Kelly criterion / dynamic bankroll how to calculate expected profit

[Redwing41]

Similar to roulette with no green and no vig, when you play 60 spins and you win 42 out of 60 you can leave the table because you are above the expect profit of 0. Besides the point, with those odds I guess you could actually make a profit by flat betting and leaving when you are up, if you start to lose you need a big enough bankroll of 500X the flat bet to get your self to even money and immediately leaving the table.


How would I attempt to calculate the expect value of kelly criterion w/ X odds, X percentage and X amount of games. With an infinite amount of games I would carry on, however if I win 7 in a row with 10 matches remaining out of 60 I would have reason to believe that I would be above the expected profit, although do not know how to put this into an equation to measure.

[Thunderground]

There's a Kelly calculator for simultaneous events among the betting tools on this site.

Be aware, though, that you won't find a single pro player who bets full Kelly. The theory is never questioned, but they all turn to fractional Kelly sooner or later. The reason for that is simple. The primary challenge in sports betting is not winning percentage, but sample size. As long as the sample size requirement (300-500 samples per model) is not met, models remain vulnerable to short term fluctuations. In fact, there are many players who swear by a so-called model, without realizing that the model itself is based on a short term fluctuation. That's why it is very common in, for instance, online NFL contests to see a player peak one season only to be back to average (or worse) the next season. For this reason many pros turn to MLB. A MLB team plays ten times the regular season games of a NFL team, so the sample requirement would be met ten times faster.

From this perspective Kelly is not the best answer for sports betting, unless the sample size requirement is met. When people recognize this they turn to fractional Kelly, but that still doesn't address the sample size problem. It merely sidesteps it. What is needed is a formula that includes the sample size. Personally, I combine winning percentage and Z-score (standard deviations). I simply multiply the two, and use that as percentage of BR to wager.

Thunderground Criterion: winning percentage * Z-score * BR.

In which Z-score = (wins-losses)/SQRT(wins+losses). I bet nothing with a Z-score below 2.0.

If the resulting number is too conservative for some, it can be adjusted just as the Kelly criterion is adjusted for sports betting, only this time the sample is included. Feel free to bet 'Double Thunderground'.

That didn't answer your question, but I thought I might as well add it, before you place too much faith in full Kelly.

[Redwing41]

There's a Kelly calculator for simultaneous events among the betting tools on this site.

Be aware, though, that you won't find a single pro player who bets full Kelly. The theory is never questioned, but they all turn to fractional Kelly sooner or later. The reason for that is simple. The primary challenge in sports betting is not winning percentage, but sample size. As long as the sample size requirement (300-500 samples per model) is not met, models remain vulnerable to short term fluctuations. In fact, there are many players who swear by a so-called model, without realizing that the model itself is based on a short term fluctuation. That's why it is very common in, for instance, online NFL contests to see a player peak one season only to be back to average (or worse) the next season. For this reason many pros turn to MLB. A MLB team plays ten times the regular season games of a NFL team, so the sample requirement would be met ten times faster.

From this perspective Kelly is not the best answer for sports betting, unless the sample size requirement is met. When people recognize this they turn to fractional Kelly, but that still doesn't address the sample size problem. It merely sidesteps it. What is needed is a formula that includes the sample size. Personally, I combine winning percentage and Z-score (standard deviations). I simply multiply the two, and use that as percentage of BR to wager.

Thunderground Criterion: winning percentage * Z-score * BR.

In which Z-score = (wins-losses)/SQRT(wins+losses). I bet nothing with a Z-score below 2.0.

If the resulting number is too conservative for some, it can be adjusted just as the Kelly criterion is adjusted for sports betting, only this time the sample is included. Feel free to bet 'Double Thunderground'.

That didn't answer your question, but I thought I might as well add it, before you place too much faith in full Kelly.

This sounds interesting. Kelly criterion it would work but there are two issues I find. There are times there are 3-4 games playing at the same time. Second issue, that you stated is that there is not a large enough sample size.

How does fractional kelly work? I have not understood it yet.

Can you break down Z score further. What does Z score mean exactly and where do I find SQRT (my understanding of standard deviation is limited). What is the result of a z score below 2 vs one above?

[Thunderground]

Z-score determines the standard deviation. SQRT is the square root. In this case the square root of the sample size (i.e. wins + losses).

Example. Which win/loss record is stronger: 11-5 (62.5%) or 82-62 (56.9%)? The percentages, used for Kelly, suggest that the first record warrants a larger wager. But compare the Z-Scores. The first record is (11-5)/SQRT(11+5) or 6/4 = 1.5. The second Z-Score is (82-62)/SQRT(82+62) or 20/12 = 1.67. Now look at a record with more than 300 results, such as 180-140. Now the Z-score is more than 2, even though the winning percentage is lowest of the three examples.

Fractional Kelly is the fraction of Kelly that sports bettors will use. So 1/2 Kelly or 1/3 Kelly. It's used to lower risk. You can play around with it here: http://www.sportsbookreview.com/bett...ly-calculator/

Bookmakers get rich off people who don't understand the impact of short term fluctuations. A player may have a great NFL season, and believe he has mastered that league. But the bookmaker can't wait for him to give it back the next season, because in most cases the player was riding a positive fluctuation and isn't nearly as good as he believes. Add Kelly to that scenario, and it's a pretty bad recipe.

[Redwing41]

Z-score determines the standard deviation. SQRT is the square root. In this case the square root of the sample size (i.e. wins + losses).

Example. Which win/loss record is stronger: 11-5 (62.5%) or 82-62 (56.9%)? The percentages, used for Kelly, suggest that the first record warrants a larger wager. But compare the Z-Scores. The first record is (11-5)/SQRT(11+5) or 6/4 = 1.5. The second Z-Score is (82-62)/SQRT(82+62) or 20/12 = 1.67. Now look at a record with more than 300 results, such as 180-140. Now the Z-score is more than 2, even though the winning percentage is lowest of the three examples.

Fractional Kelly is the fraction of Kelly that sports bettors will use. So 1/2 Kelly or 1/3 Kelly. It's used to lower risk. You can play around with it here: http://www.sportsbookreview.com/bett...ly-calculator/

Bookmakers get rich off people who don't understand the impact of short term fluctuations. A player may have a great NFL season, and believe he has mastered that league. But the bookmaker can't wait for him to give it back the next season, because in most cases the player was riding a positive fluctuation and isn't nearly as good as he believes. Add Kelly to that scenario, and it's a pretty bad recipe.

How do you get around using Kelly when there are multiple bets at the same time? The closest thing I have found is to do parlays on the next match. Game 1 parlays with game 2, game two parlays with game 3 and so on. Each parlay getting slightly bigger, also you have to be flat on every game and they progessively get smaller in percentage each bet.

I havent found the right ratio, this is just an example where you can play around with the ratio.

For kelly criterion where games are in parallel (games overlap in timeline) :
game1:Bet 50 dollars, parlay game 1 + 2 50 dollars
game2:Bet 40 dollars, parlay game 2 + 3 60 dollars
game3:Bet 30 dollars, parlay game 3 + 4 70 dollars

So it is similar to kelly where if you win all the bets it is a big pay day much like the kelly, if you win the first two matches out of 5 and lose the next 2, you will lose similar percentage as you would with kelly. I would need to tweak the number so that it will earn X% per bet.

The same goes with the loss ratio,
every time you win 1 and lose the next 1 = loss
win 2 in a row and lose 1 = slight gain
win 3 in a row = larger gain than average


Can you explain win percentage X Z score X bankroll.
= .62 X 2.26 X 1000
= 1401.2


Should of paid more attention in data class.. Ok I am getting a 2.26 z score over 195 games. The larger the Z score the less fluctuations?

[Redwing41]

Really appreciate the help. I think I would of possibly had some troubles with small sample size if you didnt explain this.

Say I had a bet that was 55% win rate at +110 for 60 matches. When I plug it into the calculator you linked, the expected bankroll = 8k and the median 3k. How can the median be 3k if the expected bankroll is 8k, as that would mean the lowest would be -2k bankroll. With a 55% win rate it would mean 55 percent +/- few percent(z score plays a role in calculating this?)

So would it be safe to say a z score of 2 or over is safe for betting full kelly or 80 percent kelly? With 195 games it would seem like it is enough to bet fully kelly.

I still am having trouble quantifying the difference of a 1.5 z score with a 2.26. A z score of 4 would be very stable where a .5 would be up and down? 2 being fairly stable?

[Thunderground]

Ok I am getting a 2.26 z score over 195 games. The larger the Z score the less fluctuations?

The bigger the Z-score the smaller the impact of short term fluctuations. They're still there, but their ability to change the value of a model decreases.

Two standard deviations is often used as separation. Of course, it can still dip back below 2 in the earlier stages. Once the sample size increases it can get up to 3 and more. The higher the Z-score, the more stable the model. Once a model reaches 300-500 results, and the Z-score is 2.5 or higher, it's looking good for long term income.

[Redwing41]

Thunderground Criterion: winning percentage * Z-score * BR.

Your criterion does not account for the odds? If you had -200 odds it would be different than 100.

Can you explain this further? If I plug in .60 * 2.26 * 1000 = 1356.
I should bet 356 flat bet? next bet will be 500 roughly? 35% per bet.
Kelly criterion in this situation tells me to bet 130. Next bet 146. 13% per bet.

[Thunderground]

Your criterion does not account for the odds? If you had -200 odds it would be different than 100.

Can you explain this further? If I plug in .60 * 2.26 * 1000 = 1356.
I should bet 356 flat bet? next bet will be 500 roughly? 35% per bet.
Kelly criterion in this situation tells me to bet 130. Next bet 146. 13% per bet.

I do plug in the odds. It's all done in Excel. My way is more conservative than Kelly. For instance 60% * 2.26 would be 1.36% of BR.

The idea behind the Kelly Criterion is to maximize profit. It's one approach. We all have to find our own. I split it up into three levels, each with a different purpose: base income (baseball), icing on the cake (NHL and NFL), and splash income (horses). At the first level I'm interested in paying the bills. Nothing more. But at the splash income level I may look for insane ROI. Horses are ideally set up for that, but those big exotic payouts are irregular.

[Redwing41]

http://www.sportsbookreview.com/forum/handicapper-think-tank/3268410-progressive-system-im-getting-ev-check-my-math.html#post25807036

Do you mind looking over this +EV progressive Baccarat system, you seem to have a good eye for detail and are open minded to analyze the data without bias towards progressive systems.

Interesting my betting is broken up the same way too. MLB, NBA are the two main sports ncaab for consistency and mma overs.
I think I may look to add nba play offs + horses.


Any tips on horses? I never got into it. Are the higher paying horses not trained well or bad jockeys(does this play a big role)? Not enough steroids? In mma there was a big problem so if they did it to themselves I cant imagine they wouldnt give their horses a big advantage.