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I have just learned about Kelly gambling from Chapter 6 of Cover & Thomas' Introduction to Information Theory. The mathematical setup is that we have a horse race, with horse $i$ winning with probability $p_i$. If horse $i$ wins, you receive $o_i$ dollars for every dollar you bet on it. You imagine this horse race repeated infinitely many times, and you optimize over ``betting strategies'' $b$ which repeatedly bet a proportion $b_i$ of your total wealth on horse $i$.

In this case, your wealth will grow (or decay) exponentially. If you want to maximize the exponent, Cover & Thomas prove the optimality of the Kelly strategy which chooses $b_i=p_i$ for all horses $i$. For a detailed writeup of this, see for example here.

This seems extremely counterintuitive to me -- I would have thought the optimal strategy would depend on the odds $o_i$. I follow all the math in the derivation, but is there a conceptual explanation for what is going on here?

Let me give an example illustrating the phenomenon.

Example 1: Suppose there is a horse race with two horses, the first of which will win the race with probability $1/3$ and the second of which wins with probability $2/3$. For horse $1$, you receive $2$ dollars from the racetrack for every dollar you bet on it if it wins (and, of course, $0$ dollars if it loses). The odds given by the racetrack -- namely $2$ dollars in the case of victory - are the same for horse $2$.

The theorem says to bet $1/3$ of your wealth on horse $1$ at every step (and the rest on horse 2) to maximize the exponent in the long-term exponential growth rate of your wealth.

Example 2: Suppose now everything remains the same, except that if you bet a dollar on horse $2$, you now receive $1,000,000 dollars if you win.

The theorem states that the best strategy is exactly the same as in Example 1, which seems very strange -- intuitively, why aren't you betting more on horse 2 in this case?

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  • $\begingroup$ First of all a warm welcome to Quant.SE! But the odds are included?!? en.wikipedia.org/wiki/Kelly_criterion#Statement $\endgroup$ – vonjd Mar 8 '18 at 18:43
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    $\begingroup$ This is a slightly different setup than what I am asking about. For the independence of odds, besides the link in my question, also see en.wikipedia.org/wiki/Gambling_and_information_theory under "Kelly Betting" and then "Doubling Rate" $\endgroup$ – Michael S. Mar 8 '18 at 18:58
  • $\begingroup$ Yes, but in your question you state the optimality criterion $b_i=p_i$ which is just another way of using the odds. $\endgroup$ – vonjd Mar 8 '18 at 19:03
  • $\begingroup$ To clarify, the odds are the numbers $o_i$ -- I am asking why the optimal betting is independent of $o_i$. $\endgroup$ – Michael S. Mar 8 '18 at 19:10
  • $\begingroup$ Yes, but odds and probability are just two sides of the same coin! See e.g. here: ctspedia.org/do/view/CTSpedia/OddsTerm $\endgroup$ – vonjd Mar 8 '18 at 19:25
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Kelly gambling does not ignore odds, the optimality criterion given as probabilities is just another way of stating the odds.

See e.g. here: https://www.ctspedia.org/do/view/CTSpedia/OddsTerm

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    $\begingroup$ I don't think this answers it. In my question, the word "odds" has a specific meaning -- it refers to the numbers $o_i$; and the word "probabilities" has a specific meaning -- it refers to the numbers $p_i$. There is no relationship whatsoever between the numbers $p_i$ and $o_i$, and I'm asking why a certain strategy is independent of $o_i$. I have added two examples illustrating the question which may help clarify matters. $\endgroup$ – Michael S. Mar 8 '18 at 19:42
  • $\begingroup$ I initially upvoted, but @MichaelS. is right that his question is different. $\endgroup$ – Matthew Gunn Mar 9 '18 at 14:58
  • $\begingroup$ @MatthewGunn: Thank you. To be honest with you I am still not convinced. I still think that I gave the correct answer, but I will think about it once again... $\endgroup$ – vonjd Mar 9 '18 at 15:58
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    $\begingroup$ Perhaps a rephrasing is, "Why does the optimal growth portfolio among a set of arrow securities not depend on the securities' payoffs?" (Defining the optimal growth portfolio as the portfolio that maximizes log wealth.) $\endgroup$ – Matthew Gunn Mar 9 '18 at 16:26
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This rather unintuitive result is a result of the log objective, of maximizing the expected growth rate of wealth rather than wealth itself.

A different problem to build some intuition [Update 2019]

To build some intuition about working with changes in logs rather than changes in levels, let's imagine a world without uncertainty.

Imagine you save fraction $b$ of your wealth each period and earn interest rate $r$, hence wealth $w_t=[b(1+r)]^t$. Your growth rate is basically the change in log wealth. More precisely, your instantaneous growth rate $g$ is: $$ \begin{align*} g &= \frac{\partial \log w_t}{\partial t} \\ &= \log b + \log (1+r) \end{align*}$$.

Observe that working in logs, the savings term $\log b$ and the interest rate term $\log(1+r)$ are linearly separable!

  • $ \frac{\partial g}{\partial r} = \frac{1}{1+r} > 0$ : A higher interest rate indeed raises the growth rate of wealth.
  • $ \frac{\partial g}{\partial b} = \frac{1}{b} > 0$ : Saving more also raises the growth rate of wealth. The effect of saving more is $\frac{\partial g}{\partial b}$.
  • BUT $ \frac{\partial ^2}{ \partial b \partial r} = 0$! A higher interest rate doesn't change the effect of saving more!

This is what's behind the result in your linked problem. When working in logs instead of levels, the payoff on the horse is linearly separable from your bet on the horse. Both terms affect the growth rate, but there's no interaction between the two.

The relation between (1) your linked problem and (2) classic Kelly problem [Original answer]

In both problems, the objective is the same: maximize expected log wealth. What differs is the set of securities you can buy.

  • The problem you linked to is a portfolio allocation problem between $m$ arrow securities for $m$ states of the world. E.g. in the case $m=2$, you have two securities with payoffs: $$ \begin{bmatrix} o_1 \\ 0 \end{bmatrix} \quad \quad \quad \begin{bmatrix} 0 \\ o_2 \end{bmatrix} $$
  • In the binary, classic Kelly betting problem the security payoffs are:

$$ \begin{bmatrix} o_1 \\ 0 \end{bmatrix} \quad \quad \quad \begin{bmatrix} 1 \\ 1 \end{bmatrix} $$ The second security is from holding cash (i.e. not betting). The solutions to these problems are entirely compatible. In both problems, the share of $t=0$ wealth you spend to buy $t=1$ payoffs in state $i$ is simply $p_i$. In neither problem does it depend on odds $o_i$. (It appears to depend on $o_i$ in the classic Kelly problem because of obfuscation due to the risk free security.)

Your problem: setup

  • There are $m$ different horses (i.e. outcomes).
  • You bet fraction $b_i$ of wealth on horse $i$.
  • Horse $i$ pays $o_i$ if you win and has a $p_i$ chance of winning.
  • You must bet all your wealth: $\sum_i b_i = 1$.

Hence, if horse $i$ wins, you will have $o_ib_i$ times your original wealth. Your expected log wealth is $\sum_{i=1}^m p_i (\log o_ib_i)$.

Maximize expectation of log wealth:

This is equivalent to maximizing your growth rate. The problem is:

$$\begin{equation} \begin{array}{*2{>{\displaystyle}r}} \mbox{maximize (over $b_i$)} & \sum_{i=1}^m p_i (\log o_i + \log b_i) \\ \mbox{subject to} & \sum_{i=1}^m b_i = 1 \end{array} \end{equation} $$ This is a convex optimization problem where Slater condition holds hence the first order conditions are necessary and sufficient. Lagrangian is $ \mathcal{L} = \sum_{i=1}^m p_i (\log o_i + \log b_i) - \lambda \left(\sum_i b_i - 1\right)$

First order conditions are: $$ \frac{p_i}{b_i} = \lambda \text{ for all $i$} \quad \quad \quad \sum_i b_i = 1 $$

Hence $\lambda = 1$ and $b_i = p_i$.

Relation to classic Kelly criterion problem (case $m=2$)

You are deciding how much wealth to allocate to a risky bet that pays odds $o_1$ and a riskless bet that pays odds 1 (i.e. net odds 0). This is equivalent to a portfolio allocation to securities with payoffs $\begin{bmatrix} o_1\\0\end{bmatrix}$ and $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$ (both of whose price is 1).

The classic Kelly criterion solution is that the agent allocates:

  • $p_1 - \frac{p_2}{o_1 - 1}$ to the security with price 1 and payoff $\begin{bmatrix}o_1 \\ 0 \end{bmatrix}$
  • $1 - p_1 + \frac{p_2}{o_1 - 1}$ to the security with price 1 and payoff $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$

What is the agent's payoff in state 1?

\begin{align*} \left( p_1 - \frac{p_2}{o_1 - 1}\right) \cdot o_1 + \left( 1 - p_1 + \frac{p_2}{o_1 - 1} \right) \cdot 1 &= p_1o_1 \end{align*}

What is the agent's payoff in state 2?

$$\left( 1 - p_1 + \frac{p_2}{o_1 - 1} \right) \cdot 1 = p_2 \left( \frac{o_1}{o_1 - 1} \right) $$

How much $t=0$ wealth does the agent spend at price $\frac{1}{o_1}$ to get the state 1 payoff?

\begin{align*} b_1 = o_1p_1 \frac{1}{o_1} = p_1 \end{align*} From the existing securities, you can construct an security with price $1$ and payoff $\begin{bmatrix}1\\o_2 \end{bmatrix}$ where $o_2 = \frac{o_1}{o_1 - 1}$.

How much wealth does the agent spend at price $\frac{1}{o_2} = \frac{o_1-1}{o_1}$ to get the state 2 payoff?

\begin{align*} b_2 = \left( p_2 \left( \frac{o_1}{o_1 - 1} \right) \right) \frac{o_1-1}{o_1} = p_2 \end{align*}

So the solutions are entirely compatible, you choose $b_1 = p_1$ and $b_2 = p_2$ in the classic Kelly problem as well.

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Kelly DOES reflect the odds!

The simple binary bet form of Kelly is:

Kelly Fraction = (p(win) * (odds + 1) - 1) / odds

So for a 60% chance of a 50% risk, ie 1:1 equals odds 1, that’s 20% of your capital at risk.

More formally, Kelly seeks to maximise log-wealth (LW)

LW = sum ( Pi * ln(1 + Stake * Payoffi)

Maximise LW, then dLW/dStake = 0

For each event i, that derivative is Payoff/(1+ stake*payoff)

Work through the maths and any variation of that will end up with a multiple of the payoffs on the denominators. Which is the net size reflecting the odds!

The important caveat here is that’s the Kelly bet size is only optimal if you are 100% certain about your informational advantage. If you are confident that you have an informational advantage but are not certain about its magnitude, then Kelly will be sub-optimal; and more likely than not detracting from returns because of over-betting. Kelly is a MAXIMUM bet size, that is only optimal if you are 100% certain.

Allowing a 50% chance that the market is right versus 50% I’m right, and the stakes and associated profits will reduce pro-rata to the odds. That’s the bit about Kelly that people tend to ignore in haste, and repent in leisure :-)

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