Using Kelly to Determine Optimal Hedging Strategy

[Ganchrow]

Secondly you maximize your profits, once again flat betting doesn't do this.

I'll just point out that this is technically untrue. Kelly does not do this. Kelly maximizes expected bankroll growth -- not expected bankroll value. If the player were only interested in maximizing return (in other words, if he were risk neutral) he'd bet his entire bankroll on the single best positive expectation bet he could find, and keep doing that until broke or at his target bankroll level.

Lastly, if you are going to implement a kelly method you need to use parlays in your betting when you have events that are going off simultaneously. You use these to help mimic the results of events that went off sequentially. Rough numbers is reduce your straight wager by 30% and use that for a parlay (two team parlay).

30% is in general way too high for most most single-bet Kelly stakes of reasonable magnitude. For example, assuming uncorrelated events:

• given a 10% single-bet Kelly stake at -110, the 2-bet simultaneous stake would be 9% for each single and 1% for the double
• given a 5% single-bet Kelly stake, the 2-bet simultaneous stake would be 4.75% for each single and 0.25% for the double
• given a 10% single-bet Kelly stake, the 2-bet simultaneous stake would be 9% for each single and 1% for the double
• given a 5% single-bet Kelly stake, the 3-bet simultaneous stake would be 4.5125% for each single, 0.2375% for each double, and 0.0125% for the treble.

In fact it isn't until the single-bet Kelly stake reaches a whopping 30%, that the 2-bet straight wager size would shrink by 30% to 21%.

I'll also point out that this methodology doesn't work if the bets are correlated (positively or negatively).

Why bother with the clever (although unnescesary) approximate-Kelly parlay formulation? Why not just use the actual Kelly weights (which may or may not also include exposure to some set of parlays)?

[Ganchrow]

My bad. Juicy mistake. You didn't apply the juice at first, but I applied it to the entire bankroll. lol.

Yeah, it's always those silly errors that get you ...

[nosuzieno]

Interesting article that turned me into a flat SPORTS bettor years ago:

the Kelly criterion as applied to sports betting would be better called the Kamikaze criterion. You can prove it for yourself, and here’s how:

Here’s what you’ll need, along with at least a half-hour of time:
1. A hand calculator
2. Two decks of ordinary playing cards
3. Lined paper
4. Pen or pencil
5. A ‘Thank You’ note to send me after you complete this exercise and realize how much money I've saved you.....

Pick any size fantasy bankroll to use as your total bankroll. Why not $10,000?

Thoroughly shuffle the 2 decks of playing cards together and place them face down in front of you. We’re going to turn one card at a time and count it as a win, loss, or tie. Everything 7 through King will be a ‘winner,’ everything 2 through 6 will be a ‘loser,’ the Aces will be ties. With those rules, the double deck contains 56 ‘winners,’ 40 ‘losers,’ and 8 ‘ties.’ That makes an overall 'winning' expectation of 58.3 percent, and that would be a great long term winning percentage against sports betting.

But that expectation will vary widely as you remove cards from the deck. As you turn the cards and remove them from the remaining deck the deck will turn ‘positive’ or ‘negative'; - that is, if you remove more 'losers' (2's through 6's) from the deck than 'winners' (7's through K's), the remaining deck will offer a higher expectation of 'winning' on the next draw, and vice-versa.

Figure the sizes of your Kelly bets accordingly. If the first card is a ‘loser,’ there are only 39 losers left in the deck, but still 56 winners. Your winning expectation for the second draw (’bet’) increases to 56 out of 95, or 58.9 percent. If the first card is a ‘winner,’ your winning expectation for the second draw drops to 55 of 95 or 57.9 percent. This, of course, is where the hand calculator comes in.

Be sure to record whether you won or lost the first bet, and how much you won or lost. As a Kelly bettor, of course, your bet sizes will vary up and down as your winning expectation goes up and down. Go ahead and do this 50-or-so times before reshuffling the deck and starting over. (Don’t do it more than 50 or 60 times without reshuffling. Always reshuffle when you've been through about half the double deck. Don't go through the entire double deck.)

Remember, according to the Kelly criterion if the deck goes ‘negative’ and you do not have a positive expectation don’t bet anything. Just flip the next card and the next until you do have a positive expectation. Size your Kelly bets exactly as you do against sports, and to make the exercise more realistic, as when actually betting against sports, flip several cards at once. After all, NFL, NBA, MLB and NHL games often go off several at a time and cannot be bet sequentially. You have to lay several bets at once. Try flipping 3 or 4 or more cards at once - maybe even a dozen or so - just like when you're betting on sports.

After doing another 50-60-or-so observations with the reshuffled deck, it’s time to compare your results using the Kelly criterion against so-called ‘flat’ bets.

Right here is precisely where Kelly promoters always screw up. Let's say they have 100 actual bets wherein they win, say, 58 and lose 42. That's a great winning percentage of 58%, of course. Now, they'll explain that their basic bet is, say, $100, but if their expectation is higher than such-and-such percentage they risk $120, or $130, or whatever, and if their expectation is even higher than such-and-such they might risk $200 or more.

Then they compare what they won by using the Kelly system to what they would have won had they been risking only $100 on each of the 100 bets.

.....Duhhh!.....They risked more money with the Kelly system and they made more money after going 58-42. I hate to burst their balloon, but when you go 58-42, the more money you risk the more money you figure to make. Sorry, boys, but that is not news.

The only way to fairly compare the Kelly system (or any other progressive betting scheme) to flat betting is to use a flat bet the same size as the average size of all the Kelly bets. That way, you’re risking the same total amount against the same overall won-lost results. No fair risking more money overall with one system than the other. That obviously skewers the results.

Or, another way to fairly compare betting systems is to keep track of the winnings as a percent of the total amount risked. If Betting System A wins 8 percent of all monies risked while Betting System B wins 12 percent of all the monies risked, Betting System B is obviously better than Betting System A.

This is precisely what Kelly-promoters choose to ignore. Comparing flat betting against a "1-star, 2-star, 3-star" system, for example, and going 58-42, if all your flat bets are only as big as your "1-star" bets, of course you will win more with the star system. You're risking more and you're winning 58% of your bets.

All right, back to our double-deck of cards. Time to check the profits from flat betting against the record of the Kelly criterion, and ta-daa! There’s your proof. Using the average size of your Kelly bets as your flat bet, the Kelly loses, and it loses every time. In fact, using most forms of the Kelly criterion, I would be surprised if after 70 or 80 ‘bets’ you are not - for all intents and purposes – broke.

You can use the same results to compare the "1-star, 2-star, 3-star" system. You don’t have to flip the cards again, you can use the same won-lost progression you got while testing the Kelly criterion. Set your own parameters concerning when to use a "1-star" bet, a "2-star" bet or a "3-star" bet. Perhaps between 55 and 58 percent you could use a "1-star" bet, etc. Of course, when your winning expectation is less than 53 or 54 percent, there is no reason to bet at all. Bet any system - including flat betting - only when you have an acceptable winning expectation

----------J.R.Miller "Debunking the Kelly System"

[Dark Horse]

Yeah, it's always those silly errors that get you ...

In the spirit of self-inflicted punishment:

After 10 straight losses, betting 16%, starting with 100K, bankroll is down to 17,490.

Fifteen straight wins:
17,490.12, bet size 2,798.41 to win 2,543.76
20,033.88, bet size 3,205.42 to win 2,913.73
22,947.61, bet size 3,671.62 to win 3,337.50
26,285.11, bet size 4205.62 to win 3,822,91
33,930.93, bet size 5,428.95 to win 4,934.91
38,865.84, bet size 6,218.54 to win 5,652.65
44,518.49, bet size 7,122.96 to win 6,474.77
50,993.26, bet size 8,158.92 to win 7,416.46
58,409.72, bet size 9,354.56 to win 8,495.11
66,904.83, bet size 10,704.77 to win 9,730.64
76,635.47, bet size 12,261.67 to win 11,145.86 (50% bettor betting 16% loses 23.3K)
87,781.33, bet size 14,045.01 to win 12,766.92
100,548.25, bet size 16,087.72 to win 14,623.74 (12-10 w/l, 54.5% bettor up 548)
115,171.99, bet size 18,427.52 to win 16,750.61 (13-10 w/l, 56.5% bettor up 15,2K)
131,922.60, bet size 21,107.62 to win 19,186.82 (14-10 w/l, 58.3% bettor up 31.9K)
151,109.41 ----------------------------------- (15-10 w/l, 60% bettor up 51.1K).

Huh!

So, if the sequence of wins and losses makes no difference, the break even point is about 54.5%.

There's that Miller article.. Hmmm.

[TLD]

Actually Miller has an excellent point, and it is applicable to any and all sportsbooks that require Kelly bettors to decide their wager amounts before each wager, while allowing flat bettors to wait and declare their flat bet amount after all the wagers are completed.

[Arilou]

Miller does not get it. Let me try another way to explain it than TLD's simple mocking - not that TLD isn't right, because he is. By using Kelly here, you are doing two things:

1. You are varying your wager size depening on your probability of success, whereas the flat gambler does not. It is trivial to see that this means that the Kelly man will get a better return on money wagered, because he gets his money in better. In particular, think about when the deck 'goes negative' and expecation is less than 0. Miller would say to keep betting, and it should be trivial to see that wagering any money with the worst of it is suicide.

2. Kelly tells you to increase the size of your wagers when you win, decrease when you lose. This lets you bet more, because it prevents you from going broke when you have a bad run. As a flat gambler, are you really going to keep the same unit size if your bankroll halved? What if it doubled? What if it shrunk to 0 because you didn't adjust your bet size and got unlucky?

In fact, the whole point of Kelly is that what matters is your ability to bet and grow a bankroll. Kelly is the way to bet the most money possible over the long run, in addition to getting it in with the best odds. If you overbet, you risk crippling your ability to bet later, and Kelly prevents that.

The times order of wins/losses matter is if there are simultaneous games: When you go 0-4 at 1pm on a Sunday, you can't adjust your bankroll between bets, so you lose an unfortunate % of your roll with a bad run. Kelly reduces bet size to help with this, but not enough to make it not hurt. If the games are one at a time, order doesn't matter because every wager is a % increase or decrease in your bankroll and multiplication doesn't care about order.

That's all in English. I've also looked at Kelly's math and proven to myself that his thesis is correct on that level. There are only three reasons I know of to not use Kelly. The first is that Kelly requires you to do a lot of math every time you wager, and there's a cost to that, so doing an aproximation of Kelly (and rounding down because it's better to underbet than overbet if you're full Kelly) can be better. Your time is money. The second reason is if Kelly is too risky for you, and you don't want those kinds of fluctuations. The third reason is if you don't trust your estimates of your edge and can't get good inputs.

Early in my gambling, I was full Kelly, even though I didn't know the math behind it, and risked large amounts (relative to bankroll) when I thought I had a large edge. Those were wild times, and I doubled up on the Super Bowl by betting on a middle that came in (since the maximum loss wasn't that huge). But that was when I was willing to lose: I was there to find out if I was good enough and if I lost my roll I was ready to pack it in. Now my roll isn't something I'm prepared to risk, so I'm much closer to half-Kelly.

[Ganchrow]

The times order of wins/losses matter is if there are simultaneous games: When you go 0-4 at 1pm on a Sunday, you can't adjust your bankroll between bets

If Kelly is properly implemented, then irrespective of game simultaneity, ordering of wins and losses will not impact results.

[Arilou]

Wow. Interesting, I'll take another look - seems like it should, but I could be wrong.

[Dark Horse]

Miller does not get it.

2. Kelly tells you to increase the size of your wagers when you win, decrease when you lose. This lets you bet more, because it prevents you from going broke when you have a bad run. As a flat gambler, are you really going to keep the same unit size if your bankroll halved? What if it doubled? What if it shrunk to 0 because you didn't adjust your bet size and got unlucky?

Flat betting uses plateaus. The idea is to vary the bet size as little as possible. When a certain plateau is reached the bet size is adjusted. It is still based on a percentage of bankroll, so the whole argument that you can go broke with flat betting and not with Kelly is utter nonsense. As I tried to point out, you may not go broke with Kelly, but you can definitely race towards zero at breakneck speed. Flat betting is the much safer approach.

If Kelly was more than an elegant theory, there would be many very rich sports bettors. If a person could truly keep his head level and make the same quality decisions after a losing streak dropped him from 100K to 18K, it might work. The reality is that most bettors at that point have their nose open.

I would love to know the survival rate of full Kelly bettors. And I'm willing to bet it is shockingly low. Unlike those who argue that Kelly can't be blamed for the shortcomings of individual gamblers, I do hold those who advocate Kelly without a loud and clear warning accountable for misguiding the public. There's a responsibility that comes with teaching something like this (I'm using 'teaching' as in exposing the public to). Even if Kelly were indeed the ideal theory, it would still be as useless to the vast majority of sports bettors as Shakespeare to kids in elementary school. The lessons in gambling are many.

I should probably qualify this in terms of real money versus fun money. Full Kelly is obviously no problem if one bets with fun money (i.e. small amounts). In that sense it might be useful to measure one's bankroll against one's net worth. If you were a 60% bettor and had a house worth six brand new cars, would you truly bet one of them on each qualifying sports game? If you're going for the golden chariot in this way, get ready to drive a pumpkin instead. Why? Because, no matter how much your value increases, if you bet that much of your bankroll, you are simply never going to be out of the reach of one mean losing streak. After you drop from 1 million to 25K, you may not be broke, but your (gambling) spirit will be broken.

[Wheell]

Any defense of flat betting over Kelly theory should seem to start and end with uncertainty theory. If you handicap games and find your success rate doesn't noticably increase when games are far off your numbers as oppesed to merely somewhat off your numbers then simply flat betting all games at the same rate can be the best play available. However, Kelly would STILL be the stronger play if you had an ability to accurately annalyze your edge in the games where you disagree with the pointspread to a large degree. For a good example of this check out Jay Edgar and his work in the NBA.

[Dark Horse]

If you can show me even one multi-millionaire who made his money betting full Kelly in sports, I'd have to reevaluate my position. Until then, I wish you happy theories.

Perfect mathematical formulas are one thing. When you launch enough space shuttles, sooner or later one will explode. Call it chaos theory.

[Jay Edgar]

Great thread. Keep it up, men (and women?) You're elevating the dialog around here by several orders of magnitude.

I'm agnostic at this point, though I'm reading with interest.

As for my own NBA stuff, that an edge of some kind exists seems clear so far this year. Still, the results also seem to demonstrate the real-world difficulty in accurately quantifying that edge bet by bet. I haven't yet found a reliable indicator of greater-edge plays and lesser-edge plays, though I have spent a lot of time trying. Indeed, as I keep noting in the thread, the biggest "edge" plays in terms of a pure gap between my numbers and the BM's numbers do not yield the best results.

(By the way I score the results on a flat-bet basis as an endorsement of nothing -- it simply means the least extra work for me.)

[Ganchrow]

Wow. Interesting, I'll take another look - seems like it should, but I could be wrong.

It actually makes perfect sense when you consider that in implementing simultaneous bet Kelly, the player is precisely duplicating the payout structure (under each and every outcome) that would theoretically exist were the events sequential. So in other words, simultaneous Kelly results aren't a function of ordering because sequential Kelly results are identical to simultaneous Kelly.
<hr>
Demonstrating with a simple example:

Bet 1
Win Prob: 65%
Odds: 1.6000 (or ~ -166.67)
Single-bet Kelly stake: 6²/₃%

Bet 2
Win Prob: 55%
Odds: 1.9000 (or ~ -111.11)
Single-bet Kelly stake: 5%

Were the bets sequential, these would be the four possible outcomes:

1. both bet 1 & bet 2 win
   result = (1+.6*6²/₃%)*(1+.9*5%) - 1 = +8.68%
2. bet 1 wins & bet 2 loses
   result = (1+.6*6²/₃%)*(1-5%) - 1 = -1.20%
3. bet 1 loses & bet 2 wins
   result = (1-6²/₃%)*(1+.9*5%) - 1 = -2.47%
4. both bet 1 & bet 2 lose
   result = (1-6²/₃%)*(1-5%) - 1 = -11.33%

Were the bets simultaneous, the Kelly-optimal weights would be:
Bet 1: 6¹/₃%
Bet 2: 4²/₃%
Parlay 1&2: ¹/₃% (payout odds = 1.6*1.9 = 3.0400)

And of course it's easy to see that the simultaneous-Kelly results are equivalent to sequential-Kelly under each and every outcome:

1. bet 1 wins & bet 2 wins & parlay wins
   result = (.6*6¹/₃%) + (.9*4²/₃%) + (2.04*¹/₃%) = +8.68%
2. bet 1 wins & bet 2 loses & parlay loses
   result = (.6*6¹/₃%) + (4²/₃%) + (¹/₃%) = -1.20%
3. bet 1 loses & bet 2 wins & parlay loses
   result = (6¹/₃%) + (.9*4²/₃%) + (¹/₃%) = -2.47%
4. bet 1 loses & bet 2 loses & parlay loses
   result = (6¹/₃%) + (4²/₃%) + (¹/₃%) = -11.33%

[trustbutverify]

Isn't the problem with EXPECTED WIN %? In blackjack, a counter knows exactly his/her edge. A stock market trader who arbs and uses uses various hedge type strategies isnt predicting stock performance- He's working 2 (or more) numbers against eachother, and knows exactly his advantage- to the nth decimal point. A communications engineer is dealing with pretty much known quantities- # of nodes, # of calls/unit time, etc.. He knows, to a very high degree of accuracy, what his numbers are going to be.

When handicapping sports, I've found, the hard way, thats it's an entirely different situation. Our expected winning pct is based on analysis of past situations- and its a rough estimate, AT BEST. I think most cappers who have actually made significant profit over the long term would agree that the best we can say about our strategies- be they tech, fund, situational or supernatural- is that we have some advantage high enough over the break even to be worth playing. For various reasons- knowing what that edge is with a great deal of accuracy is, at least for me, impossible.

Kelly extracts highest value from a bankroll if you meet your projected winning pct. Thats math and I don't think anyone's going to argue that. But if you don't get close to that goal- you're in trouble. Big trouble.

[Ganchrow]

I'm agnostic at this point, though I'm reading with interest.

For those comfortable with quantitative principles, agnosticism with respect to Kelly makes about as much sense as agnosticism towards the color red.

The fact that Kelly, when properly implemented, does what it claims (specifically that it maximizes the expected growth rate of a bankroll) is no more open to debate than the fact that the ratio of the circumference of a circle to its diameter is pi. And that's not hyperbole either -- mathematics is just as certain of either one of these facts as the other, and that's 100%.

Sure, one might point out that were you to draw a circle, the ratio of the circumference of your freehand "circle" to its diameter would be demonstrably different (with fine enough measuring equipment) from pi -- but just as that criticism of pi should in no way dissuade you from belief therein, neither should the similarly trivial arguments against Kelly dissuade you either.

One could fairly argue that Kelly might be exceedingly difficult to implement in practice (perhaps because a given bettor might have a hard time estimating his edge on any given bet), or that logarithmic utility (the utility curve assumed by Kelly) doesn't adequately describe the preferences of 99.99+% of bettors. Those are fair and reasonable arguments that can be logically discussed and debated. But the Kelly-phobic cargo-cultism that some might bandy about apparently without either the slightest regard for or understanding of the centuries-old mathematical principles that lie at its underpinnings? That's anti-intellectualism at its most parochial.

[Jay Edgar]

Ganch, I think you're pummeling an imaginary foe there.

I believe in the mathematics.

The math is great.

But I can't reliably quantify my edge.

So I doubt whether I can make the math work for me.

I am 'agnostic' as to whether a workable Kelly tool exists to help me in this particular endeavor.

[Ganchrow]

I can't reliably quantify my edge.

So I doubt whether I can make the math work for me.

I am 'agnostic' as to whether a workable Kelly tool exists to help me in this particular endeavor.

And that's an example of a perfectly valid criticism of the practical implementation of Kelly.

But that's very different from arguing that because Kelly wasn't a gambler he doesn't account for losing streaks; or due to the fact that Kelly often recommends fairly large bets that have a chance of losing it somehow doesn't serve to maximize bankroll growth; or (as per JR Miller), using Kelly is symptomatic of problem gambling because it recommends betting more after a string of victories (as is true with any percentage staking strategy; or that Kelly is only a theory and doesn't work in practice; or that Kelly is a kamikaze-strategy that will only lead to a bettor's demise. I could go on and on.

I'm not recommending that everyone use Kelly ... I'm not even recommending that anyone use Kelly. All I'm trying to do is approach Kelly as scientifically and analytically as possible, which will necessarily put me in opposition (on this one particular topic) to those who would argue against Kelly viscerally and by appeal to the pseudo-scientific.

It's OK not to use Kelly ... just make sure you're not using it for the right reasons.

[pico]

seems to me, a gambler should always be risk neutral.

[durito]

Perfect mathematical formulas are one thing. When you launch enough space shuttles, sooner or later one will explode. Call it chaos theory.

This may be old news, but NASA originally estimated 1% of all space shuttles would crash/explode. At 2 events in approx 160 flights, that's well within reason.

[pico]

This may be old news, but NASA originally estimated 1% of all space shuttles would crash/explode. At 2 events in approx 160 flights, that's well within reason.

FYI, the estimate is only based on machine failure...does not factor in human error

[Ganchrow]

seems to me, a gambler should always be risk neutral.

See Maximizing Expected Growth for a discussion of why risk-neutrality is generally inconsistent with long term bankroll growth.

[durito]

Anything to add here fishhead?

[Flying Dutchman]

Kelly Bettors will burn in Hell!

And certainly get there before the flat bettors.

[cala56]

Top post by far.