Kelly-Optimal Allocation Across Independent Bets

Optimization · Hard · Free problem

You have three independent bets you can make each round, staking fractions of your current wealth on each.

Bet 1 (Coin flip): A fair coin is flipped. You win your stake (net $+1$ per dollar) on Heads, and lose it (net $-1$ per dollar) on Tails.

Bet 2 (Dice sum): Two fair dice are rolled. You win $+4$ per dollar staked if the sum is exactly $4$, and lose $-1$ per dollar otherwise.

Bet 3 (Card product): Three cards are drawn from a standard 52-card deck. You win $+9$ per dollar staked if the product of the first two cards' ranks exceeds

0$, and lose $-1$ per dollar otherwise. (Ranks are
$ through
3$.)

You allocate fractions $f_1, f_2, f_3 \geq 0$ of your wealth to the three bets, subject to $f_1 + f_2 + f_3 \leq 1$.

  1. Write down the expected log-growth rate $g(f_1, f_2, f_3) = E[\log(W_{\text{end}} / W_{\text{start}})]$ as a function of the allocations.
  1. Derive the first-order (KKT) conditions for the Kelly-optimal allocation that maximizes $g$.
  1. Without solving the system numerically, explain qualitatively which bets should receive nonzero allocation, and how the optimal fractions compare to a naive strategy of betting a fixed small fraction $c$ on every positive-EV bet.

Hints

  1. Start by computing the expected value of each bet. Kelly never allocates to a non-positive-EV bet, so you can screen out losers immediately.
  2. The growth rate is $g = E[\log(1 + f_1 X_1 + f_2 X_2 + f_3 X_3)]$ summed over all
^3 = 8$ joint outcomes. The first-order condition for bet $j$ is $E[X_j / (1 + \sum_i f_i X_i)] = \lambda$.
  • For a single bet with win probability $p$ and net payoff $+b$ vs. $-1$, the Kelly fraction is $f^{*} = p - (1-p)/b$. Check which of the three bets yields $f^{*} > 0$.

    • Worked Solution

    How to Think About It: The Kelly criterion tells you to maximize expected log-growth, not expected return. The first thing to do in any Kelly problem is screen each bet: does it have positive expected value? If $E[X_i] \leq 0$, Kelly says don't touch it. Then for the positive-EV bets, you need to figure out the optimal fraction -- and because all bets draw from the same bankroll simultaneously, the allocations interact through the log.

    Quick Estimate: Let us compute the expected value of each bet per dollar staked.

    Kelly screening: only Bet 3 has positive EV. Bets 1 and 2 should get $f_1 = f_2 = 0$ at the optimum. For Bet 3 alone, the single-bet Kelly fraction is $f^{*} = p_3 - (1 - p_3)/b_3 = 0.692 - 0.308/9 \approx 0.658$. That is a large fraction, reflecting the very favorable odds.

    Approach: We write the expected log-growth rate over all ^3 = 8$ joint outcomes, then take partial derivatives and enforce the KKT conditions.

    Formal Solution:

    Let $p_1 = 1/2$, $p_2 = 1/12$, $p_3 \approx 0.692$, and let $b_1 = 1$, $b_2 = 4$, $b_3 = 9$ be the net win payoffs (with net loss always $-1$). Each bet independently pays $+b_i$ with probability $p_i$ and $-1$ with probability $q_i = 1 - p_i$.

    The wealth after one round is

    $W_{\text{end}} = W_{\text{start}} \cdot \left(1 + f_1 X_1 + f_2 X_2 + f_3 X_3\right)$

    where $X_i \in \{+b_i, -1\}$. Since the bets are independent, the expected log-growth rate is a sum over all $8$ joint states:

    $g(f_1, f_2, f_3) = \sum_{(s_1, s_2, s_3)} \left(\prod_{i=1}^{3} P(X_i = s_i)\right) \log\left(1 + \sum_{i=1}^{3} f_i s_i\right)$

    where each $s_i \in \{+b_i, -1\}$.

    To maximize, take the partial derivative with respect to $f_j$:

    $\frac{\partial g}{\partial f_j} = \sum_{(s_1, s_2, s_3)} \left(\prod_{i=1}^{3} P(X_i = s_i)\right) \frac{s_j}{1 + \sum_{i} f_i s_i}$

    This can be written compactly as

    $\frac{\partial g}{\partial f_j} = E\left[\frac{X_j}{1 + f_1 X_1 + f_2 X_2 + f_3 X_3}\right]$

    The KKT conditions for the constrained optimum (with $f_j \geq 0$ and $\sum f_j \leq 1$) are:

    $E\left[\frac{X_j}{1 + \sum_i f_i X_i}\right] \leq \lambda, \quad \text{with equality if } f_j > 0$

    where $\lambda \geq 0$ is the multiplier for the budget constraint $\sum f_j \leq 1$ (and $\lambda = 0$ if the budget is not binding).

    Additionally, for each $j$:

    $f_j \geq 0, \quad \text{and} \quad f_j \cdot \left(\lambda - E\left[\frac{X_j}{1 + \sum_i f_i X_i}\right]\right) = 0$

    Qualitative Analysis:

    1. *Bet 1 gets zero allocation.* Since $E[X_1] = 0$, the marginal growth from a small stake is $E[X_1/(1 + \cdots)]$. At $f_1 = 0$, this equals $E[X_1 / (1 + f_2 X_2 + f_3 X_3)]$. For a fair bet ($+1$ and $-1$ equally likely), this marginal value is slightly negative (by Jensen's inequality, the loss state -- where wealth is lower -- gets more weight in the ratio). Zero or negative edge bets never receive Kelly allocation.
    1. *Bet 2 gets zero allocation.* With $E[X_2] = -7/12$, the expected value is strongly negative. No amount of payoff asymmetry (the $+4$ upside) rescues it -- the win probability
      /12$ is too low. The marginal log-growth at $f_2 = 0$ is negative.
    1. *Bet 3 gets all the allocation.* With $p_3 \approx 0.692$ and a $9{:}1$ payoff, this bet has enormous edge. The single-bet Kelly formula gives $f_3^{*} = p_3 - q_3 / b_3 \approx 0.658$. Since no other bet warrants allocation, the multi-bet optimum reduces to the single-bet Kelly on Bet 3 alone.
    1. *Comparison with naive fixed-fraction betting.* A common heuristic is to bet a fixed small fraction $c$ (say $c = 0.05$) on every positive-EV bet. Here that would mean $f_3 = 0.05$ (only Bet 3 qualifies). This is far more conservative than the Kelly fraction of $\approx 0.658$, but it avoids the risk of overbetting. In practice, the Kelly fraction is often scaled down (half-Kelly or quarter-Kelly) because the input probabilities are estimated, not known. The naive strategy is a crude version of fractional Kelly -- it protects against estimation error but leaves significant growth on the table.

    Answer: Only Bet 3 (card product) receives nonzero Kelly allocation, with $f_3^{*} \approx 0.658$. Bets 1 and 2 are excluded because their expected values are non-positive. The expected log-growth rate at the optimum is $g(0, 0, f_3^{*}) = p_3 \log(1 + 9 f_3^{*}) + (1 - p_3) \log(1 - f_3^{*})$. The first-order conditions require $E[X_j / (1 + \sum_i f_i X_i)] \leq \lambda$ for all $j$, with equality for bets receiving positive allocation.

    Intuition

    The Kelly criterion is the bridge between expected value and bankroll management. A bet can have positive EV but still be a bad idea at large stake sizes -- the log utility penalizes ruin far more than linear utility rewards upside. The screening rule is simple: if $E[X] \leq 0$, the Kelly fraction is zero or negative, meaning you should not bet (or should take the other side if possible). This is why a fair coin flip at even odds gets zero allocation -- there is no edge to exploit, and any nonzero stake only adds variance.

    In practice, the Kelly fraction is an upper bound on how aggressively you should bet. Most traders use fractional Kelly (half or quarter) because the true probabilities are never known exactly, and overbetting relative to your actual edge is far more damaging than underbetting. The naive fixed-fraction approach ($c$ on every positive-EV bet) is a rough-and-ready version of this conservatism -- it works reasonably well when you have many small bets, but it ignores the huge difference in edge across bets. The full Kelly framework tells you to size each bet in proportion to its edge-to-odds ratio, which is the core principle behind professional bankroll management.

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