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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)<sup>p</sup> * (1 - X)<sup>1-p</sup> - 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 edge<sup>1</sup>.
Mathematically , the formula for the Kelly stake is derived using calculus<sup>2</sup>. 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:<sup>3</sup>
Kelly Stake as percentage of bankroll = (Prob * Odds – 1) / (Odds – 1) for Probability * Odds ≥ 1
Let’s take a look at a few 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 $500.
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.
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.
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 +400<sup>4</sup>.
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%)<sup>69.33%</sup>*(1-8%)<sup>1-69.33%</sup> -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%)<sup>22%</sup>*(1-2.5%)<sup>1-22%</sup> -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 bankroll<sup>5</sup>. 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.<hr>
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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)<sup>p</sup> * (1 - X)<sup>1-p</sup> - 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:
<sup>dlog(G)</sup>/<sub>dX</sub> = p*(O-1)/(1 + (O-1) * X) - (1-p)/(1 - X)
setting to zero and solving yields:
X = (Op-1)/(O-1)
with <sup>d2log(G)</sup>/<sub>dX</sub><sup>2</sup> ≤ 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 P<sub>W</sub>, a loss probability of P<sub>L</sub>, and a push probability of P<sub>T</sub> (where P<sub>W</sub> + P<sub>L</sub> + P<sub>T</sub> = 1), then the probability of winning conditioned on not pushing would be:
P*<sub>W</sub> = P<sub>W</sub> / (1 - P<sub>T</sub>)
and the probability of losing conditioned on not pushing would be:
P*<sub>L</sub> = P<sub>L</sub> / (1 - P<sub>T</sub>)
So assuming decimal odds of O, Edge would be:
Edge = O × P<sub>W</sub> / (1 - P<sub>T</sub>) - 1
-or-
Edge = O × P<sub>W</sub> - (1 - P<sub>T</sub>)
which in either case is just the same as:
Edge = O × P*<sub>W</sub> - 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.
