Every result in this series so far says "given this distribution, do this": maximise growth, size by Kelly, sell, swap, one session at a time. None of them tells you what the distribution is. Estimating your own shape is the only part of bankroll management that is not solved.
You don't hold one shape, you hold a belief over shapes: a distribution over distributions. Each possible ROI is a whole shape, a chance for every place from first to busting. For simplicity, the video takes every ROI from 5 % to 25 % as equally likely: 21 shapes, one belief. On the series' tournament, Kelly asks for 151 buy-ins if you are really a 25 % player, 287 at 15 %, 978 at 5 %.
Bayes says: weight every shape by your belief, average the growth, take the best bet on that average (Savage, DeGroot, Berger). Computed exactly, the answer is 287 buy-ins, the 15 % player's bankroll. Your chance of each finish is the average over the shapes you might be, so the bet follows the middle of your belief, not its spread. Getting the middle wrong is lopsided: size for 25 % and be a 5 % player, and a typical 1,000 tournaments leave 0.61× your roll; size for 5 % and be a 25 % player, and you still grow 1.25×.
Bayes plus Bellman: carry the belief from session to session. Each session, bet Kelly for the whole belief, play, update. Your finishes don't depend on your stake, so learning never changes today's bet. Take a player who is really 20 % and cashes at exactly that rate. The bet goes 287 → 234 → 202 → 202 buy-ins after 0, 1,000, 10,000 and 100,000 tournaments (202 is Kelly for 20 %). Against knowing your shape, the spread costs 16 %, 8 %, 1.5 %, 0.2 % of your growth. Less spread, sharper bet: not bigger, not smaller, yours. And it took 10,000 tournaments to get within a few points.
So there are two levers: estimate your shape better, and make it better by studying. And a critic of someone else's selling has none of the inputs: their bankroll, risk target, shapes in both games, the market's price for their action, their swap partners. "Just play smaller" is a claim about two shapes. With one $5,500 roll: if Player X is 15 % in both games, selling 55 % of a $55 at 1.13 typically grows the roll 2.54× in 1,000 tournaments against 1.24× for $25s at 100 %; if X is 5 % in the $55 and 25 % in the $25s, it flips, 1.69× against 1.88×.
The payouts are real: PokerStars WCOOP 114-L, $215 NLHE, 1,028 entries, 175 paid. The skill model is the assumption. Every number is an exact calculation, not a simulation.
In the video
0:00 Everything settled starts with "given the shape"
1:00 A distribution over distributions
1:51 Twenty-one possible yous
2:30 Bayes: bet on the whole belief
3:28 Wrong one way costs more
4:21 Bayes plus Bellman: learn as you play
5:06 The more data, the sharper the bet
6:15 Two levers
7:01 The critic has no inputs
7:50 "Play smaller" is a shape claim
8:54 Playlist 1, closed
Sources
Leonard J. Savage (1954), The Foundations of Statistics, Wiley.
Morris H. DeGroot (1970), Optimal Statistical Decisions, McGraw-Hill. Wiley Classics reprint (2004): https://doi.org/10.1002/0471729000
James O. Berger (1985), Statistical Decision Theory and Bayesian Analysis, 2nd ed., Springer. https://doi.org/10.1007/978-1-4757-4286-2
Richard Bellman (1957), Dynamic Programming, Princeton University Press.
John L. Kelly Jr. (1956), "A New Interpretation of Information Rate", Bell System Technical Journal 35(4), 917–926. https://doi.org/10.1002/j.1538-7305.1956.tb03809.x
Payout table: PokerStars WCOOP 2026 Event 114-L, $215 NLHE, 29 September 2026.
A note for readers of the paper: Part 10 says uncertainty about your edge widens the effective distribution and makes Kelly "usually strictly smaller". In this model it does not: the chance of each finish is linear in your skill, so averaging over your belief gives exactly the middle player's shape, and the bet follows the middle (287 buy-ins, the same as the point estimate). What the spread costs is growth, and data removes it. The Bayes-plus-Bellman loop and the sharpening example are new in this video.
The series
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