Expected Rollover Profit Given Max Betsize

[Ganchrow]

(While I do realize that the vast majority of readers of don’t care much about all this, I figured I’d post my thoughts here anyway for the handful of those who might happen to be interested.)

In another thread I mentioned that the expected profit from running through the rollover on a deposit + bonus will be higher the greater the bet size. This is due to the fact that that as bet size increases relative to deposit size the more likely it becomes that the player will lose his entire deposit before completing the rollover.

Because an unadvantaged bettor expects to lose money on every bet he makes, the chance that he might not have to bet through the full rollover increases his expected profit. Hence, the standard formula for expected profit given theoretical hold and bonus + rollover:

= bonus - (1+bonus) * rollover * theoretical_hold

will be too low when there’s a realistic chance of going bankrupt before completing the rollover.

I got to thinking about this and tried to come up with a formula for expected profit (or loss) given a -110 line set and a bonus + rollover under the condition that the player makes the largest bet he can given his current stake (we’re assuming that book imposed wagering limits are never a binding constraint) without betting any more than is required by the rollover.

I found that there’s a rather substantial difference between expected P&L’s when betting the max and when betting the min. Although there’s no closed form solution to the problem for an arbitrary rollover, I did work out the formulas for expected P&L for rollovers from 2x to 52x (the formulas do follow a certain pattern). Clearly for a rollover of 1x or less there’s no difference between the two.

Here are a few of the formulas:

Rollover       E(Profit)
_______________________________

             41 * bonus - 3
  2x        ---------------
                  44
 
             881 * bonus - 87
  3x         ----------------
                   968

             870 * bonus - 98
  4x         ----------------
                   968

             859 * bonus - 109
  5x         ----------------
                   968

             18,105 * bonus - 3,191
 10x         ----------------------
                    21,296

             380,326 * bonus - 88,186
 20x         ------------------------
                     468,512

             7,828,457 * bonus - 2,478,807
 50x         -----------------------------
                     10,307,264

A few examples of the dramatic difference between expected profits with small and large bet sizes:

10%/3x:
Small bets E(profit): -5.000%
Large bets E(profit): 0.114%

15%/3x
Small bets E(profit): -0.682%
Large bets E(profit): 4.664%

25%/5x:
Small bets E(profit): -3.409%
Large bets E(profit): 10.925%

30%/6x
Small bets E(profit): -5.455%
Large bets E(profit): 13.884%

50%/15x
Small bets E(profit): -52.273%
Large bets E(profit): 23.897%

[isetcap]

I thought I was in Sportsbetting 101, but it appears I accidentally walked into a Master's course with Professor Ganchrow.

Read and learn, people. He should be making everyone pay for this stuff and we get it for free.

Thanks, Ganchrow. I really appreciate you posting high quality material at SBR.

[isetcap]

By utilizing the "risk of ruin" factor detailed by Ganchrow above, a player can save him/her self a great deal of time and ultimately make more money.

Utilizing this methodology will heighten a player's exposure to deviation, though. Some players try to limit their risk and aim to cash every bonus they receive. In fact I suspect most players that are playing for bonuses think this way. You can be sure this has been well calculated by the books offering these bonuses. When a player introduces "risk of ruin" into the way he/she approaches cashing these bonuses, then he/she starts riding a rollercoaster and runs the risk of putting a lot of zeroes on the board. On the flip side, a fortunate streak can yield profits far beyond just the bonuses. In the long run everything will flatten out if you have the bankroll to handle the ups and downs, but you can be sure to make more money than the person who bets minimum and eliminates all risk.

[imgv94]

ganchrow you are a mastermind!! I thought I knew something. I know nothing!

ganchrow I am part of that handful. You make alot of sense.

[pags11]

he really is a mastermind...you a math professor or something?...

[raiders72001]

good stuff Ganch

[raiders72001]

Ganch- Feel free to hit my tech section any time that you want. We keep all this type of information in one forum making it easy access without having to do an exhaustive search. I'll copy/post this one over there.

[bigboydan]

now this is what i call one strong post !!!

Everyone should take time to read this post of ganchrow's.

this is the type of post where everyone that reads it should give you rep points for IMHOO

[datek23]

I thought I was in Sportsbetting 101, but it appears I accidentally walked into a Master's course with Professor Ganchrow.

Read and learn, people. He should be making everyone pay for this stuff and we get it for free.

Thanks, Ganchrow. I really appreciate you posting high quality material at SBR.

Ganchrow did put up top dollar stuff.

[raiders72001]

Ganchrow-As Isetcap just explained the greater the bet size the greater the standard deviation making both busting out and big profits easier to reach.

You lost me a little here when you showed expected profits at max and min bet amounts. What formula are you using to determine theoritical hold at min and max amounts?

What I'm getting at is why do you thing the E(profit) is different at different bet sizes? It seems to me that the E(profit) is the same and that the only thing that changes is the standard deviation.

Maybe I haven't had enough sleep but what am I missing here?

[raiders72001]

Ganch- The reason that I'm asking is that I'd like to make this interactive inputting winning percentage along with vig. The step I'm missing is the E(profit) at different bet sizes.

[Ganchrow]

Ganchrow-As Isetcap just explained the greater the bet size the greater the standard deviation making both busting out and big profits easier to reach.

You lost me a little here when you showed expected profits at max and min bet amounts. What formula are you using to determine theoritical hold at min and max amounts
What I'm getting at is why do you thing the E(profit) is different at different bet sizes? It seems to me that the E(profit) is the same and that the only thing that changes is the standard deviation.?

Theoretical hold isn't a function of bet size but just a function of the line set. At a -110 line set theoretical hold is 1/22, at a -105 line set it's 1/42, etc.

What does vary with bet size, however, is the pathwise expected profit, which is to say the expected profit given not only a line set, but also a betting path. In the case of the "small" bet size, the betting path is just a constant. Call it e, arbitrarily close to zero. In the case of the "large" bet size, the betting path is equal to the min of the current stake and the dollar amount remaining on the rollover.

Example: Deposit $100 at a 4x rollover with 25% bonus. With "small" bet size, player will bet in the smallest increment possible until he's bet a total of 4 x ($100+25%*$100) = $500. With "large" bet size player will first bet $125 (his entire stake), if he loses, he's bust and bet size will equal zero, if he wins, bet size will equal his new total stake which would be $125 + $125 *10/11 = $238.64. After this second bet he willl have bet a total of $125+$238.64 or $363.64. If he loses his second bet he's bust and bet size will equal zero, if he wins his second bet he bets $136.36 which is all he needs to reach his rollover.

The important point here is that for "small" bet size, player expects to bet a total of $500. For "large" bet size, player expects to bet considerably less (50%*$125+25%*$363.64+25%*$500=$278.41). For every dollar bet player expects to lose theoretical hold % of that, but because he expects to bet less with "large" bet size than with "small" his expected profit is higher with "large" than with "small".

Does this make sense?

[Ganchrow]

Ganch- The reason that I'm asking is that I'd like to make this interactive inputting winning percentage along with vig. The step I'm missing is the E(profit) at different bet sizes.

For the "large" bet size there's no closed form solution for arbitrary rollover, vig, and win %. It shouldn't be overly difficult, however, to write a script to algorithmically determine a solution given certain inputs. I'll try to work on this at some point or if anyone feels like tackling this, I'd definitely guide him along the right path.

For "small" bet size it's very straightfoward and a closed form solution does exist.

[wageringblog]

Fantastic thread. I was just wondering about this the other day, actually, about the optimum bet amt to get through a rollover. I need to get a spreadsheet going, but when I was just going through what it would take in my head, I kept coming up with too many variables. What I was most interested in was to compute the optimum bet amount while preserving x% of the bonus+deposit (ie, guaranteeing some profit, or, I should say, 'guaranteeing' based on a theoretical 50/50 chance of winning bets, and also taking into account the 4.55% vig - in other words, assuming 47.62% winnings or so, what should the bet amt be to get through the rollover and still have x amount of profits left?).

Unless you have something similar handy, I still need to fiddle around with this, although I do agree (and obviously the math proves this out), that the optimum strategy is to bet the whole shabang each time even if it results in going bust - but for a 3x rollover, theoretically the odds of walking out with money are only 1 in 8, or 12.5%, so 87.5% chance of losing the whole deposit+bonus.

Obviously, over the long haul, though, the expected value does make it worthwhile to keep at this strategy, but therein lies the problem - are there really enough A+ to B+ rated book (or whatever reliable cutoff we would use) to keep at this strategy, or would one run through all the reliable books in a matter of weeks? So for someone who doesn't want to risk losing $100's at once, or enjoys the action in going through the rollover, yada yada, where is that happy medium? This is my current "research project".

[marc]

I think any good bonus whore would tell you that the ideal bet size is always your entire balance. The problem plyaers run into is taht when you are a small player it's very easy to make big percentage gains. You could bet your full balance, and if you win, you could bet your full balance again, and repeat the proccess until the roll is done, or you go bust. But for the larger players, their balances might exceed the bet limit. So they have to start breaking up thier wagers wihc eats into thier profit. So on a percentage basis, thier perfomance will decline.

This is also why when people get these so called "bailout packages" inmy mind its always important that lagrer players find multiple books to bail them out (e.g. if you need a 5k bailout, find 5 books that will give you a 1k bailout, instead of 1 book bailing you out for the full 5k) If someone offers you a 100% bonus 20x roll, it could be a great deal if you deposit $500, maybe even 1k. But as you start to deposit more than 1k, unless the book has large online limits, the bonus could go from being great to being awful.

[Ganchrow]

amount while preserving x% of the bonus+deposit (ie, guaranteeing some profit, or, I should say, 'guaranteeing' based on a theoretical 50/50 chance of winning bets, and also taking into account the 4.55% vig - in other words, assuming 47.62% winnings or so, what should the bet amt be to get through the rollover and still have x amount of profits left?).

Good points here but I have to say that I think you're getting slightly bogged down. What you seem to be asking for ("guaranteeing some profit ... based on a theoretical 50/50 chance of winning bets, and also taking into account the 4.55% vig") doesn't really make sense mathematically. That being said, the "large bet" strategy explained above provides not only the highest expected profit among all possible rollover strategies but also the largest probability of going bust. At the opposite extreme, the "small bet" strategy explained above yields the lowest expected profit among all possible rollover strategies but the lowest probability of going bust.

So what it seems that you're looking for is an "optimal" manner to combine these two strategies. But of course there is no readily attainable optimal strategy that can apply to all players because this a risk-reward issue. Different players with with different levels of risk aversion will all elect to combine the strategies differently (with perfectly risk neutral individuals electing for the pure large bet strategy).

Another possible "natural" betting strategy would be the one that implies the highest probability of yielding any profit (or alternatively of simply not going bust). That, however, is a rather complicated mathematical problem that (I believe) can really only be approximated using monte carlo simulations. (Although there could be a simple thought experiment that yields a generic solution as well -- please do let me know if you can come up with one.)

Unless you have something similar handy, I still need to fiddle around with this, although I do agree (and obviously the math proves this out), that the optimum strategy is to bet the whole shabang each time even if it results in going bust - but for a 3x rollover, theoretically the odds of walking out with money are only 1 in 8, or 12.5%, so 87.5% chance of losing the whole deposit+bonus.

Allow me to point out that with a 3x rollover (and in fact for any rollover up to about 6.5x) the probability of going bust is only 75% not 87.5% (because after two rounds of betting everything, one would only need to bet a portion of the resultant stake.) Still, that the large bet strategy yields an outsize chance of going bust is right on point (and is in fact the very rationale as for why expected profit is so large under such a betting structure).

So for someone who doesn't want to risk losing $100's at once, or enjoys the action in going through the rollover, yada yada, where is that happy medium? This is my current "research project".

The happy medium is certainly a function of the player's risk tolerance (and the utility, if any, gained through "action"). First come up with a proper specification for your risk tolerance and then you'll be halfway there.

[Ganchrow]

I think any good bonus whore would tell you that the ideal bet size is always your entire balance. The problem plyaers run into is taht when you are a small player it's very easy to make big percentage gains. You could bet your full balance, and if you win, you could bet your full balance again, and repeat the proccess until the roll is done, or you go bust. But for the larger players, their balances might exceed the bet limit. So they have to start breaking up thier wagers wihc eats into thier profit. So on a percentage basis, thier perfomance will decline.

This is also why when people get these so called "bailout packages" inmy mind its always important that lagrer players find multiple books to bail them out (e.g. if you need a 5k bailout, find 5 books that will give you a 1k bailout, instead of 1 book bailing you out for the full 5k) If someone offers you a 100% bonus 20x roll, it could be a great deal if you deposit $500, maybe even 1k. But as you start to deposit more than 1k, unless the book has large online limits, the bonus could go from being great to being awful.

Excellent points.

[Ganchrow]

Here's a spreadsheet I came up with that allows the user to input a bonus, a rollover, a pick line, and the probability of that pick winning. This then yields the small and large bet expected profits (and the corresponding equivalent line sets for E(profit) <= 50%).

If anyone finds any errors please let me know.

[rm18]

This made me wonder how stupid bodog's cliential was when they had 20% bonus. 20% bonus actually gives you 7.2/1 odds to hit 3 winers in a row, not to mention what you could do with the dimeline in baseball

[raiders72001]

Ganch- Thanks for making this interactive. Before I dig into this I want to be exact on what you are talking about. Can you define what you mean by pick line and opp line.

[Ganchrow]

Can you define what you mean by pick line and opp line.

Were the market:
Team A +120
Team B -140

and you bet on Team B, the pick line would be -140 and the opp line would be +120.

[raiders72001]

Thanks- This really is great stuff and user friendly.

[raiders72001]

Ganch- I see exactly what you are doing with the Big Bet. Using default settings of 15% bonus, 5x at -110 E(profit) = 2.051% but chance of cashing $538 is 12.5% using amount risked as rollover %.

I'm still not sure your logic behind the small bets. You're multiplying the amount of money that you have after making one max bet and winning by the expected loss at vig used, -110 default, 4.545%. What's the logic of this?

[raiders72001]

Ganch- One more thing, once again using your default settings and big bet method.

6 out of 8 times you lose $100
1 out of 8 times you win $79
1 out of 8 times you win $538

In 8 tries you win $79 + $538 - $600 = $17

so doesn't E(profit) = 2.125% ?

[Ganchrow]

Ganch- One more thing, once again using your default settings and big bet method.

6 out of 8 times you lose $100
1 out of 8 times you win $79
1 out of 8 times you win $538

In 8 tries you win $79 + $538 - $600 = $17

so doesn't E(profit) = 2.125% ?

Rounding error.

6 out of 8 times you lose $100
1 out of 8 times you win $78.6777
1 out of 8 times you win $537.7273

In 8 tries you win $78.6777 + $537.7273 - $600 = $16.4050
so E(profit) = 2.051%

[Ganchrow]

I'm still not sure your logic behind the small bets. You're multiplying the amount of money that you have after making one max bet and winning by the expected loss at vig used, -110 default, 4.545%. What's the logic of this?

Using defaults:

15% bonus 5x rollover => 1.15 * 5 = 5.75

5.75x initial deposit must be bet to complete rollover.

Each bet costs 4.545 ... %.

Hence expected profit = amount bet * expected loss per bet + bonus
= 5.75 *-4.545 ... % + 15%
=--11.136%

[JoshW]

Great thread, good info ganchrow, ty

[raiders72001]

Thanks Ganch- I guess I should have just looked at the formulas instead of just taking the numbers. :doh1:

[tacomax]

Thanks Ganch- I guess I should have just looked at the formulas instead of just taking the numbers. :doh1:

Glad you managed to understand it.

Eventually.

[raiders72001]

This is great. First Taco pretends to know about gambling and now he's a math genius.

[raiders72001]

Taco crapping up another good thread. This jackass knows nothing about gambling and less about math.

[raiders72001]

I'd like to propose the "Taco Rule" . In any thread where some of the better posters are such as ganch, marc, lakersfan, daringly, JC, Roberto, pier and a couple of others are involved in that you must have an IQ of at least 50 to participate. Can we put this to a vote?

I'd would like an icon that says, "Taco rule invoked, Leave now"