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](http://octavia.zoology.washington.edu/teaching/429/lecturenotes/Lecture8.pdf).

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?