[Ganchrow]

A quantitative introduction to the Kelly criterion

Part II -- Maximizing Expected Growth

In Part I of this series we introduced the concept of expected growth, where we discussed why a bettor might reasonably choose to gauge the relative attractiveness of a given bet by considering its expected growth. In Part II of the series we'll look at how a bettor might use the notion of expected growth to determine how large a bet to place on a given event. This is the very essence of the Kelly criterion.

There are two extremes when it comes to placing positive expectation bets. On the one hand you have people like my aunt, who’s so afraid of risk that I doubt she’d even bet the sun would rise tomorrow (“But what if it didn’t? I could lose a lot of money!”). On the other hand you have people like my old college buddy Will, whose gambling motto was “Get an advantage, and then push it.”

One Saturday night during the spring term of my sophomore year, Will decided he was going to run a craps game. He put the word out to a number of the bigger trust fund kids and associated hangers-on and let the dice fly. After maybe 4 or 5 hours, Will was up close to $8,000, which was far from an insignificant amount for us at the time. One player, an uppity gap-toothed British guy named Dudley, whose own losses accounted for most of that $8K, loudly proclaimed that he was sick of playing for small stakes and wanted some “real” action. He told Will he was looking to bet $15,000 on one series of rolls. Will paused for a moment and then quickly agreed. He just couldn’t back down from the challenge. It didn’t matter that this represented all of Will’s spending money for the entire semester -- the odds were in his favor and he knew it and as far as he was concerned the choice was clear.

So what happened? Well to make a long story short, the guy picked up the dice and without a word silently rolled himself an 11. Will paid him the next Monday and wound up having to work at the campus bookstore for the rest of the semester. I remember a few weeks later I ran into Will at work and we got to talking while he moved boxes around trying to look busy. I asked him if he and Dudley and were still friends.

“Sure,” he said, “But the guy’s a moron. Didn’t he realize the odds were in my favor?”

So there you have it. Will was quick to label Dudley a moron because he made a negative EV bet. What Will failed to realize, however, was that this guy certainly had the means to make $15,000 bets, and ultimately wouldn’t have been all that impacted by the result were he to have lost. Will on the other hand, had no business making a $15,000 bet that he stood to lose close to half the time. It didn’t matter that if he made the same bet 10,000 times over he’d almost certainly have come out well ahead, it only took making the bet one time to bankrupt him for the semester and render him incapable of staking any more craps games at all.

Dudley might very well have been foolish for having offered to make the negative EV bet, but Will on the other hand was foolish for having risked such a large chunk of his bankroll on the positive EV bet in the first place. Never mind that losing the bet forced Will to work in the bookstore, never mind that losing the bet forced Will to switch from his Heineken bottles to Milwaukee’s Best cans, losing the bet had probably the worst effect possible on an advantage bettor – decimating his bankroll.

Hopefully, this example helps illustrate a key concept that was touched on in the last article. Specifically, that expected value and expected growth are both key components of proper long-term wagering. Most bettors instinctively recognize the importance of expected value – most everyone realizes that betting 2-1 odds on a fair coin flip is “smart”, while betting 1-2 odds on a fair coin flip is not. But very few people consider as much as they should the expected growth of their bankroll due to their wagers they make. When a bettor places too much importance on the expected value and not enough on expected growth, he puts himself in danger of winding up in the same predicament as Will – pushing around boxes at the Brown Bookstore and trying to look busy, despite having made a indisputably “smart” bet when only considering EV alone.

But let’s go back to Will’s initial decision to make the $15,000 bet. Certainly it’s pretty clear that making the bet was a mistake, but it should also be clear that because the bet had positive EV there was obviously a certain (lower) risk amount for which Will would have been making the right decision in accepting the wager. For a person with unlimited access to funds, the decision of how much to bet on a positive EV wager is easy – bet as much as possible. But for a person with a limited bankroll who wants to survive until the next day so he can continue staking craps games, the decision isn’t quite so obvious. That’s where Kelly comes in.

You’ll recall from Part I of this article the equation for expected growth:

E(G) = (1 + (O-1) * X)^p * (1 - X)^1-p - 1

Where X represents the percentage of bankroll wagered on the given bet and O the decimal odds.

For a player like Will, who has his basic necessities already paid for (food, shelter, clothing), his only real goal is to grow his bankroll as much and as quickly as possible. As such, Will’s objective would be to maximize the expected growth of his bankroll. The size of the bet (always given as a percentage of the player’s total bankroll) is known as the “Kelly Stake” and is a function of the bet’s payout odds and either win probability or edge¹.

Mathematically , the formula for the Kelly stake is derived using calculus². The actual mechanics are rather unimportant, but the result is that in order to maximize the growth of one’s bankroll when placing only one bet at a time, one should bet a percentage of bankroll equal to edge divided by decimal odds minus 1. (This is assuming the player has a positive edge. If he doesn’t his optimal bet is zero.) In other words:

Kelly Stake as percentage of bankroll = Edge / (Odds – 1) for Edge ≥ 0

Put in terms of win probability the equation becomes:³

Kelly Stake as percentage of bankroll = (Prob * Odds – 1) / (Odds – 1) for Probability * Odds ≥ 1

Let’s take a look at a few examples:

1. Given a bankroll of $10,000 and an edge of 5%, then on a bet at odds of +100 one should wager 5% / (2-1) = 5% of bankroll, or $500.
2. Given a bankroll of $10,000 and a win probability of 55%, then on a bet at odds of -110, one should wager $10,000 * (55% * 1.909091 - 1) / (1.909091-1) = 5.5% of bankroll, or $550.
3. Given a bankroll of $10,000 and a win probability of 25% then on a bet at odds of +350, one should wager $10,000 * (25% * 4.5 - 1) / (4.5-1) ≈ 3.57% of bankroll, or about $357.
4. Given a bankroll of $10,000 and a win probability of 70% then on a bet at odds of -250, one should not wager anything because edge = win prob*odds = 70%*1.4 = 98% < 1.

Let’s look at all this a little more closely. Consider a bet at even odds (decimal: 2.0000) -- in this case, the bankroll growth maximizing Kelly equation simplifies to:

K(even odds) = Edge/(2-1) = Edge for Edge ≥ 0

In other words, when betting at even odds, the expected bankroll growth maximizing bet is equal to the percent edge on that bet. So if you have an edge of 5% on a bet at +100, then you should be wagering 5% of your bankroll. If your edge were only 2.5% then you should be wagering 2.5% of your bankroll. Now let’s consider a bet at -200, or decimal odds of 1.5:

K(-200 odds) = Edge/(1.5-1) = 2*Edge for Edge ≥ 0

So this means that for a bet at -200, the expected bankroll growth maximizing bet size would be twice the edge on the bet. Similarly, for a bet at -300, one should bet three times the edge, and for a bet at -1,000 one should bet ten times the edge.

This fits rather well with the manner in which many players size their relative bets on favorites. For a bet at a given edge if they were to bet $100 at +100, they’d bet $150 at -150, $200 at -200, $250 at -250, etc.

Now let’s consider bets on underdogs (that is, bets on money line underdogs -- bets paying greater than even odds). In the case of a bet at +200:

K(+200 odds) = Edge/(3-1) = ½*Edge for Edge ≥ 0

The optimal bet size is only half the edge. Similarly at a line of +300, the optimal bet size would be a third of edge, at +400 a quarter the edge, etc.

Now this is quite different from the manner in which many players choose to structure their underdog bets. If they were to bet $100 on a line of +100, they might also bet $100 on a bet with the same edge at +400. For a player wanting to maximize his bankroll growth, this is inappropriate behavior because it attributes, relatively , excessively large amounts to underdog bets. Assuming constant EV an expected growth maximizing player should only bet half of his +100 bet size at +200, and only a quarter his +100 bet size at +400⁴.

So what we see in the case of any bet (be it on an underdog or a favorite) is that the player should bet an amount such that the percentage of his bankroll he stands to win is the same as his percent edge. In other words, a player betting at an edge of 2% should place a bet to win 2% of his bankroll. This means that at -200 he’d be risking 4% of his bankroll, while at +200 he’d only be risking 1% of his bankroll. The rationale behind this should be clear when you consider the following example:

For a player betting at an edge of 5% and odds of -200, the proper Kelly stake is 10%. Over 100 bets, he has an expected return of 64.7% with a 36.7% probability of not turning a profit and a 3.4% probability of losing two-thirds or more of his stake.

For a player betting at the same 5% edge but at odds of +400, were he to bet the 10% stake of the -200 player, while he’d have the identical 64.7% expectation, he’d have a 73.5% probability of no profit, while his probability of losing two-thirds or more of his stake would be 55.8%.

Generalizing, for two same-sized bets of equivalent (positive) EV repeatedly made over time, there’s a higher probability associated with losing a given amount of money when making the longer odds bet.

Once again, we keep returning to the same simple but often overlooked point – expected value isn’t everything. Due to the fact that longer odds (for a given edge) imply greater a probability of loss, the Kelly bettor will bet less on longer odds and more on shorter odds. Any time an advantage player loses money he’s giving up opportunity cost as that represents money he can’t wager on +EV propositions down the line. As such the Kelly player will (for a given edge) always seek to minimize his loss probability over time by selecting the shorter odds bet, even though that necessitates risking more to win the same amount.

Taking the logic a step further, a Kelly player should be willing to even accept lower edge in order to play at shorter odds. For example:

• At odds of -200 (decimal:1.500) and an edge of 4%, the win probability would be p = (1+4%)/1.5 ≈ 69.33%, and Kelly stake would be K = 4%/(1.5-1) = 8%. This represents expected bankroll growth of:
  (1+(1.5-1)*8%)^69.33%*(1-8%)^1-69.33% -1 ≈ 0.1624%
• At odds of +400 (decimal: 5.0000) and an edge of 10%, the win probability would be p = (1+10%)/5 = 22%, and Kelly stake would be K = 10%/(5-1) = 2.5%. This represents expected bankroll growth of:
  (1+(5-1)*2.5%)^22%*(1-2.5%)^1-22% -1 ≈ 0.1221%

So what this tells us is that a Kelly player would prefer (and by a decent margin) 4% edge at -200 to 10% edge at +400.

In this article we’ve introduced Kelly staking. This represents a methodology for sizing bets in order to maximize the expected future growth rate of a bankroll⁵. The bet sizes determined by Kelly will necessarily not maximize expected value, because doing so would require betting one’s entire bankroll on every positive EV wager that presented itself. This would eventually lead to bankruptcy and the inability to place further positive EV wagers.

We’ve seen that Kelly may also be utilized to gauge the relative attractiveness of several bets. What we see is that for a given edge, an expected growth maximizing bettor will prefer the bet with shorter odds (in other words, the bigger favorite). This result, derived entirely from first principles, may be surprising to some advantage players who’ve come to find wagers on underdogs generally more profitable than bets on favorites. While our conclusion in no way precludes the possibility that underdogs may in general provide superior return opportunities than favorites, the fact that for two bets of equal expected return the bet on the favorite will yield greater expected bankroll growth is indisputable and needs to be acknowledged by all those seeking to manage bankroll risk.

In Part III of this series we’ll discuss how one may generalize Kelly so it may be applied to a greater range of circumstances including multiple simultaneous bets, multi-way mutually exclusive outcomes, and hedging.

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Click to hide footnotes

Technically, because odds, edge, and win probability are linked by way of the equality Odds * Prob = 1 + Edge, any two of these variables could be used to determine the Kelly stake. The calculus is rather simple. We need to maximize E(G) = (1 + (O-1) * X)p * (1 - X)1-p - 1 with respect to X, subject to X lying on the unit interval [0,1]. To simplify the analysis, however, we can take the natural log of both sides of the equality and seek to maximize the log of expected growth. This is equivalent because the log function is monotonically increasing. So our problem becomes: Maximize wrt X: log(Growth) = p*log(1 + (O-1) * X) + (1-p)*log(1 - X) s.t. 0 ≤ X ≤ 1 which gives us: dlog(G)/dX = p*(O-1)/(1 + (O-1) * X) - (1-p)/(1 - X) setting to zero and solving yields: X = (Op-1)/(O-1) with d2log(G)/dX2 ≤ 0 for all feasible 0 ≤ X < 1 This may also be extended to include bets that include a third push outcome where the at-risk amount is returned to the bettor in full (such as in the case of an integer spread or total). In order to generalize this article to include bets with ternary outcomes, one need only consider the "probability of winning conditioned on not pushing" instead of pure "win probability". In general, given a win probability of PW, a loss probability of PL, and a push probability of PT (where PW + PL + PT = 1), then the probability of winning conditioned on not pushing would be: P*W = PW / (1 - PT) and the probability of losing conditioned on not pushing would be: P*L = PL / (1 - PT) So assuming decimal odds of O, Edge would be: Edge = O × PW / (1 - PT) - 1     -or- Edge = O × PW - (1 - PT) which in either case is just the same as: Edge = O × P*W - 1 And the Kelly stake would remain unchanged as: Kelly Stake as percentage of bankroll = Edge / (Odds – 1) for Edge ≥ 0 So why do so few players do this? It’s my opinion that the only explanation for this inconsistent behavior (risking the same amount on all underdogs while betting to win the same amount on favorites) is the manner in which US-style odds are quoted. Odds of -200 imply one would need to bet $200 to win $100 so it would seem to make sense to bet in increments of that $200. Odds of +200 imply one would need to bet $100 to win $200, and so it would seem to make sense to bet in increments of that $100. What if, however, US odds on under dogs were also quotes as negative numbers? What if a +200 underdog were written as a -50 underdog (meaning a player would need to risk $50 to win $100) and a +400 dog as a -25 dog? The two methods for expressing odds are obviously identical, but it’s my belief that if odds were quoted in this manner you’d have far fewer bettors undertaking the questionable practice of betting an equivalent dollar amount on all underdogs. An equivalent way of looking at this is that Kelly maximizes both the bettor’s median and modal future bankroll over a large number of bets. In other words, applying expected bankroll growth to the current bankroll yields both most likely bankroll outcome (the mode) and the outcome which has an equal likelihood of being outperformed and underperformed.

[Willie Bee]

Good stuff, Ganch Thanks for continuing my education. If there's anything you ever need to know about growing vegetables in rocky ground, don't hesitate to ask me

[sports_quant99]

Ganchrow...thanks for the writeup....great stuff.

The thing i'm not following you on is "expected growth". It doesnt make sense ... expected *value* is how your bankroll will truly grow in the long run.

I know what you're saying, in that the shape of the bell curve will not be even/symmetrical, but how can that be represented by one number?

The example you gave in your Rx thread, whereby raising your bet to 25% of bankroll, your expected growth rate then becomes negative, defies common sense. The only thing that changes is your volatility, but your long term bankroll will grow.

Not trying to flame you here, just trying to understand. Thanks

[Ganchrow]

The thing i'm not following you on is "expected growth". It doesnt make sense ... expected *value* is how your bankroll will truly grow in the long run.

...

The example you gave in your Rx thread, whereby raising your bet to 25% of bankroll, your expected growth rate then becomes negative, defies common sense. The only thing that changes is your volatility, but your long term bankroll will grow.

First off, welcome to SBR.

I take it you've already read the first article in this series where I tried to illustrate the difference between expected value and expected growth? Guess I didn't do such a good job. huh?

The difference between expected value and growth is the same as that between an arithmetic and a geometric mean.

You can think of the expected value of a bet as the arithmetic mean of all outcomes were you to repeat the same dollar-value bet an infinite number of times.

Expected growth, on the other hand, corresponds to the geometric mean outcome you'd obtain were you to repeat the same percentage-of-bankroll bet an infinite number of times.

This is a subtle but extremely importance difference.

The best way to see the difference is by considering a bet of 100% of one's bankroll. The win probability and payout odds are irrelevant to the discussion, just long as they're less than 100% and infinity respectively. For the sake of this discussion we'll assume the win probability is 99% and the bet is made at +100 (decimal: 2.0000). This bet has an expected value of 2*99% - 1 * 100% = 98% of bankroll, and corresponds to expected growth of (1+(2-1)*100%)^99% * (1-100%)^1% = -100% (this latter figure implies that as your number of sequential bets increases, your probability of going bankrupt approaches certainty). We'll assume the starting bankroll is $100.

After 1 bet, there's a 99% probability of winning and ending up with a bankroll of $200, and a 1% probability of ending up with a bankroll of zero (in which case betting would stop as the player would have no more money with which to play).

After 10 bets, there's a 1-99% ≈ 90.4% probability of ending up with a bankroll of 2¹⁰* $100 = $102,400 and a roughly 9.6% probability of ending up bankrupt.

After 1,000 bets, there's a 1-99%^1,000 ≈ 0.00431712% probability of ending up with a bankroll of 2^1,000* $100 ≈ $1.07151 × 10³⁰³ and a 99%^1,000 ≈ 99.995683% probability of ending up bankrupt.

What we see is that the expected return per bet is always constant at 98%. However, the expected average growth rate per bet is -100%.

Another way to think of expected growth after a large number of bets is that it represents the single most likely outcome. Expected value, on the other hand, represents the average outcome regardless of its relative likelihood.

[Ganchrow]

I know what you're saying, in that the shape of the bell curve will not be even/symmetrical, but how can that be represented by one number?

..

. The only thing that changes is your volatility, but your long term bankroll will grow.

If you're more accustomed to thinking in terms of standard deviation, I'd point out that standard Markowitz utility (the utility function implicitly assumed by financial economists when considering the Markowitz efficient frontier of the traditional CAPM model) is actually a small variable approximation of Kelly.

To wit, if X is our random variable representing return, and μ represents the mean of X, and σ the standard deviation of X, then recalling that the Taylor expansion of ln(X) = X - X²/2 + X³/3 - X⁴/4 + ..., for small values of X ≈ 0, expected Kelly utility as a function of X is given by:

K(X) = E(ln(1+X))
≈ E(X) - E(X²)/2
≈ μ - σ²/2

which is simply Markowitz utility with a coefficient of risk aversion of 1.

In fact, Kelly is really a generalization of Markowitz that includes all higher-order moments.

[3put]

Very interesting Ganchrow.

There are situations where the goal is to Minimize Expected Growth with respect to certain restrictions on betsize.

Take Pointbet as an example.
They closed after not paying in months and many players tried to lose their balance by arbing.

I guess big Pointbet-underdogs were used often in this process.
But quite counter-intuitive, at least to me initially, using your formulas, it seems that betting on big favorites is much better in this situation.

I would appriciate your thoughts on this subjebt.

[Breaker]

No doubt.
Trying to do the same with my Cascade account, any thoughts on how to do this in the most efficient way?

[Ganchrow]

Trying to do the same with my Cascade account, any thoughts on how to do this in the most efficient way?

Unless you can find a positive EV arb between Cascade and another book AND you think there's a non-zero probability of Cascade repaying then don't bother.

[Ganchrow]

Take Pointbet as an example.
They closed after not paying in months and many players tried to lose their balance by arbing.

I guess big Pointbet-underdogs were used often in this process.
But quite counter-intuitive, at least to me initially, using your formulas, it seems that betting on big favorites is much better in this situation.

The article above only addresses a single bet at a time. Once you start considering multiple bets simultaneously, whether the underlying events' outcomes are mutually exclusive (as would be the case when considering a perfect hedge), independent (as would be the case with bets on different games), or non-mutually exclusive dependent^* (for example a side and a money line on the same game), the equations in the above article no longer strictly hold. This will be addressed in the next article in the series.

There are situations where the goal is to Minimize Expected Growth with respect to certain restrictions on betsize.

In mathematics these are known boundary conditions and they can often complicate problems immensely. Fortunately, in the case of single-bet Kelly the solution is simple. Just bet the smaller of the Kelly stake and the maximum bet.

Take Pointbet as an example.
They closed after not paying in months and many players tried to lose their balance by arbing.

You don't need Kelly to tell you that this is a bad idea. If a book won't pay you if you win, then a bet at the book isn't a hedge but rather a waste of time and bandwidth.

Of course, if a player thinks the book in question has a non-zero probability of repayment and he can find a profitable arb involving that book, then he certainly could use his funds at the distressed book as a semi-hedge of sorts, provided he discounted any holdings at that book by a reasonable estimate of repayment probability.

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^* Dependent includes both correlated and uncorrelated bets. After all, it is possible for two non-mutually exclusive bets to be uncorrelated, but not independent. Consider for example, a bet on Team X winning a game versus Team Y and a bet that Team X and Team Y's finals scores will be within 1 of each other. If Team A's win and lose probability are equal, as are the win-by-1 and lose-by-1 probabilities, then while the two bets are uncorrelated, they're obviously not independent.

[3put]

I look forward to the next article in the serie.

In mathematics these are known boundary conditions and they can often complicate problems immensely. Fortunately, in the case of single-bet Kelly, the solution is simple. Just bet the smaller of the Kelly stake, and the maximum bet.

This is true when there is a max risk amount.

However, most A+ books use a max base amount.
In this case there is no fixed rule, but I think that when comparing bets with equal edge you must bet the bigger of the Kelly stake for the maximum bet in order to minimize the expected bankroll growth.

[Ganchrow]

This is true when there is a max risk amount.

However, most A+ books use a max base amount.

I'm not sure I'm understanding you here.

Obviously, for a given edge there's one-to-one correspondence between a "base" amount and a risk amount, so if a book quotes you a max base that's no different from quoting a max risk.

In this case there is no fixed rule, but I think that when comparing bets with equal edge you must bet the bigger of the Kelly stake for the maximum bet in order to minimize the expected bankroll growth.

I'm not following you at all.

[3put]

However, most A+ books use a max base amount.

With a max base amount of $1000 you can risk $1000 to win $4000 if the line is +400 and risk $4000 to win $1000 if the line is -400.

Thats the way Pinnacle, Cris, Greek, 5Dimes etc. operates.

Now let us compare two bets at odds +400 and -400 both with an edge of 2%.

At odds of -400 (decimal:1.25) and an edge of 2%, the win probability would be p≈ 81.6%, and Kelly stake would be K = 8%. E(G) ≈ 0.0817%

At odds of +400 (decimal:5) and an edge of 2%, the win probability would be p≈ 20.4%, and Kelly stake would be K = 0.5%. E(G) ≈ 0.0050%

So if I want to maximize E(G) I will bet 8% on -400

But I want to minimize E(G)

With a bankroll of $10.000 and max risk amount of $1000 my anti-Kelly stake would be 10% with E(G)~0.0763% at -400 and -1.511% at +400.
The underdog at +400 is still the 'best' bet.

If max base amount is $1000 the +400 bet is the same.
But now for the -400 bet my anti-Kelly stake would be 40% with E(G)~-1.609%

Now I should bet the favorite.

[RickySteve]

With a max base amount of $1000 you can risk $1000 to win $4000 if the line is +400 and risk $4000 to win $1000 if the line is -400.

Thats the way Pinnacle, Cris, Greek, 5Dimes etc. operates.

Now let us compare two bets at odds +400 and -400 both with an edge of 2%.

At odds of -400 (decimal:1.25) and an edge of 2%, the win probability would be p≈ 81.6%, and Kelly stake would be K = 8%. E(G) ≈ 0.0817%

At odds of +400 (decimal:5) and an edge of 2%, the win probability would be p≈ 20.4%, and Kelly stake would be K = 0.5%. E(G) ≈ 0.0050%

So if I want to maximize E(G) I will bet 8% on -400

But I want to minimize E(G)

With a bankroll of $10.000 and max risk amount of $1000 my anti-Kelly stake would be 10% with E(G)~0.0763% at -400 and -1.511% at +400.
The underdog at +400 is still the 'best' bet.

If max base amount is $1000 the +400 bet is the same.
But now for the -400 bet my anti-Kelly stake would be 40% with E(G)~-1.609%

Now I should bet the favorite.

I have no clue why you're talking about minimizing your expected growth rate, but it probably comes from the same part of your reptile brain that insists on paying my bills.

Since you're curious, the most effective way to minimize your expected growth rate is to first convert your entire bankroll into hard currency, preferably bahts. After rolling around naked in the cash, place each bill individually into a shredder. (If you don't have a shredder, I recommend Staples or perhaps a Quik-N-EZ Minimizerz foreclosure sale in your area.) Periodically empty the receptacle into a toasty fire. Repeat, as necessary.

Presto! You're optimized!

[Ganchrow]

With a max base amount of $1000 you can risk $1000 to win $4000 if the line is +400 and risk $4000 to win $1000 if the line is -400.

Thats the way Pinnacle, Cris, Greek, 5Dimes etc. operates.

Now let us compare two bets at odds +400 and -400 both with an edge of 2%.

At odds of -400 (decimal:1.25) and an edge of 2%, the win probability would be p≈ 81.6%, and Kelly stake would be K = 8%. E(G) ≈ 0.0817%

At odds of +400 (decimal:5) and an edge of 2%, the win probability would be p≈ 20.4%, and Kelly stake would be K = 0.5%. E(G) ≈ 0.0050%

So if I want to maximize E(G) I will bet 8% on -400

But I want to minimize E(G)

With a bankroll of $10.000 and max risk amount of $1000 my anti-Kelly stake would be 10% with E(G)~0.0763% at -400 and -1.511% at +400.
The underdog at +400 is still the 'best' bet.

If max base amount is $1000 the +400 bet is the same.
But now for the -400 bet my anti-Kelly stake would be 40% with E(G)~-1.609%

Now I should bet the favorite.

I just reread your earlier post and saw that you were talking about minimizing expecting growth wrt to boundary conditions there as well. My apologies.

Your numbers are are all correct on the face and indeed if a bettor's objective were to minimize expected growth and could only choose between those two bets, your conclusion would hold. However:

1. Minimizing expected growth seems a decidely foolish objective. Even if one believed that bookmaker XYZ, at which one had a balance, had some probability of not repaying, the logical conclusion would not be to start minimizing bankroll growth. Rather the way to proceed would be either to discount the value of dollars held at XYZ (saying, for instance that $1 at XYZ corresponds to $0.10 in the real world) and then redo Kelly given this wrinkle, or to set up a larger outcome tree that included branches for payment and non repayment (independent of the primary bets) and run multi-outcome Kelly as usual.
   
   Without running the numbers myself it still seems pretty obvious to me that the -400 bet would be the best choice -- but that has nothing to do with any concept of bankroll growth minimization. (And ff course if the probability of repayment were deemed 0%, it wouldn't matter which bets you placed at bookmaker XYZ -- all would be equally useless.)
   
   
2. If for some reason minimizing expected growth were truly the player's objective, then assuming hold of 4.5455% we'd also have these two bets:
   
   
   1. a bet at ~ +303.8 (decimal: ~4.0385) with an edge of ~ -25.69% and a max risk of $1,000
   2. a bet at ~ -556.2 (decimal: ~1.1798) with an edge of ~ -6.09% and a max risk of ~$5,562.47
   
   
   Bet A would correspond to the flip side of the -400 +2% bet, and Bet B to the flip side of the +400 +2% bet.
   
   Of these, the -556.2 max bet at growth -8.5960% is "best", and the +303.8 max bet at growth of -3.6474% "worst". Both bets are still "better" (meaning worse) than either of the two +2% edge bets.

[3put]

Without running the numbers myself it still seems pretty obvious to me that the -400 bet would be the best choice

Try running the numbers for -200 and +200 under the same conditions. Now the +200 bet is the best choice!

I have no clue why you're talking about minimizing your expected growth rate, but it probably comes from the same part of your reptile brain that insists on paying my bills.

Minimizing expected growth seems a decidely foolish objective.

Well, say bookmaker X takes excessively high fees for withdrawels (and using Kelly you make many withdrawels!), bookmaker Y gives a nice bonus for reloads, bookmaker Z has a very profitable cashback program on losses.
Using arbitrage methods you would like to 'move' funds from these bookmakers as fast as possible.

It's fascinating that the equations leading to Kelly stakes also can be used in this anti-Kelly way.

[Ganchrow]

Try running the numbers for -200 and +200 under the same conditions. Now the +200 bet is the best choice!

Under reasonable assumptions, this seems highly dubious to me. +200 2% might be superior to -200 +2% were your goal purely to minimize expected growth, but I think I've demonstrated why not only is this a silly objective, but even if it were your objective, you'd do better taking the negative EV side of one of these bets, and even better than that by just calling the sports book and relinquishing claim on your balance.

Well, say bookmaker X takes excessively high fees for withdrawels (and using Kelly you make many withdrawels!), bookmaker Y gives a nice bonus for reloads, bookmaker Z has a very profitable cashback program on losses.
Using arbitrage methods you would like to 'move' funds from these bookmakers as fast as possible.

You certainly would like to do this, but using some form of "anti-Kelly" that minimizes expected bankroll growth is most assuredly not the answer.

It's fascinating that the equations leading to Kelly stakes also can be used in this anti-Kelly way.

I certainly do appreciate your interest and enthusiasm on this topic, but the fact is that using the Kelly equations in this manner serves no reasonable economic purposes of which I'm aware. You could certainly could use Kelly utility maximization to help determine optimal behavior given such complications as high withdrawal fees, reload bonuses, and cash back on losses, but minimizing expected bankroll growth as you've outlines above, would not be an effective way to accomplish this.

[TLD]

I’m not sure if this has already been addressed (implicitly perhaps) or will be in future chapters, but one thing that comes to mind for me in considering Kelly or other betting strategies is how should one take into account anticipated future opportunities?

Let’s say I come across an opportunity to make an even money play where I have a 60% chance to win. Intuitively it seems like my most rational behavior would differ if this were the kind of thing I come upon once or twice a year, compared to a few times a week. If it’s a once in a blue moon type opportunity, it seems like I need to take a shot while I can, whereas if these opportunities are a dime a dozen, I can bet it a lot more conservatively and be content to build my bankroll gradually with very little risk.

Does Kelly somehow account for this? Can it incorporate my expectations based on past experience of what sorts of wagering opportunities I’ll likely face in the future?

[Ganchrow]

I’m not sure if this has already been addressed (implicitly perhaps) or will be in future chapters, but one thing that comes to mind for me in considering Kelly or other betting strategies is how should one take into account anticipated future opportunities?

The only way in which this would factor in would be when considering the opportunity cost of a wager. For example, if you were faced with a huge edge bet with a Kelly stake of 50% it might make sense to reduce your bet amount if there existed the possibility of an another strong bet appearing prior to your 50% bet were decided.

Let’s say I come across an opportunity to make an even money play where I have a 60% chance to win. Intuitively it seems like my most rational behavior would differ if this were the kind of thing I come upon once or twice a year, compared to a few times a week. If it’s a once in a blue moon type opportunity, it seems like I need to take a shot while I can, whereas if these opportunities are a dime a dozen, I can bet it a lot more conservatively and be content to build my bankroll gradually with very little risk.

Recall that "full" Kelly maximizes expected bankroll growth. The fact that another good bet might manifest itself tomorrow (after today's bet has been decided) is of no consideration. All you can do is attempt to grow your bankroll today and then address tomorrow's decision tomorrow.

Contrast this with the opportunity cost issue I mentioned above. In that case, the likelihood of future opportunities manifesting themselves was a consideration only because the associated bet sizing decision might need to made before the current bet was settled. On the other hand, when there's no contemporaneity in bet sizing decisions then the distribution of future wagering opportunities may be entirely ignored.

Click to show mathematical example

To illustrate with an example. Let's say that tomorrow you know you'll have the opportunity to place a bet where you can double your money with 100% certainty. How should that impact the decision about whether or not to make a bet today? Let X = stake size on today's bet Let O = decimal odds on today's bet Let p = win probability of today's bet If tomorrow's bet didn't exist, our objective would be to maximize with respect to X: U(x) = p*log(1+(O-1)*X) + (1-p)*log(1-x) However, because any money we can carry over until tomorrow we know we can double, our objective function in light of this becomes: U*(x) = p*log[2*(1+(O-1)*X)] + (1-p)*log[2*(1-x)] Which due to the nature of the logarithm: U*(x) = log(2) + p*log(1+(O-1)*X) + (1-p)*log(1-x) This implies: U*(x) = U(x) + log(2) and dU/dX = dU*/dX Hence, the existence of tomorrow's bet only serves to make the player a constant amount happier (equal to the log of 2) and doesn't figure into the maximization decision at all.

[TLD]

I’m not yet convinced.

As you pointed out about Will, his decision to go for the big score against Dudley was foolhardy because he was taking too great a risk that he would be left with insufficient funds to continue staking his profitable craps games.

So future opportunities matter. Will needs to take into account whether losing to Dudley will put him above or below the line of what he needs to continue staking his profitable craps games. If the semester were ending today and there were no prospects for such craps games (or comparably favorable wagering opportunities) in the future, that would alter the costs and benefits of Will’s decision. (Probably not enough to make it wise to take Dudley’s bet in this extreme example, but it would make it slightly less bad.)

Maybe there will be zero opportunities after this, because Will is getting married tomorrow and has vowed to his bride to never gamble again, though she has no objections to his “getting it out of his system” the night before and gambling as much or as little as he wants. Then there’s no “long run” or “staying in the game” to worry about; maximizing bankroll growth will collapse into maximizing expected value.

Maybe there’s a Super (retarded) Dudley who is willing to bet $150,000 against Will’s $25,000 instead of $15,000 against $15,000, but it has to be tonight, and Will has no chance of coming up with $25,000 except by trying to first beat Dudley for the $15,000. Now not only is there a certain value to “staying in the game,” but also a much higher value than before to “staying in the game with a bankroll of at least $25,000.”

In the original example, losing meant working at the bookstore and missing out on the opportunity to keep staking craps games. In general, the value/disvalue of winning or losing a certain portion of your bankroll will depend in part on what future opportunities you anticipate arising and where your bankroll needs to be to be able to take advantage of such opportunities.

“Staying in the game” is always good, and “decimating your bankroll to where you either can’t bet any more or at least will need to go through a grueling period of grinding out tiny profits on tiny bets to rebuild it” is always bad, but how good and how bad they are can vary depending on upcoming prospects, which to me implies in some circumstances you should accept a little more risk than standard Kelly, and in other circumstances you should be even more risk-averse about those crushing losses than standard Kelly.

[Ganchrow]

I think you may be over thinking this. Kelly does one thing. It maximizes expected future bankroll growth, which is the equivalent of maximizing logarithmic utility. If an individual's utility were not logarithmic (meaning that his goal were not solely to maximize expected bankroll growth) then at best Kelly would only represent an an approximation of his optimal staking strategy.

If your goal is to maximize expected bankroll growth you can't do any better than Kelly. But if your goal were to get in one last bet
before giving up gambling forever, or if it were to ensure that your total bankroll were always evenly divisible by 7, then Kelly will not strictly appropriate.

You aren't bring invalid examples, you're just adding complications, which insofar as they imply non-logarithmic utility can't be answered directly by Kelly. This doesn't mean that you couldn't set up a new utility function and maximize that, but that wouldn't strictly represent expected bankroll growth maximizing preferences.

Maybe there’s a Super (retarded) Dudley who is willing to bet $150,000 against Will’s $25,000 instead of $15,000 against $15,000, but it has to be tonight, and Will has no chance of coming up with $25,000 except by trying to first beat Dudley for the $15,000.

This can easily be solved by Kelly. If we assume that Will wins either bet with 51% likelihood then this simply reduces to a 51%² = 26.01% probability of winning $162,500 and a 1 - 51%² = 73.99% probability of losing $12,500 (decimal odds of 14.000) . Assuming a $15,000 bankroll, the growth rate implied by the pair of bets would be

(1 + $162,500/$15,000)^26.01% * (1 - $12,500/$15,000)^73.99% - 1 ≈ -49.49%

So making this bet would not be wise for a player attempting to grow his bankroll over time at the maximum possible rate.

The point of indifference, incidentally, between making the bet and not making the bet would occur at an initial bet size of about $7,660.36. That this is the case can be seen by:

(1 + 13 * $7,660.36/$15,000)^26.01% * (1 - $7,660.36/$15,000)^73.99% - 1 ≈ 0.00%

[TLD]

You stated:

“For example, if you were faced with a huge edge bet with a Kelly stake of 50% it might make sense to reduce your bet amount if there existed the possibility of an another strong bet appearing prior to your 50% bet were decided,” and

“[T]he likelihood of future opportunities manifesting themselves was a consideration only because the associated bet sizing decision might need to made before the current bet was settled. On the other hand, when there's no contemporaneity in bet sizing decisions then the distribution of future wagering opportunities may be entirely ignored.”

This didn’t seem to go far enough to me, because I think an anticipated future opportunity can be relevant to your present bet sizing decision even for a bet that will be settled before that later bet has to be made.

Your treatment of the Super Dudley example appears to concede this. The added information of a future wagering opportunity (which will occur after the Dudley bet is settled) means Will should be willing to bet up to $7,660.36 against Dudley, which is a different figure than what he should be willing to bet without that added information (or if Super Dudley was willing to bet $500,000 against $25,000 instead of $150,000 against $25,000).

So the existence of the non-simultaneous Super Dudley opportunity is not just relevant to other silly goals like keeping your bankroll divisible by 7, but precisely to the Kelly goal of maximizing bankroll growth.

Beyond that, in real life, I’m not faced with many Super Dudley opportunities with very specific odds and outcomes like that. Instead, it’s more like I stated in my earlier post where from past experience I’d guess I’ll come across such-and-such positive EV bets of very approximately this level very roughly this frequently.

One of the knocks on the applicability of Kelly (and not the mathematical validity of it) is that you have to use estimates of things that are difficult to estimate. So you have to take something that the evidence indicates probably has about a 55%-65% chance of occurring (e.g., the Steelers covering –2.5) and treat it as precisely 60% so you’ll have a number to plug into the Kelly formula.

But if, as I contend, anticipated future non-simultaneous wagering opportunities are also relevant in determining present bet sizing decisions for the purposes of maximizing bankroll growth, then figuring out what numbers to plug in becomes unimaginably more complicated and the applicability of Kelly is even more in doubt than at first glance.

Which is not to say something else would better maximize bankroll growth than Kelly when you are in a state of such uncertainty. I suppose you still should just “Kelly it” as best you can to maximize your bankroll growth, and if that means using numbers that are little better than wild guesses then so be it. (Because “little better” is preferable to “not better at all.”)

[Ganchrow]

The reason why the "Super Dudley" bet was relevant to the sizing decision for the first bet was due solely to the large minimum bet size of the Super Dudley. This corresponds to what's known as a (lower) boundary condition which is a topic I have yet to address. Minimum bets can do all sorts of weird things to the decision making process. Get rid of the minimum bet size and the second bet becomes irrelevant in sizing the first. (Maximum bets are actually a lot easier mathematically in that they preserve continuity of bet sizes. With a max bet, valid choices range from zero to the max, while with with a min bet, valid choices range from the min bet to the max bet AND also include zero. Solving these types of problems requires what's known as "mixed integer programming".)

The idea here is that the two bets are linked and are correctly considered in aggregate. This is because bet sizes are determined by Kelly as a percentage of bankroll while the minimum bet size is given in dollar terms. Super Dudley is only possible if the resulting bankroll after bet 1 is at least the size of the minimum bet (and only desirable if it's least 120% of the minimum bet). In short, this makes the Super Dudley bet dependent on any bet that precedes it and creates a situation where the outcome of the previous bets determines the feasibility of the Super Dudley bet.

To be fair, minimum bets aren't the only future conditions that might impact current bet sizing decisions. Further examples of when you'd want to consider future opportunities when making current bet size determinations:

• A book offers to place you in a higher cash-back tranche if you satisfy a certain monthly betting minimum,
• transferring money to a given post-up account might be an expensive and/or time-consuming process,
• a book might refuse your action if you won too much betting table tennis,
• you may be limited at a certain book you if you started winning on Thursdays,
• a bookie might have you roughed up if you won too much in a given month,
• a book might put you on a payment plan if your balance ever exceeded a certain dollar value,
• you'd be flown down to Costa Rica for an all-expenses pad vacation if your winnings (or losses) in any calendar month ever exceeded a set dollar amount
• a player with OCD might have a difficult time concentrating and making proper capping decisions if the first two digits of his bankroll ever totaled a multiple of 7

The point is that while complications such as these will certainly make anticipating future bets necessary, under standard Kelly assumptions which leave out externalities such as minimum bets sizes (which may serve to make you more or less aggressive) and the issues at play in the above examples, you do not have to consider the distribution of future betting opportunities.

As long as today's betting decisions won't impact the feasibility of tomorrow's bets, then tomorrow's bets can be ignored. Using my earlier example, if tomorrow you'll be faced with the opportunity to place a bet to double your money with 99% certainty, you wouldn't want to adjust today's decisions at all.

This didn’t seem to go far enough to me, because I think an anticipated future opportunity can be relevant to your present bet sizing decision even for a bet that will be settled before that later bet has to be made.

Your treatment of the Super Dudley example appears to concede this. The added information of a future wagering opportunity (which will occur after the Dudley bet is settled) means Will should be willing to bet up to $7,660.36 against Dudley, which is a different figure than what he should be willing to bet without that added information (or if Super Dudley was willing to bet $500,000 against $25,000 instead of $150,000 against $25,000).

I actually oversimplified my treatment of Super Dudley in that I assumed a priori that Will would only choose to bet his proceeds from the first bet on the second. (As I mentioned before, coming up with an answer to a problem that includes minimum bets requires mixed integer programming.)

Relaxing this (incorrect) assumption, we'd find that the minimum bet for Super Dudley to be indifferent would need to be roughly $16,248. In order to get there, Will would wager about $6,014 on the first bet and then if he won would wager the minimum on Super Dudley. He'd be equally happy to do nothing.

[betso]

Very interesting post and discussion!

In my basic understanding Kelly is good for bet sizing when faced with a "single period" problem.

The question on how to extend it to analyze a multi period scenario as well as taking a Kelly approach to reach a certain bankroll goal when faced with a limited number of betting opportunities is very interesting.

I have gotten some ideas on how to approach these problems from in which chapter 9 talk about how to use Kelly from a practical perspective and chapter 5 presents a general framework for solving the multi period problem with a general utility function.

[rjp]

Great set of posts here, but a quick question: how do you take into account the probability of a push for calculating the optimal bet amount? As it stands now the formula simply assumes no win is a loss, but when dealing with point spreads without a half point this isn't always the case.

[Ganchrow]

Great set of posts here, but a quick question: how do you take into account the probability of a push for calculating the optimal bet amount? As it stands now the formula simply assumes no win is a loss, but when dealing with point spreads without a half point this isn't always the case.

In the case of a single bet it's very straightforward. Every time win probability is used it's simply replaced with the probability of winning conditioned on not pushing.

In general, given a win probability of P_W, a loss probability of P_L, and a push probability of P_T (where P_W + P_L + P_T = 1), then the probability of winning conditioned on not losing would be:

P*_W = P_W / (1 - P_T)

and the probability of losing conditioned on not pushing would be:

P*_L = P_L / (1 - P_T)

So assuming decimal odds of O, Edge would be:

Edge = O × P_W / (1 - P_T) - 1
    -or-
Edge = O × P_W - (1 - P_T)

which in either case is just the same as:

Edge = O × P*_W - 1

And the Kelly stake would remain unchanged as:

Kelly Stake as percentage of bankroll = Edge / (Odds – 1) for Edge ≥ 0

[rjp]

Ganchrow, thank you very much for the response. I used to follow the flat betting rationale, which I now realize has very little (logical/mathematical) reasoning.

I've been trying to read as much as I can about Kelly betting, and I've found your articles to provide the most help when relating this to sports betting.

The two books I've found the most useful are: The Theory of Gambling and Statistical Logic and Extra Stuff: Gambling Ramblings. I've got Finding the Edge on the way, but I'd love to hear of any resources you'd consider worthwhile to read (books, papers, etc.)

Thanks.

[Ganchrow]

Other than the , the canonical work on the Kelly Criterion is the 1997 paper by Edward Thorpe (of Beat the Dealer fame) entitled "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market".

The abstract and Chapters 1-10 of Thorpe paper are available and the associated figures, appendices, and references are avilable .

I should be releasing part III of my own series on Kelly in the (fairly) near future.

[rjp]

The Thorpe paper is also in Finding the Edge, so thanks for that. The references link isn't working though. Looking forward to part III.

[Ganchrow]

The Thorpe paper is also in Finding the Edge, so thanks for that. The references link isn't working though. Looking forward to part III.

Fixed the link, thanks.

[Ganchrow]

On a considerably more elementary level than Thorpe's paper you may also want to check out by Jospeh Buchdahl and by Stanford Wong.

[rjp]

What's your take on Wong's book? Let me just say that reading his chapter about money management in Sharp Sports Betting didn't convince me (which is too bad, because I read that some time ago). Maybe he was using too much Blackjack talk for my liking.

[Ganchrow]

What's your take on Wong's book? Let me just say that reading his chapter about money management in Sharp Sports Betting didn't convince me (which is too bad, because I read that some time ago). Maybe he was using too much Blackjack talk for my liking.

It is what it is. The book certainly wasn't written with Economics PhDs in mind, but (like Buchdhal's book) it nevertheless provides an introduction to quantitative modeling methodology within a sports betting framework.

[ugard]

Thanks a lot for this. This series is the first time I have actually printed out forum posts to read.

I'm sure the editor would have picked up on these before this goes that book you keep talking about:

• In the examples: "Given a bankroll of $10,000 and an edge of 5%, then on a bet at odds of +100 one should wager 5% / (2-1) = 5% of bankroll, or $10,000.", I think "$10,000" should be "$500".

• In footnote 3: "then the probability of winning conditioned on not losing would be:", I think "losing" should be "pushing"

Also, whilst I'm at it , in the Kelly calculator help:

"Mathematically speaking, the utility function for a Kelly multiplier of κ>0 is U(x;κ)=(1-1/κ)*x^(1 - 1/κ) for κ≠1, and U(x;κ)=loge(x) for κ=1. This implies dU/dx=x^(-1/κ) for all κ > 0"

"U(x;κ)=(1-1/κ)*x^(1 - 1/κ)" looks a lot like the isoelastic utility function. Shouldn't it be 1/(1 - 1/K)*x^(1 - 1/κ) in order for the derivative to be x^(-1/κ)? I think I may have misunderstood this, as you use the same utility function in this thread.

Whilst on the Kelly divisors subject, in the calculator help, you make of point of stating it uses "true" Kelly. I assume this refers to the use of the different utility function for K does not equal 1, rather than simply dividing the ln(x) stake by the multiplier/divisor as many people seem to do. I suppose that simpler method deviates from the "true" result (which is to maximise a particular percentile of bankroll growth) the larger the divisor? If all this is covered in the next part, can I add a nudge to to those above for its 'publication'.

Once again, thanks for putting this up, and I'm sorry if I sound like I'm nit picking.

[Ganchrow]

Thanks a lot for this. This series is the first time I have actually printed out forum posts to read.

I'm sure the editor would have picked up on these before this goes that book you keep talking about:

• In the examples: "Given a bankroll of $10,000 and an edge of 5%, then on a bet at odds of +100 one should wager 5% / (2-1) = 5% of bankroll, or $10,000.", I think "$10,000" should be "$500".

• In footnote 3: "then the probability of winning conditioned on not losing would be:", I think "losing" should be "pushing"

Also, whilst I'm at it , in the Kelly calculator help:

"Mathematically speaking, the utility function for a Kelly multiplier of κ>0 is U(x;κ)=(1-1/κ)*x^(1 - 1/κ) for κ&ne1;, and U(x;κ)=loge(x) for κ=1. This implies dU/dx=x^(-1/κ) for all κ > 0"

"U(x;κ)=(1-1/κ)*x^(1 - 1/κ)" looks a lot like the isoelastic utility function. Shouldn't it be 1/(1 - 1/K)*x^(1 - 1/κ) in order for the derivative to be x^(-1/κ)? I think I may have misunderstood this, as you use the same utility function in this thread.

Correct on all counts. Good catches, all. The offending statements have been corrected. You looking for a job as an editor?

Whilst on the Kelly divisors subject, in the calculator help, you make of point of stating it uses "true" Kelly. I assume this refers to the use of the different utility function for K does not equal 1, rather than simply dividing the ln(x) stake by the multiplier/divisor as many people seem to do. I suppose that simpler method deviates from the "true" result (which is to maximise a particular percentile of bankroll growth) the larger the divisor? If all this is covered in the next part, can I add a nudge to to those above for its 'publication'.

That's pretty much it. I'll try to come up with the limits when I have more time later in order to show the precise divergence, but in the-two variable case using the naïve method we have:

Kelly_naïve = (w*p - (1-p) ) / (κ(wp +w*(1-p)))

where w is the net win amount on a 1-unit bet (i.e., decimal odds - 1), p the win probability, and κ the Kelly multiplier (e.g., 0.5 for half-Kelly, 1 for full-Kelly, 2 for double-Kelly, etc.). The exact Kelly stake would be given by:

Kelly_exact = ((w*p)^κ - (1-p)^κ) / ((wp)^κ +w*(1-p)^κ)

I'm sorry if I sound like I'm nit picking.

Not all all. I appreciate the keen eye.

[clcoyle93]

Thanks for the education and for taking the time to break it down for us.

Chris