When your bankroll is under the line for the game you want, there are three ways out: play smaller, don't play, or sell action. This video is the third. Keep half your action and you keep a quarter of the variance (the square of your share), but you lose EV only on what you sold below its value. Sell half at a markup of 1.10: you keep 25 % of the variance and 83 % of the EV.
Same tournament and player as video 02: real PokerStars WCOOP 114-L payouts ($215, 1,028 entries, 175 paid), an assumed skill of +15 % ROI after rake, and a 50-buy-in roll ($10,750). Played in full, that roll shrinks: −0.23 % per tournament, about a tenth of the roll left after 1,000.
Near zero, growth is the bet size times the edge (Thorp), so some sale always makes a +EV seat grow. At par, selling is exactly playing smaller: sell 62 % and you sit on the zero-growth line (130 buy-ins); sell 83 % and you are at Kelly (287). At 1.10, below your true value of 1.15, growth turns positive from 36 % sold and peaks at 95 % sold, +0.20 % per tournament.
Selling below value: sell half at 1.10 and your EV per tournament drops from $32.25 to $26.88. You gave away $5.38. Your roll goes from ×0.10 to about ×2.1 after 1,000 tournaments. That is insurance. To the market the seat is worth +$32; to your roll (its certainty equivalent) −$25. The $57 gap is the risk premium (Pratt 1964, Arrow 1965). How much to keep: what you keep is a bet on your edge minus the markup, sized by Kelly: 17 % at par, 5 % at 1.10, nothing at true value. Below a markup of about 0.99, no sale helps this roll.
In the video
0:00 Under the line: a winning player, a shrinking roll
0:51 What a sale does: a covered write
1:44 Variance falls quadratically, EV linearly
2:31 Some sale always works (Thorp)
3:29 Growth against the share you sell
4:19 Selling below true value
5:07 It is insurance: the risk premium
6:04 How much to keep
7:04 Next: swaps
Sources
Edward O. Thorp (1969), "Optimal Gambling Systems for Favorable Games", Review of the International Statistical Institute 37(3), 273–293. https://doi.org/10.2307/1402118
Edward O. Thorp (2006), "The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market", Handbook of Asset and Liability Management, Vol. 1, 385–428. https://doi.org/10.1016/S1872-0978(06)01009-X
John W. Pratt (1964), "Risk Aversion in the Small and in the Large", Econometrica 32(1/2), 122–136. https://doi.org/10.2307/1913738
Kenneth J. Arrow (1965), Aspects of the Theory of Risk-Bearing, Yrjö Jahnsson Lectures, Helsinki.
Peter Carr & Liuren Wu (2009), "Variance Risk Premiums", Review of Financial Studies 22(3), 1311–1341. https://doi.org/10.1093/rfs/hhn038
Payout table: PokerStars WCOOP 2026 Event 114-L, $215 NLHE, 29 September 2026.
A note for readers of the paper: Part 6's example (20 % ROI, sell 60 % at 1.1, growth −0.5 % → +0.8 %) cannot be reproduced. On any payout table, +0.8 % would need a roll of at most 17 buy-ins; on the WCOOP table, playing in full at that roll loses 1.35 % per tournament, not 0.5 %. The direction holds; the video uses its own, checked numbers.
The series
Previous: Kelly: why it sometimes says "not at this bankroll"
Next, on 2026-11-06: Swaps, the bigger menu, one session at a time
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