Tournament variance

Variance: why identical players get different results

Variance is the variance of results around the average. In tournaments, it is huge: one player can win several tournaments in a row and pull ahead, while another with exactly the same strategy will slowly lose buy-ins. After a thousand tournaments, their bankrolls will diverge so much that an outside observer will conclude that they are playing differently.

The simulator creates hundreds of 'identical' players: everyone has the same prizes, probabilities, and number of tournaments. Then each player plays their own series of independent random tournaments. The output is a distribution of bankrolls, which shows how wide the variance can be with zero or small advantage.

The more tournaments there are and the higher the variance of a single tournament, the wider the distribution. If the prize for first place is tens of times higher than the buy-in, most players will end up at a loss, and the entire profit will be collected by a few lucky ones — even though the expected value is the same for everyone.

How to read a histogram

On the horizontal axis are bankrolls in buy-ins, broken down into ranges, on the vertical axis is how many players fell into each range. A tall column on the left means many negative results; a long 'tail' on the right means rare large wins. It is this tail that creates the illusion that winners play better.

What quartiles show

• Q1 — the bankroll below which a quarter of the players fell;
• median — the 'typical' result, the middle of the distribution;
• Q3 — the threshold above which only a quarter of players;
• IQR is the width of the interval in which half of all the results lie.

If the average bankroll is significantly higher than the median, the distribution is skewed to the right: the average is pulled up by a few very lucky players, while the typical result is more modest.