How Finite-Horizon Utility Changes Optimal Kelly Bets

# Kelly Criterion for finite rounds


# Kelly Criterion for finite rounds












I understand that the Kelly criterion maximizes long-term utility under a log-wealth function, but from what I've gathered, this is only true if we have an arbitrarily high number of betting rounds. My question is: Is there a way to adapt the Kelly strategy for a finite number of rounds? For example, say, a game offering 2:1 odds, with a 50% probability of winning; Kelly's formula gives (2 * 0.5 - 0.5)/2 = 0.25 of our wealth to be bet on this game, per round.

Specifically, I am asking two things:

- If we had something like 10 rounds of betting, how should our bets change under the underlying framework of maximizing "long-term" wealth over the course of 20 rounds? Should we be more conservative, using a half-Kelly bet or quarter-Kelly bet, or should we be more aggressive, since intuitively we have no more potential gains after the 10th round of betting?

- If we had a single round of betting, how would our single bet change? At this point, does it depend solely on one's individual utility function and risk aversion? For example, I'm thinking that someone with no risk aversion would just bet their entire bankroll on this bet in a single round, since the expected value is positive. Are all responses that are consistent with a certain risk-utility framework then rational?

Thanks!

## Answer by Kermittfrog (score 3)

https://quant.stackexchange.com/a/80219

Given independently and identically distributed betting outcomes $G$ (gain) with probability $p$ and $L$ (loss) with probability $1-p$, whether or not the optimal betting fraction $f$ is a function of the total number of bets $n$ depends on the structure of the utility function.

Under logarithmic and power utility, the fraction is independent of $n$ as we will show below. For exponential power utility, I cannot prove the dependency in closed form, but provide a quick way to check for yourself.

### Logarithmic utility

Under logarithmic utility and given independent bets, the optimal betting size is independent of the number of rounds.

Let's take the thoughts in Wiki's proof of the Kelly criterion as a starting point.

Assuming logarithmic utility, $u=\ln(w)$ and $n$-round gambling with fixed win probability per round $p$,a fixed investment fraction per round $f$, as well as gains $G$ and losses $L$, we have a binomially distributed final wealth - and hence log utility. The expected utility is:

$$ \begin{align} \mathrm{EU}(f)&=\sum_{k=0}^n{n\choose k}p^k(1-p)^{n-k}\ln\left((1+fG)^k(1-fL)^{n-k}\right)\\ &=\sum_{k=0}^n{n\choose k}p^k(1-p)^{n-k}\left[k\ln(1+fG)+(n-k)\ln(1-fL)\right]\\ &=\ln(1+fG)\sum_{k=0}^nk{n\choose k}p^k(1-p)^{n-k}+\ln(1-fL)\sum_{k=0}^n(n-k){n\choose k}p^k(1-p)^{n-k}\\ &=np\mathrm{ln}(1+fG)+n(1-p)\ln(1-fL)\\ &=n\left(p\mathrm{ln}(1+fG)+(1-p)\ln(1-fL)\right) \end{align} $$

From here on we see that the derivation of the optimal $f$ is independent of the number of rounds played, $n$:

$$ \begin{align} \max_f \mathrm{EU}&\Rightarrow\frac{\partial \mathrm{EU}}{\partial f}\stackrel{!}{=}0\\ &\Rightarrow p\frac{G}{1+fG}-(1-p)\frac{L}{1-fL}\stackrel{!}{=}0\\ &\Rightarrow f^*=\frac{p}{L}-\frac{1-p}{G} \end{align} $$

### Power utility

Under power utility, $u=\frac{w^{1-\alpha}}{1-\alpha}$. Plugging this into the expected utility, we have

$$ \begin{align} \mathrm{EU}(f)&=\frac{1}{1-\alpha}\sum_{k=0}^n{n\choose k}p^k(1-p)^{n-k}\left((1+fG)^k(1-fL)^{n-k}\right)^{1-\alpha}\\ &=\frac{1}{1-\alpha}\sum_{k=0}^n{n\choose k}\left(p(1+fG)^{1-\alpha}\right)^k\left((1-p)(1-fL)^{1-\alpha}\right)^{n-k}\\ &\equiv \frac{1}{1-\alpha}\sum_{k=0}^n{n\choose k}P^kQ^{n-k}\\ &=\frac{1}{1-\alpha}\sum_{k=0}^n{n\choose k}\left(\frac{P+Q}{P+Q}P\right)^k\left(\frac{P+Q}{P+Q}Q\right)^{n-k}\\ &=\frac{1}{1-\alpha}\left(P+Q\right)^n\sum_{k=0}^n{n\choose k}\left(p^*\right)^k\left(1-p^*\right)^{n-k}\\ &=\frac{1}{1-\alpha}(P+Q)^n\\ &\equiv\frac{1}{1-\alpha}\left(p\left(1+fG\right)^{1-\alpha}+(1-p)\left(1-fL\right)^{1-\alpha}\right)^n \end{align} $$

Again, the optimum will be independent of $n$:

$$ f^*=\frac{\left[(1-p)L\right]^{-1/\alpha}}{G\left[pG\right]^{-1/\alpha}+L\left[(1-p)L\right]^{-1/\alpha}}-\frac{\left[pG\right]^{-1/\alpha}}{G\left[pG\right]^{-1/\alpha}+L\left[(1-p)L\right]^{-1/\alpha}} $$

### General formulation

In general, the optimal betting fraction maximizes the expected utility of future wealth $w(x)=(1+fG)^x(1-fL)^{n-x}$ given a random number of realizing a gain $x$ and corresponding number of losses $n-x$:

$$ f^*:\max_f\mathrm{E}\left(u\left[(1+fG)^x(1-fL)^{n-x}\right]\right) $$

Deriving and rearranging the first order condition yields

$$ \begin{align} f^*&=\frac{1}{L}\frac{\mathrm{E}\left[xw(x)u'(w(x))\right]}{n\mathrm{E}\left[w(x)u'(w(x))\right]}-\frac{1}{G}\frac{\mathrm{E}\left[(n-x)w(x)u'(w(x))\right]}{n\mathrm{E}\left[w(x)u'(w(x))\right]}\\ &\equiv \frac{\tilde{p}}{L}-\frac{1-\tilde{p}}{G} \end{align} $$

where $\tilde{p}$ can be interpreted as some pseudo probability. Note that the expectations depend on $f^*$ again, hence the solution can be understood as a fixed point equation.

Under exponential utility I find that the number of rounds is relevant for the optimal $f$. Alas, I cannot derive the relationship algebraically - only by solving the above fix point equation for various levels of $n$ and different assumptions on utility.