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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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The relation between (1) your linked problem and (2) classic Kelly problem.

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.)

Some possible intuition? 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 $\frac{\partial \log w}{\partial t} = \log b + \log (1+r)$. The growth rate is linearly separable in savings rate and the interest rate. A higher interest $r$ makes your wealth grow faster, but it doesn't enter into the marginal effect of saving more $\frac{\partial^2 \log w}{\partial t \partial b} = \frac{1}{b}$.

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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