Edge and Bet Sizing

Every trade or bet asks you two separate questions, and beginners answer only the first. Question one: do I have an edge? — is the expected value in my favor, as computed in Expected-Value Games. Question two, which decides whether the edge ever turns into money: how much do I bet? You can have a real, positive edge and still go broke by betting too much, and you can waste a fine edge by betting too little. Sizing is not a footnote to the edge; it's half the game.

Your edge is your expected profit per dollar staked. For a bet that wins with probability $p$ at $b$-to-1 odds and loses your stake otherwise, the edge is

$$\text{edge} = p\,b - (1 - p).$$

That's just the expected value of a one-dollar bet: probability of winning times what you win, minus probability of losing times what you lose. A positive edge means "over many repetitions this makes money." It says nothing yet about how much to put at risk.

Sizing: bet edge over odds

The sizing rule that maximizes long-run growth is the Kelly criterion: bet the fraction

$$f^\star = \frac{\text{edge}}{b} = \frac{p\,b - (1-p)}{b}$$

of your bankroll. In words, bet your edge divided by the odds. Bigger edge, bet more; longer odds (bigger $b$), bet less of your roll because each loss is a bigger fraction of what you risked. For a simple even-money bet ($b = 1$) this collapses to just betting your edge, $f^\star = 2p - 1$.

Worked example: the 55% coin

You can bet any fraction of your $1,000 bankroll on a coin that lands your way 55% of the time, at even money. The edge is $2(0.55) - 1 = 0.10$ — a healthy 10 cents of expected profit per dollar. Kelly says bet a fraction $f^\star = 0.10$, i.e. $100 on the first flip, then 10% of whatever your bankroll becomes after each result.

The temptation is to think "I've got a big edge, let me press it." Watch what over-betting does to the same favorable coin:

Fraction bet each flip	Long-run outcome
5% (half-Kelly)	grows steadily at ~¾ the top rate, gentle swings
10% (full Kelly)	fastest possible growth — but stomach-churning drawdowns
20% (twice Kelly)	growth rate collapses to zero — you drift nowhere
30%+ (over-betting)	growth turns negative — near-certain ruin over time

The edge never changed — the coin is favorable in every row. What changed is that betting too much lets variance compound against you faster than the edge compounds for you. Past twice Kelly, a favorable bet becomes a wealth destroyer.

Why real traders bet a fraction of the "right" size

Kelly assumes you know your edge exactly. You never do. Your win probability is an estimate, and estimates are optimistic exactly when it matters — the bets you found most attractive are the ones where you most likely overestimated the edge, a cousin of the winner's curse. Because over-betting is punished so much harder than under-betting, the sane response to uncertainty is to shrink. Most practitioners bet half Kelly or less, which keeps roughly three-quarters of the growth for far less than half the variance. The same logic drives real Position Sizing on trading desks: size to a risk budget, not to a fragile point estimate of the edge.

The interview version

The canonical prompt is exactly the biased coin: "a coin lands heads 60% of the time, you can bet any fraction of your bankroll at even odds — what fraction maximizes long-run growth?" The edge is $2(0.6) - 1 = 0.20$, so full Kelly is 20% and the slick answer notes you'd more likely run half that in practice. The trap the interviewer is watching for is the person who says "bet it all" — that maximizes expected wealth but guarantees eventual ruin, the whole reason sizing exists as a separate discipline from finding the edge.