The Kelly Criterion
Suppose we’re offered an even money bet on a coin toss. If the coin is fair, then our expected winnings is zero.
What if we secretly know the coin is unfair, and shows heads 60% of the time? Obviously we should play as many times as we can and always bet on heads, but what betting strategy should we use? How much of our wealth should we risk?
Let’s say we start with $100 and play 300 times. If we simply bet $1 each time, it’s unlikely we’ll go bust, as that would require 100 more tails than heads, and we expect to make \((0.6 - 0.4) \times $300 = $60\).
Thus we can easily make money with high probability. But can we do better than $60?
Try your ideas here!
Centuries ago, Bernoulli unlocked the secret to solving this problem: maximize the expected log wealth.
In 1956, John Larry Kelly Jr. arrived at the same conclusion in A New Interpretation of Information Rate. Although he was far from being the first to consider log wealth, his paper is still valuable because:
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Kelly walks through illuminating examples resembling real-life scenarios, like Bernoulli.
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Kelly studies series of bets rather than particular bets in isolation as Bernoulli did, and derives an optimal strategy: the Kelly criterion.
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Kelly reveals a startling connection with information theory.
Sure Thing
Let us first consider a simpler case: suppose we’re offered an even money bet on a coin that always shows heads. (Perhaps it has heads on both sides.)
Then we should play as much as we can, and always bet our entire fortune on heads, doubling our wealth each time. In other words, if \(V_n\) denotes our wealth after playing \(n\) times:
\[ V_n = 2^n V_0 \]
Biased Coin
Now consider an even money bet on a less biased coin. Suppose it shows heads with probability \(q\) and tails with probability \(p = 1 - q\). If we bet one dollar, we expect to profit by \(q - p\) dollars, thus it makes sense to play if \(q > p\).
If we gamble our entire bankroll every time we play, then after \(n\) rounds, our expected wealth is:
\[ E[V_n] = (2q)^n V_0 \]
We have maximized the expected value, but what would Bernoulli do? He’d remind us log wealth is what matters. We have:
\[ E[\log V_n] = q^n (n \log 2 + \log V_0) + (1 - q^n) \log 0 = -\infty \]
This captures Warren Buffett’s words: "to make money they didn’t have and didn’t need, they risked what they did have and did need, and that’s foolish."
One might raise concerns about negative infinity stomping over the rest of the calculation. Bernoulli counts things like "productive capacity" as wealth, and here we’re not gambling away productive capacity and truly losing everything; we’re merely losing all our money. Thus the following may more accurate:
\[ E[\log V_n] = q^n \log (2^n V_0 + X) + (1 - q^n) \log X \]
where \(X\) is a reserve that is never gambled. For large \(n\), the left term is approximately:
which vanishes. The remaining term approaches \(\log X\), thus we wind up losing everything except for what we cannot gamble. Either way, log wealth teaches us this is a bad strategy.
What can log wealth teach us about good strategies? This is precisely what Kelly answered (likely without having read Bernoulli’s paper). Kelly defines the gain:
\[ G = \lim_{n\rightarrow\infty} \frac{1}{n} \lg \frac{V_n}{V_0} \]
where \(\lg\) denotes the base-2 logarithm. Compared with \(\log V_n\):
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We divide by \(V_0\) before taking logarithms to normalize by our initial wealth. This is harmless: maximizing a multiple of our wealth is the same as maximizing our wealth. The resulting expressions may be more intuitive as studies suggest that even after major changes, humans tend to return to a relatively stable level of happiness. That is, win or lose, apart from an initial adjustment period, we make judgements relative to what we have right now rather than on absolute numbers.
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We divide by the number of rounds \(n\) so the limit makes sense. While our wealth may grow unbounded as \(n\) increases, in our example we can at most double our money each round. Bernoulli studied one-off bets, while Kelly studied long-run behaviour over a series of bets.
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Instead of natural logarithms, we pick base 2 logarithms so we can instantly state the change in wealth per game as a fraction. For example, a gain of 0.42 means that in the long run, it’s as if our wealth grows by 42% every game.
Our goal is to maximize \(G\). Betting all our wealth each time yields \(G = 1\) for the coin that always shows heads, otherwise almost surely \(G = -\infty\). Betting zero each time yields \(G = 0\); nothing ventured, nothing gained.
If we instead bet a constant fraction \(f\) of our wealth each time, then after \(W\) wins and \(L\) losses:
so the gain is:
\[ G = \lim_{n\rightarrow\infty} \left( \frac{W}{n} \lg (1 + f) + \frac{L}{n} \lg (1 - f) \right) \]
Thus almost surely:
\[ G = q \lg (1 + f) + p \lg (1 - f) \]
due to the law of large numbers. Differential calculus shows that at \(f = q - p\), this attains the maximum value:
\[ G_\max = 1 + p \lg p + q \lg q \]
When \(p = 0.4, q = 0.6\) this implies we should bet 20% of our wealth each time, giving a gain of 2.9%.
Kelly’s Paradox?
The St. Petersburg paradox is a dull affair because it can be resolved by observing that the supply of rewards is finite in real life. In contrast, Kelly’s coin game is worthy of our attention. If we only look at expected wealth, then we conclude that in real life, we should repeatedly stake our entire fortune on a bet where our edge is razor-thin. This defies common sense so perhaps Kelly’s game deserves to be called a paradox.
Unlike the St. Petersburg paradox, log wealth is given a proper chance to shine, as it protects us from recklessly endangering all we own.
Loaded Dice
For an ideal die, if we succesfully bet on the number 6, then we should receive our stake back plus 5 times our stake, and otherwise forfeit our stake. In other words, the odds are 5 to 1, because on winning, our profit is 5 times our bet.
What if we secretly know the die is loaded? Suppose the die shows 6 with probability \(q\) and some other number with probability \(p = 1 - q\). Similarly to above, if we bet one dollar, we expect a profit of \(5 q - p\) dollars, thus it makes sense to predict 6 if \(5 q > p\).
If \(q = 0.2\) then the expected profit on a one dollar bet is 0.2 dollars. Above, a loaded coin that shows heads with probablity 0.6 also has an expected profit of 0.2 dollars. Should we then use the same betting strategy as for our coin? Namely, bet 20% of our funds every roll?
We aim to maximize the gain:
\[ G = q \lg (1 + 5f) + p \lg (1 - f) \]
This occurs when:
\[ f = \frac{5q - p}{5} \]
For \(q = 0.2\), this is 0.04, that is, if the die is loaded so that 6 shows with probability 0.2, we should always bet 4% of our wealth for a maximal gain of just under 0.55%. Although the expected profit is the same, we must reduce our bet size because of the increased risk of losing.
A die with \(q = 1/3\) leads to the Kelly bet size of 20%. The risk of losing is the same, but now the expected reward is 5 times larger.
Risk versus reward
Investors often talk about "risk versus reward". Kelly pinned down this delicate trade-off mathematically.
The technical details match our intuition beautifully. One might guess our stake ought to be proportional to the good thing, namely reward, and inversely proportional to the bad thing, namely risk.
\[ \frac{\text{reward}}{\text{risk}} \]
Kelly showed this is indeed optimal, where:
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"reward" means the expected profit, also known as the edge.
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"risk" means the odds against us.
That is, we should always bet the following fraction of our wealth:
\[ \frac{\text{edge}}{\text{odds}} \]
For our biased coin, the edge is 0.2 and the odds are 1 (to 1), leading to the Kelly fraction 0.2. For our loaded die, the edge is again 0.2 but the odds are 5 (to 1), leading to the Kelly fraction 0.04.
Speaking of investors, instead of Kelly’s "gain", they say rate of return, and instead of a single coin flip or die roll, they typically speak of an asset’s performance over a single year.
Conspiracy Theory
Kelly framed the above gambles in terms of communication.
Imagine betting on a coin, fair or foul, but an accomplice somehow tells us the result before we put any money down. If the communication channel is flawless, then it’s like a coin that always shows heads. That is, we should always bet our entire fortune on what we’re told by our insider, as we’re guaranteed to double it.
What if the communication channel (or our conspirator) is less reliable? Then if \(p\) is the probability we receive a bad tip, then it’s just like the biased coin discussed above.
The expression:
\[ G_\max = 1 + p \lg p + q \lg q \]
turns out to be what Shannon called the transmission rate \(R\) of a noisy binary channel which flips bits with probability \(p\), and as usual \(q = 1 - p\). It is also known as the mutual information between the input and output of such a channel.
Kelly showed that more generally:
\[ G_\max = R \]
How did this happen?! Claude Shannon defined information as base-2 logarithms of probabilities for high-minded reasons that had nothing to do with grubby concerns like money, utility, or gambling. Yet somehow it lines up perfectly with these games.
Never tell me the odds!
The Kelly criterion tells us how much of our wealth to put in a promising investment opportunity, yet finance professionals seem to have their heads in the sand, with a level of ignorance bordering on criminal negligence.
Victor Haghani and Richard Dewey performed an experiment which simulated a biased coin that shows heads with probability 0.6, and gave $25 each to 61 "financially and quantitively trained individuals". They could stake whatever they wanted on as many even-money bets as they could make for 30 minutes, which worked out to roughly 300 flips.
Total winnings were capped at $250 each to bound the cost of the study. If Kelly’s advice were heeded, we expect about 95% of the participants to top out. But only 21% managed to do so. and worse still, one third of the players lost money, failing to beat the infantile strategy of betting the same small amount each time. Indeed, 28% of the players went bust!
Perhaps only a few players took the game seriously enough to do their best, but these results suggest that if you give control of investments to supposed experts, even if they excel at finding favourable gambles, only a minority know how to size bets to properly exploit them, and they are outnumbered by oafs who lose money despite having an edge.
Economists appear to be responsible for this sorry state of affairs as they have historically disparaged Kelly (and hence Bernoulli), and others. See William Poundstone, Fortune’s Formula. See also his webpage about Kelly.