's even higher precision than was required.

QED

[Data]

Sir, I am serious about that. Let's not confuse a rounding error and an error in rounding. You have 6 significant digits in your probability figures. The resulting bet size number deserves to have at least 3 significant digits. If Excel presents the numbers in certain way, that does not make that way, that happens to be rounding to the 3 decimal places, applicable to the problem at hand. Actually, I am pretty sure it is incorrect here.

[Ganchrow]

Sir, I am serious about that. Let's not confuse a rounding error and an error in rounding. You have 6 significant digits in your probability figures. The resulting bet size number deserves to have at least 3 significant digits. If Excel presents the numbers in certain way, that does not make that way, that happens to be rounding to the 3 decimal places, applicable to the problem at hand. Actually, I am pretty sure it is incorrect here.

This is completely insane. I asked for a stake of 0.01% and that's exactly for what I was looking ... 0.01% ... significant digits and all.

Listen, if it really, really makes you happy then plase feel free to amend the full-Kelly probability I stated above from 79.1984% to 79.1985%.

But come on.

I'm not going to argue about this any more ... it's just absurd.

[Ganchrow]

Realize that because we're dealing with very small numbers fairly close to the convergence level if the above proves ineffective you might want to try manually zeroing out the previous bet size solutions before running an optimization (it uses whatever bet sizes are already in there as a "best guess" on how to seed the optimization -- not doing this can drastically slow down cause more complex optimizations).

To address the above I added a "Clear Stakes" functionality to the optimization spreadsheet. To wit:

There's a "Clear Stakes?" dropdown box beneath these two buttons. Setting the value to TRUE will clear stakes from previous optimizations (set to zero) when clicking either "Read Odds" or "Solve".

Should this be done? If you're running a new optimization then you almost certainly should. But if the optimization you're running is very similar to the previous one, the previous results may be provide the optimizer "clues" on starting values and speed up the process. Occasionally, however, this may work against you as these clues may fallaciously convince the optimizer it's seeing convergence when in reality it isn't. So if you don't trust some results, try setting "Clear Stakes?" to TRUE (but in the case of a large optimization you might have to be prepared for it to take longer).

[Wheell]

For the record there is another way of comparing the two teasers that might be instructive. Please look at these three games

Miami -8.5 Dallas on Saturday
Buffalo +1.5 Denver on Sunday
Chicago -7.5 Tennessee on Monday

You really want to play Miami and Buffalo in a teaser, you are lukewarm on teasing Chicago. As you have noted before the two team teaser odds (assuming a six point tease) are -110 and the three team teaser odds are +180. Let's look at both teasers:

Miami -2.5, Buffalo +7.5, 1100 to win 1000
Miami -2.5, Buffalo +7.5, Chicago -1.5, 1100 to win 1980.

If either Miami or Buffalo were to lose their leg of the teaser both teasers are identical (a loss of 1100). If both Miami and Buffalo were to cover their legs the two teasers simplify to this:

Two teamer: +1000
Three teamer: 1100 to win 1980 on Chicago -1.5.

We can simplify further and say the three teamer is a 2100 bet to win 980 since the three teamer forgoes the $1000 winnings that the two teamer would have already provided. 2100 to win 980 is a bet at odds of roughly -214 (or, in Data's case -214.28571).

There are a few ways one can view teasers. You can view each leg of the two team teaser as -262 (or, if Data bets the teaser, -261.98684). You can view the three team teaser as three legs at -244 (For Data -244.23408), or two legs at -262 and one leg of -214.

It really is an accounting question but in general I would advise you think about it as two legs of -262 and one leg of -214. In supermarket terms, by two, get the third at a discount.

[Ganchrow]

For the record there is another way of comparing the two teasers that might be instructive. Please look at these three games

Miami -8.5 Dallas on Saturday
Buffalo +1.5 Denver on Sunday
Chicago -7.5 Tennessee on Monday

You really want to play Miami and Buffalo in a teaser, you are lukewarm on teasing Chicago. As you have noted before the two team teaser odds (assuming a six point tease) are -110 and the three team teaser odds are +180. Let's look at both teasers:

Miami -2.5, Buffalo +7.5, 1100 to win 1000
Miami -2.5, Buffalo +7.5, Chicago -1.5, 1100 to win 1980.

If either Miami or Buffalo were to lose their leg of the teaser both teasers are identical (a loss of 1100). If both Miami and Buffalo were to cover their legs the two teasers simplify to this:

Two teamer: +1000
Three teamer: 1100 to win 1980 on Chicago -1.5.

We can simplify further and say the three teamer is a 2100 bet to win 980 since the three teamer forgoes the $1000 winnings that the two teamer would have already provided. 2100 to win 980 is a bet at odds of roughly -214 (or, in Data's case -214.28571).

There are a few ways one can view teasers. You can view each leg of the two team teaser as -262 (or, if Data bets the teaser, -261.98684). You can view the three team teaser as three legs at -244 (For Data -244.23408), or two legs at -262 and one leg of -214.

It really is an accounting question but in general I would advise you think about it as two legs of -262 and one leg of -214. In supermarket terms, by two, get the third at a discount.

I'm with you. This certainly helps with the intuition.

Of course I'd ultimately recommend that the bettor make explicit his win probability estimates for each leg and then just toss those in an optimizer such as the spreadsheet linked above.

[Bullajami]

The Think Tank forum is my abusive boyfriend that I keep running back to.

I was just happy I could figure out the cube root function using the x^y button on my Windows calculator.

[Bullajami]

At the risk of further mental emasculation, how would one** figure the value of ties reducing to the next lower payout in a teaser structure?

Is 6-point 4-team +280 that reduces to +160 in the event of a tie worth more or less than 6-point 4-team +300 where a tie loses the bet?




**Assuming that one has math skills in the human range.

[Ganchrow]

Is 6-point 4-team +280 that reduces to +160 in the event of a tie worth more or less than 6-point 4-team +300 where a tie loses the bet?

If, for simplicity, we assume only 1 leg is capable of pushing ATS then from a pure EV perspective:

Let p = probability of all 3 non-pushable legs winning
Let q = probability of pushable leg winning
Let r = probability of push

Then:

EV(+300 teaser) = p*q*4 - 1

EV(+280 teaser) = p*q*3.8 + p*r*2.6 - 1

So the +300 teaser will have higher EV iff:

p*q*4 - 1 > p*q*3.8 + p*r*2.6 - 1

which simplifies to:

q > 13*r

So in other words, if we only allow for the possibility that a single leg may push, then based on the (rather paltry) payout odds you've quoted above, the +300 teaser will be the better play from a pure EV perspective provided that the win probability on the pushable leg is greater than 13 times the push probability.

If you wanted to allow for the possibility of multiple legs pushing you'd need to further specify reduced payout levels (and optionally varying win/push probabilities for each leg), but the basic idea is the same even if the algebra is a bit more messy.

All this said, you should have no problem finding +300 on 4-teamers with ties reducing to +180 (although if for whatever reason you were getting off-market lines in may well be worth it at the reduced odds).

[Bullajami]

Thank You Good Sir!