Why a Positive Expected Value Position Usually Loses

Expected value is the probability-weighted average of all outcomes. For a two-outcome position, it is the probability of the favourable outcome times its payoff, minus the probability of the unfavourable outcome times the stake.

Consider a hypothetical outcome priced at decimal odds of 5.0, a net payoff of 4 units per unit staked, and suppose the true probability is 22 percent. The expected value is 0.22 times 4, minus 0.78 times 1, which is plus 0.10 units per unit staked. The position is positive in expectation. It also fails 78 percent of the time.

Nothing about that is contradictory. The average is pulled up by a payoff that is large and rare. Any single trial is most likely to lose.

The spread of results around the expectation depends on the price. A near-even outcome at a small edge produces results that cluster reasonably tightly. A long-priced outcome with the same expected value produces long losing runs punctuated by large gains.

In the hypothetical above, the probability of losing ten consecutive independent trials is 0.78 to the tenth power, roughly 8 percent. Losing streaks of that length are not a sign that the estimate was wrong; they are a routine feature of the distribution.

This is why intuition fails on long prices. People judge by what they see, and what they see most often is a loss.

Separating a real edge from noise requires enough trials that the standard error of the average result is small compared with the edge itself. The standard error shrinks with the square root of the number of trials, so halving it takes four times as many.

For near-even prices and small edges, that already means large samples. For long prices, where the variance per trial is much higher, it takes many more. A few dozen results at long prices say almost nothing about whether the underlying estimates are sound, in either direction.

The practical consequence: never adjust a model because of a short run of results on long-priced outcomes. Adjust it because its probability estimates are measurably miscalibrated.

Because outcomes are noisy, evaluate the quantity that drives expected value: the probability estimate.

• Calibration. Group predictions by estimated probability and check how often each group actually occurred. Predictions of about 20 percent should occur about 20 percent of the time over a large sample.
• Proper scoring rules. Log loss and the Brier score reward estimates that are both calibrated and sharp, and are far less noisy than profit and loss.
• Closing price comparison. If your estimate consistently implies better value than the final market price, that suggests your information is ahead of the market, and it converges much faster than results do.
• Segment by price range. Models are often well calibrated near even prices and poorly calibrated on long prices, where data is thin.

Every expected value calculation takes a probability as input. On long-priced outcomes, a small absolute error in that probability is a large relative error. Estimating 22 percent when the truth is 18 percent turns the hypothetical above from plus 0.10 to minus 0.10 per unit. Rare outcomes are also the hardest to estimate, because they appear least often in historical data.

So the positive expected value position that usually loses deserves two kinds of patience: patience with variance when the estimate is sound, and scepticism about the estimate itself, which should be tested with calibration rather than assumed.

This page describes how expectation and variance behave and is not betting advice.