# What Is The Basin Of Convergence For Kelly Betting?

The basic rough argument for Kelly betting goes something like this.

First, assume we’re making a sequence of T independent bets, one-after-another, with multiplicative returns (similar to e.g. financial markets). We choose how much money to put on which bets at each timestep.

Returns multiply, so log returns add. And they’re independent at each timestep, so the *total* log return over T timesteps is a sum of T independent random variables. “Sum of T independent random variables” makes us want to invoke the Central Limit Theorem, so let’s assume whatever other conditions we need in order to do that. (There are multiple options for the other conditions.) So: total log return will be normally distributed for large T, with mean equal to the sum of expected log return at each timestep.

Then the key question is: for any given utility function, will it be dominated by the typical/modal/median return, or will it be dominated by the tails? For instance, the utility function u(W) = W is dominated by the upper tail: agents maximizing that utility function will happily accept a probability-approaching-1 of zero wealth, in exchange for an exponentially tiny chance of exponentially huge returns. On the other end of the spectrum, a utility function which just wants wealth above some relatively-low threshold (i.e. utility = 0 below threshold, utility = 1 above) will be dominated by the lower tail: agents maximizing that utility function mostly care about minimizing the increasingly-tiny probability of a total return below the threshold, and will pass up exponentially larger returns in order to avoid that downside.

But in the middle, it seems like there should be a whole class of utility functions which are dominated by the typical/modal/median return. And what the hand-wavy central limit argument says is that an agent with *any* of the utility functions in that class will, for sufficiently large T, just maximize expected log return at each timestep - i.e. Kelly bet. That class of utility functions is the “basin of convergence” for Kelly betting - i.e. the class of utility functions whose asymptotic behavior converges to Kelly betting, for long time horizons (i.e. large T) when making a sequence of independent bets with multiplicative returns.

Thus the question of this post: what’s the basin of convergence for Kelly betting?

I don't know the answer to that question, despite having poked at it a little. The rest of this post will contain some more quick-and-dirty thoughts on the topic, but my main hope is that somebody else will be inspired to answer the question.

Mathematical Setup: What Precisely Are We Saying?
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Suppose that, at each timestep, our agent invests their portfolio into some assets. The proportion invested in each asset $i$ at time $t$ is $c_i^t$, and the return of asset $i$ between $t$ and $t+1$ is $R_i^t$. Then the total wealth after $T$ timesteps is

$W^T = W^0 e^{\sum_{t=0}^{t-1} ln(\sum_i c_i^t R_i^t)}$

The agent has some utility function $u(W^T)$, and chooses $c^t$ at each timestep to maximize $E[u(W^T)]$.

For the easiest version of the problem which still captures the bulk of the intent, let's assume that **returns **$R_i^t$** are independent and identically distributed (IID) over time**.

Now, if we want to use the Central Limit Theorem, we need $ln(\sum_i c_i^t R_i^t)$ to be IID over time, and also have bounded variance. Alas, both of those can definitely be false:

*   The agent can choose $c^t_i$ using whatever information is available at time t, including past returns $R_i^{\tau < t}$.
*   Some strategies lose all wealth in finite time with nonzero probability, in which case variance will typically be infinite, because $ln(0)$ is infinite.

Concrete examples where each of the above conditions fails:

*   Suppose the agent's utility function is binary: it just wants to end with wealth above some fixed amount $W^*$. Then typically, the agent's optimal strategy will hold riskier portfolios when wealth is low, safer portfolios as wealth gets close to $W^*$, and zero risk once wealth passes $W^*$ (think conventional wisdom on retirement savings). So, the allocation at each timestep depends on returns up to that timestep, breaking independence.
*   Suppose the agent's utility function is convex, i.e. the agent prefers (50% double wealth, 50% lose everything) over (100% wealth stays the same). Then when the agent takes a gamble with e.g. 50% chance of losing everything, it will have a 50% chance that $ln(\sum_i c_i^t R_i^t)$ is $-\infty$, which uh... does not make for bounded variance.

... so a central part of the challenge of proving a basin of convergence for Kelly is showing that, *within the basin*, problems of this sort do not break the argument (... or using some entirely different kind of argument for Kelly).

The other big part of proving a basin of convergence would presumably be to talk about the expected-utility-contribution from the tails of the distribution.