Optimal Betting Strategy When You Have an Edge
Most trading, gambling, and betting has a negative expected value. But suppose you find an edge: a situation with positive expected value. How do you exploit it without blowing up?
What Is an Edge?
A bet has an edge if its expected value is positive.
Let:
- \(p\) = probability of winning
- \(q = 1-p\) = probability of losing
- \(b\) = net profit per unit staked if you win
- you lose \(1\) unit if you lose
Then the expected profit per unit staked is:
\[ EV = p b - q = p b - (1-p) \]You have an edge if:
\[ EV > 0 \quad \Longleftrightarrow \quad p b > q \]More generally, if you win \(X\) and lose \(Y\), then:
\[ EV = pX + (1-p)Y \]If \(EV \le 0\), the correct bet is no bet.
Horse-racing example
Suppose there are \(N\) horses in a race. The market treats them as equally likely, so the implied probability of each horse is:
\[ m_i = \frac{1}{N} \]You are an insider. You observe many races under the same conditions. If horse \(i\) wins \(X_i\) times out of \(M\) races, your estimated probability is:
\[ \hat p_i = \frac{X_i}{M} \]The market's implied probability is still \(1/N\). Your edge on horse \(i\) is:
\[ \text{edge}_i = \hat p_i - \frac{1}{N} \]You bet on the horse with the largest estimated probability:
\[ i^* = \operatorname*{arg\,max}_i \hat p_i \]If the payout is fair at odds \(N\), then the net odds are \(b = N-1\). The EV is:
\[ EV = p(N-1) - (1-p) = Np - 1 \]So you have an edge if:
\[ p > \frac{1}{N} \]Your horse can still lose. But over many races, you should win more often than the market expects.
The question is: how much should you bet?
Kelly: How to Size an Edge
You should not go all in. A positive edge does not protect you from ruin. You need bankroll management so that the law of large numbers has time to work.
The Kelly criterion answers:
Given an edge, what fraction of your bankroll should you risk to maximize long-term growth?
The key idea is that you should not maximize expected profit. You should maximize expected log wealth. Losing half your bankroll hurts more than gaining half helps. Kelly balances growth and survival.
For a bet with net odds \(b\), win probability \(p\), and loss probability \(q = 1-p\), the optimal fraction is:
\[ f^* = \frac{bp - q}{b} \]If \(f^* \le 0\), the correct bet is zero.
For an even-money bet, \(b=1\), so:
\[ f^* = p - q = 2p - 1 \]Examples:
- If \(p = 0.55\), then \(f^* = 0.55 - 0.45 = 0.10\). Bet \(10\%\) of bankroll.
- If \(p = 0.52\), then \(f^* = 0.52 - 0.48 = 0.04\). Bet \(4\%\).
- If \(p = 0.50\), then \(f^* = 0\). No bet.
Why expected log wealth?
Suppose you bet fraction \(f\) of your bankroll on an even-money bet.
If you win, your bankroll becomes:
\[ B(1+f) \]If you lose, it becomes:
\[ B(1-f) \]Expected log wealth is:
\[ G(f) = p\ln(1+f) + q\ln(1-f) \]To maximize, take the derivative and set it to zero:
\[ G'(f) = \frac{p}{1+f} - \frac{q}{1-f} = 0 \]Solving gives:
\[ f^* = p - q = 2p - 1 \]For general net odds \(b\):
\[ G(f) = p\ln(1+bf) + q\ln(1-f) \]and the optimum is:
\[ f^* = \frac{bp - q}{b} \]The “Kelly value” of a bet is the long-run growth rate you get from betting that fraction. A bet can be +EV but still support only a small bet size.
Why Overbetting Is Dangerous
A positive edge does not protect you from ruin. If you bet too large, a normal losing streak can destroy your bankroll before your edge has time to work.
Imagine betting \(10\%\) of your bankroll on even-money bets. Ten losses in a row leave you with:
\[ 0.9^{10} \approx 0.349 = 34.9\% \]Painful, but survivable.
Now imagine betting \(50\%\) per bet. Ten losses in a row leave you with:
\[ 0.5^{10} \approx 0.00098 = 0.098\% \]You are effectively wiped out. The edge may still exist, but you are no longer there to collect it.
This is why professional bettors and investors often use fractional Kelly:
\[ f_{\text{used}} = c f^* \]where \(0 < c < 1\). Common choices are half Kelly, quarter Kelly, or smaller.
Full Kelly maximizes growth, but it also produces wild swings. Fractional Kelly gives up some growth in exchange for much lower volatility and a much smaller chance of ruin.
The Law of Large Numbers
The law of large numbers says that as the number of independent trials increases, the average result converges to the expected value.
Formally, if \(X_1, X_2, \dots, X_n\) are independent and identically distributed with mean \(\mu\), then:
\[ \frac{1}{n}\sum_{i=1}^n X_i \to \mu \]as \(n \to \infty\).
In betting terms:
- A bad bet can win in the short run.
- A good bet can lose in the short run.
- Over many repetitions, results tend toward the expected value.
This is why edge matters only if you get enough trials.
But there is a catch: to get enough trials, you must survive the variance along the way.
If you have positive expected value, the law of large numbers is your friend — but only if you bet small enough to stay in the game.
If you have negative expected value, the law of large numbers is your enemy. The more you play, the more certain it becomes that the house wins.
Final Thoughts
The math of long-run betting is not about hitting one big win. It is about direction, size, repetition, and survival.
Edge tells you which way the river flows.
Kelly tells you how big your boat should be.
The law of large numbers tells you how long the journey takes.
And betting small is what keeps you from crashing and burning from overspeeding before you get there.
Bet small enough to survive. Be honest enough to admit when there is no edge. And be disciplined enough to let the long run arrive.
The goal is not to win one hand. The goal is to still be in the game.
If you want to discuss betting strategies either on traditional instruments like stocks, options etc, or on any other instruments, feel free to connect.