Two different questions
When people talk about risk in a game, they often mix up two separate questions. One is how large the average result is. The other is how bumpy the individual results are around that average.
The first is the return: the average gain or loss per play, which is the expected value. The second is the volatility, also called variance or spread, which describes how far single results tend to land from the average. Neither number tells you the other. A game can have a small average loss and wild swings, or a similar average loss and gentle ones.
Measuring each one
Return is found by weighting each outcome by its probability and adding up. Volatility is usually measured by the standard deviation: take each outcome’s distance from the average, square it, weight it by probability, add up, and take the square root.
A shortcut for a single play is that variance equals the average of the squared results minus the square of the average result.
Two games with the same return
The games below are invented to keep the arithmetic clean. Each play stakes 1 unit.
Game A (frequent small wins). Win 0.9 units with probability 1/2, lose 1 unit with probability 1/2.
- Return: 0.5 × 0.9 − 0.5 × 1 = −0.05 units per play.
- Average of squared results: 0.5 × 0.81 + 0.5 × 1 = 0.905. Minus the square of the average (0.0025) gives variance 0.9025.
- Standard deviation: the square root of 0.9025, which is 0.95.
Game C (rare big win). Win 18 units with probability 1/20, lose 1 unit with probability 19/20.
- Return: 0.05 × 18 − 0.95 × 1 = 0.9 − 0.95 = −0.05 units per play.
- Average of squared results: 0.05 × 324 + 0.95 × 1 = 16.2 + 0.95 = 17.15. Minus 0.0025 gives variance 17.1475.
- Standard deviation: about 4.14.
Both games lose 0.05 units per play on average, so they have the same return. But Game C is more than four times as volatile (4.14 divided by 0.95 is about 4.4). Anyone looking only at the average would call them equivalent. In experience, they feel very different.
What the difference looks like in practice
In Game A, wins and losses are about equally frequent, and the balance wanders gradually. In Game C, a win arrives on average once in 20 plays, and in between there are long runs of small losses. The chance of going 20 plays in a row without a win is 0.95 raised to the 20th power, which is about 0.36, or roughly 36%. Such a drought is normal, not a sign that anything is wrong with the game, and a win does not become more likely because one is “due”.
Across a larger number of plays the totals reflect this. For 100 plays, the expected result is 100 × (−0.05) = −5 units in both games. The typical spread, however, is 0.95 × 10 = 9.5 units in Game A and 4.14 × 10 = about 41 in Game C, because spread grows with the square root of the number of plays (the square root of 100 is 10). So in Game C a finish well ahead or well behind is far more ordinary, even though the long-run average is the same. Over very many plays, the law of large numbers pulls the per-play average toward −0.05 in both, but the totals still wander widely along the way.
Why volatility matters for a fixed pool
The link to a bankroll is direct. A fixed pool has to survive the swings on the way to the long-run average. With a pool of 20 units and stakes of 1, Game A would typically lose gradually, whereas Game C could drain 20 units in 20 plays with no win, which has a chance of about 36% as shown above. The average return is identical; the practical risk of running out early is not. This is the same logic behind risk of ruin: higher volatility relative to the pool raises the chance of an early wipe-out.
Volatility also scales with stake size. Game B, a version of Game A with every number multiplied by 10 (win 9, lose 10), has a return of −0.5 per play and a standard deviation of 9.5, ten times larger on both counts. Changing the stake changes both numbers together.
Where the two ideas appear elsewhere
- In poker, the notion that good decisions can still lose for a long time is about volatility, discussed under poker variance.
- In betting, long-shot prices produce infrequent large payouts, while short prices produce frequent small ones, a pattern like Games C and A. The bookmaker margin affects return, while the choice of long shot or favourite mostly changes volatility.
- In lotteries, a very rare large prize produces extremely high volatility with a return that is typically negative.
- In investing and trading, volatility and return are the two standard descriptors of an asset, though real markets are less tidy than the games here.
Avoiding common mix-ups
- High volatility does not mean high return. A bumpy game can have a poor or negative average. Excitement and expected result are separate things.
- Low volatility does not mean safe. A game can lose steadily and predictably. Predictable losses are still losses.
- A big win does not reveal the return. One result, or even a few, tells little about the average, particularly in a high-volatility game where long samples are needed.
- Neither number predicts the next play. They describe the game as a whole.
For readers deciding how to think about a game, it helps to ask two questions: what the average result is per unit staked, and how large the swings might be relative to the money set aside. A game with a negative return loses on average, however pleasing the swings. If play has stopped being enjoyable or has started to cause strain, a local problem gambling support service can help.