Selling sheds variance at a price. Swapping sheds it without one. Same player as video 03: a $10,750 roll in the $215 WCOOP, 50 buy-ins, +15 % ROI. Alone, the seat's EV is +$32 a tournament, but to that roll it is worth −$25: the sure amount per tournament that would grow the roll as fast (its certainty equivalent).
A swap: you and another winner in the same tournament each hand the other a share of your action. Nobody pays anybody up front. In a field of 1,028, your payouts are almost independent (correlation −0.001: you cannot both win). So your EV stays at +$32.25 and your variance falls: swap half, and it halves. One partner takes the seat from −$25 to −$6. A pool, where everyone holds an equal share of every seat: 3 winners +$3, 5 winners +$12, 10 winners +$21, 20 winners +$26, climbing toward your EV.
Sell or swap? Put both on one scale. The best sale is worth +$2.51 at no markup, +$21.74 at 1.10, and your full EV at your true value, 1.15. Each pool is worth a markup of its own: 1.003 for 3 winners, 1.05 for 5, 1.10 for 10, 1.12 for 20. If the market pays more than your pool's price, sell. If it pays less, swap. Winners nobody pays a markup grow faster by swapping with each other than by selling below their value. And doing both is never worse: at 1.10 with a pool of 10, selling only +$21.74, pooling only +$21.08, both +$23.91.
Why a bigger menu can't hurt: every new option can be set to zero, so the best plan with it is at least as good as the best plan without (Boyd & Vandenberghe). And why you can plan one session at a time: with log growth, the career is a sum of sessions, so you maximise each one on its own (Mossin 1968, Hakansson 1971).
The payouts are real: PokerStars WCOOP 114-L, $215 NLHE, 1,028 entries, 175 paid. The skill is the assumption: +15 % ROI after rake. Every number is an exact calculation, not a simulation.
In the video
0:00 Selling has a price, swapping doesn't
0:42 A swap in the same tournament
1:24 Same EV, less variance
2:25 What a swap is worth to your roll
3:26 Sell or swap: the pool's price
4:35 The bigger menu
5:40 One session at a time
6:49 The only unsolved part
Sources
Harry Markowitz (1952), "Portfolio Selection", Journal of Finance 7(1), 77–91. https://doi.org/10.1111/j.1540-6261.1952.tb01525.x
Nils H. Hakansson (1971), "Capital Growth and the Mean-Variance Approach to Portfolio Selection", Journal of Financial and Quantitative Analysis 6(1), 517–557. https://doi.org/10.2307/2330126
Thomas M. Cover (1991), "Universal Portfolios", Mathematical Finance 1(1), 1–29. https://doi.org/10.1111/j.1467-9965.1991.tb00002.x
Stephen Boyd & Lieven Vandenberghe (2004), Convex Optimization, Cambridge University Press. https://web.stanford.edu/~boyd/cvxbook/
Richard Bellman (1957), Dynamic Programming, Princeton University Press.
Jan Mossin (1968), "Optimal Multiperiod Portfolio Policies", Journal of Business 41(2), 215–229. https://doi.org/10.1086/295078
Nils H. Hakansson (1971), "On Optimal Myopic Portfolio Policies, With and Without Serial Correlation of Yields", Journal of Business 44(3), 324–334. https://doi.org/10.1086/295383
Payout table: PokerStars WCOOP 2026 Event 114-L, $215 NLHE, 29 September 2026.
A note for readers of the paper: Part 8 says selling and swapping make you "almost always" strictly better off. Precisely: never worse, and sometimes exactly equal. At the Kelly bankroll, selling at par adds nothing. Part 9 cites Hakansson's 1971 Journal of Finance paper for the one-session-at-a-time result; the result is in his 1971 Journal of Business paper, listed above. Comparing a swap with a sale, the pool's price, is not in the paper; it is new in this video.
The series
Previous: Selling, and why selling below value can be right
Next, on 2026-11-10: You don't know your ROI. Size your bankroll anyway
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