Variance: why winning players lose for months

The player enters 500-player tournaments that pay the top 75 places, with 10% of each buy-in going to the site as rake. Their ROI is 20%: on average they win back 1.2 buy-ins for every one they spend, a big edge. At about 20 tournaments a week, 1,000 tournaments is roughly a year of play. The payout curve is illustrative, not a measured player pool. The model gives a winning player a higher chance of each paid finish, keeps that edge fixed, and draws each result independently. The percentages below describe that model, not every player with the same ROI.

Expected profit

+200 buy-ins

after 1,000 tournaments

Careers still behind

14%

138 of 1,000

Middle 90% finish

−93 to +512

buy-ins

On average the player makes +200 buy-ins, but almost nobody makes exactly that. One career in 7 finishes the year behind, with the same skill as the ones that finish far ahead.

A downswing here starts at a new high in the running total and lasts until that high is beaten. For this winning player:

• The median career spends a stretch of 442 tournaments in a row below its best result so far.
• One career in ten has a stretch of 821 or more, most of the year without a new high.
• The median deepest fall from a high is 109 buy-ins, and one career in ten falls 180 or more.

A losing player on a heater can feel skilled, while a winning player in a downswing may start changing a game that was working. Neither should trust a few hundred tournaments.

Most tournaments end with nothing, most cashes are small, and the profit comes from the rare deep run. In this simulation the player cashes 20% of the time. Each tournament has an average result of +0.20 buy-ins and a standard deviation of 5.9 buy-ins: the noise in a single tournament is about 30 times the edge.

Over many tournaments the edge adds up in a straight line, while the spread grows only with the square root of the number played:

That is why a year can go either way. The edge only pulls clear of the noise after several years, or several thousand tournaments.

Cash games swing less, but the same sum applies. Take a player who wins 5 big blinds per 100 hands with a standard deviation of 80 big blinds per 100, an illustrative assumption for this calculation. After 20,000 hands, the 95% confidence interval for their true win rate is:

Twenty thousand hands cannot tell this winner from a losing player. If the observed win rate remains 5, the lower end of this approximate interval clears zero at about 98,000 hands. That is not a guarantee that a winning player will establish an edge by then; the observed result will vary.

Graded decisions offer another measure. When action values are available, each decision's EV loss is measured against the best play whatever the cards did next, so it does not depend on the eventual river result. The hands dealt still affect which decisions you face, and a grade is only as reliable as its ranges, assumptions and source. The trainer labels solver answers, engine estimates and rules of thumb separately.

A bankroll is the money set aside for poker, counted in buy-ins. It reduces the risk of running out of funds during a downswing, without guaranteeing survival. There are two ways to size it.

The simulation answers it directly: the smallest roll that 95% of the 1,000 careers never ran through over the year is 157 buy-ins.

A diffusion approximation estimates the risk of ruin over an unlimited horizon, using only the mean and standard deviation:

The longer horizon contributes to the larger estimate, but these are also different models: the formula smooths discrete tournament payouts into a continuous process. Treat its result as an approximation, especially for top-heavy payouts. Both figures assume a fixed 20% ROI and the same payout model. A smaller edge needs a much bigger roll, because the average result is in the denominator.

• Judge decisions, not results. Over a few hundred tournaments the result is mostly the cards; the decisions are yours.
• Size the roll to the edge you can show, not the edge you hope for, and move down before a downswing forces you to.
• Expect the long flat stretches. Knowing that a median career goes 442 tournaments without a new high makes one easier to sit through.