Profit Chop

Bob's stable had a good month. The group chat lit up with proposed splits — equal shares, weighted by buy-in, weighted by hours played, weighted by total winnings, weighted by "who brought the most energy." Bob asked Tau which one was correct.

Bob: So five people in a pool. One of them booked a big score. One of them ran cold. Three of them grinded steady. How does the profit get split.

Uncle Tau: There's one right answer. It was proven in 1953.

Bob: Of course it was.

Uncle Tau: Lloyd Shapley. "A Value for n-Person Games." Nobel Prize in economics, shared posthumously. Shapley showed that there is exactly one way to divide the surplus of a cooperative game that satisfies four axioms no reasonable person can reject. Exactly one. The answer is called the Shapley value.

Bob: Four axioms.

Uncle Tau: I'll give them to you in a second. First I want you to understand what problem they're solving.

You have a pool of players. Each player contributes something to the pool's overall success. Some players contribute more, some less. At the end of a period, there's a surplus to distribute. "How do we split it" is not a matter of opinion — it's a math problem. Specifically, it's the fair-allocation problem in cooperative game theory. Shapley solved it. We just apply it.

The four axioms

Uncle Tau: These aren't ideology. They're so basic that if someone rejects any of them, you have to ask why they're talking about fairness at all.

Efficiency. The total of all the shares has to equal the total surplus. You can't leave money on the table. You can't fabricate money to give out. The shares sum to the pie.

Symmetry. If two players have made identical contributions to every possible coalition of the group — same edge, same variance, same everything, from the pool's perspective — they get the same share. The split can depend on what you did. It can't depend on who you are.

Null player. If a player contributed zero to any coalition — their edge and variance combine to a growth contribution of zero — they get zero. No subsidy. No charity. If you added nothing, you get nothing.

Additivity. If you run two independent games this month — maybe a week of MTTs and a week of mix games — the fair share in the combined case equals the sum of the fair shares in each game separately. One player's share in Monday doesn't depend on what's happening Wednesday.

Bob: These all sound obvious.

Uncle Tau: They are obvious. Which is why Shapley's theorem is so powerful. He proved there's only one allocation rule that satisfies all four. Everything else breaks at least one of them.

Why the popular rules break

Bob: OK. Equal split. What breaks.

Uncle Tau: Null player. If one of the five people in the pool had zero edge — say a genuinely break-even player that a friend vouched in — equal split gives them a chunk of other people's growth contribution. That violates null player. The null player in this pool is a net drain on the other four, and equal split doesn't care.

Bob: Weight by buy-in?

Uncle Tau: Breaks null player too. Two players can buy in for the same amount but have wildly different edges and variances. The pool's growth rate doesn't care what you paid at the cashier. It cares what you bring. If the split cares about dollars in and not about variance-adjusted edge, you're paying rent to buy-in size, not edge.

Bob: Weight by winnings.

Uncle Tau: Breaks additivity and symmetry at the same time. A player who booked one massive score is not necessarily a bigger contributor to the pool's growth than a player who had a flat month at the same edge — variance matters. Two players can have identical expected contributions and wildly different realized winnings in a month; raw winnings is a noisy sample of the signal Shapley is reading. Paying them proportionally to the sample error is paying them for variance, not skill.

Bob: Weight by markup.

Uncle Tau: Closer. Markup is trying to measure edge. But markup is a market price, not the player's actual Shapley contribution, and markups are noisy and sticky and shaped by negotiation. Plus it ignores variance, which Shapley doesn't. If one of your players has 15% ROI at 8 units of variance and another has 15% ROI at 25 units of variance, their contributions to the pool's growth are wildly different — the lower-variance player contributes way more. Markup will treat them as roughly equal. Shapley won't.

Bob: So every heuristic breaks at least one axiom.

Uncle Tau: Every single one. And once you see that — once you see that equal, buy-in-weighted, winnings-weighted, markup-weighted are all almost correct but violate one of the axioms most people would sign off on — the only question is: what's the rule that doesn't break any of them?

Bob: Shapley.

Uncle Tau: Only Shapley.

What the Shapley value actually is

Uncle Tau: The general Shapley formula looks scary. It involves summing over all orderings of the players and computing marginal contributions. For a group of ten, that's 3.6 million orderings. Intractable.

But here's the trick. In a poker pool of independent players — each running their own tournaments, uncorrelated outcomes — the growth rate contribution decomposes into a sum. Player i's contribution to the pool's growth is a function only of player i's own edge and variance. It does not depend on who else is in the pool.

That collapses the Shapley formula. All the orderings give the same marginal contribution. You get a single line.

For a player with edge $\mu_i$ and variance $\sigma_i^2$ — both of which come out of their posterior — the Shapley value for that player is:

$$\phi_i = \frac{\mu_i^2}{2\sigma_i^2}$$

Bob: That's it.

Uncle Tau: That's it. Every player gets a share of the pool's profits equal to their own edge squared over twice their variance, normalized by the pool's total. It is literally the Kelly growth rate contribution Kelly derived in 1956, which Shapley's axioms then single out as the unique fair split.

Bob: Hold on. Edge squared over variance? Where does that come from?

Uncle Tau: That's the Kelly-optimal log growth rate contribution. At Kelly-optimal sizing, each player's contribution to the log growth rate of the pool is $\mu^2 / 2\sigma^2$. That's not Shapley imposing it — that's Kelly. Shapley's job is to say this is the right thing to divide, not "pool's raw dollar profits" or "each player's raw winnings" or "buy-ins."

What Shapley singles out is that this is the quantity whose division satisfies all four axioms. Dividing dollars doesn't. Dividing winnings doesn't. Dividing growth-rate contributions does. Same Nobel-class economics that says a covered call has a price also says that a poker pool has a unique fair allocation.

Bob: And Kelly. And Shannon. All those guys.

Uncle Tau: All 1950s. All solved before I was born. All that's left for us to do is compute.

The player's bankroll and the stable's two services

Bob: Whose money actually plays the tournament.

Uncle Tau: Yours, with the stable on top. Start with this: your bankroll is real. It's yours. Kelly-sized from your own posterior. It's what you'd play from if no stable existed. Joining a stable doesn't replace it. The stable sits on top and offers two things.

Bob: Two things.

Uncle Tau: Two things, neither of which dissolves your bankroll into a pot. One is makeup leverage. One is a risk facility. They do different work and you should see them as different instruments.

Service one: Kelly-sized makeup leverage

Uncle Tau: Classic staking deal, same shape people have been running for decades. Stable fronts the buy-in, player clears the makeup line before profit share kicks in, profits split down the middle once you're clear. Nothing new structurally.

What's new is the sizing. The leverage isn't negotiated. It's Kelly-sized from your posterior on top of your own bankroll. Whatever additional exposure the math says you can safely carry at Kelly-optimal geometric growth — that's the makeup line the stable offers you. Computed, not haggled. Same number the app would hand any player with your shape.

Bob: So this is just "a proper version of the old makeup deal."

Uncle Tau: Exactly. Old deal, math underneath. The reason old staking collapsed into bad outcomes was that makeup amounts got set by vibes — pick a number, maybe double my roll, maybe not. This version refuses that. The makeup line is the math.

Service two: a risk facility for exposure your Kelly can't hold

Uncle Tau: Second service. When a tournament is too big for your own bankroll plus the makeup line to absorb at Kelly — the variance would wreck your growth rate even with the leverage — the stable will take the overage off your hands. You sell the excess exposure at 50% EV. Half the edge, half the variance. You don't clear anything; you sold it.

This is the "allocation pool" or the "risk pool" or whatever the stable calls it internally. It's a facility for risk your bankroll-plus-makeup can't safely carry.

Bob: And that 50% EV sale is the same as …

Uncle Tau: Three names, same cash flow, different accounting vocabulary.

One. Sell to the pool at 50% EV — half-markup. If your fair markup is 1.20, you sell at 1.10.

Two. 100% makeup with a 50/50 profit split. Stable puts up the whole buy-in, no markup negotiation, profit splits down the middle.

Three. The positive-case version: borrow half the buy-in from the stable and sell the other half at face value, zero markup. When you cash, repay the loan and hand over the sold share's proceeds.

All three give the stable 50% of your edge in exchange for 50% of the variance on the bigger buy-in. Which word the stable uses — sale, makeup, loan — is cosmetic.

Why this is good for everybody

Bob: The stable is taking 50% of my edge on that excess exposure. How is that positive-sum.

Uncle Tau: Because the stable isn't extracting. The stable is generating wages from a mathematical property, and every player gets a share of those wages.

Here's the mechanism. Your tournament outcomes and Sam's tournament outcomes are uncorrelated — you busting on Sunday doesn't affect whether Sam cashes on Tuesday. When uncorrelated variances aggregate, the total grows much slower than the sum of the parts. That's Markowitz, 1952. A stable's total variance is far less than the sum of each player's individual variance.

Lower aggregated variance means the stable's Kelly-optimal bankroll is smaller than the sum of each player's Kelly-optimal bankroll would have been, playing solo. The gap between "what everyone would need solo" and "what the stable needs pooled" is the diversification surplus. It's not extracted from anyone. It's a mathematical consequence of uncorrelated risks aggregating.

That surplus is what the stable collects through the risk facility. The Shapley chop at the end of the period divides it back out — each player gets a share proportional to their $\mu^2/(2\sigma^2)$ contribution to the pool's growth rate. Every player ends up with higher geometric growth than they'd have had playing solo, because the pool's aggregated geometric drag is lower than any individual's.

Bob: So it's positive-sum.

Uncle Tau: Positive-sum. You get leverage and variance insurance. The stable operator earns a slice for assembling the structure. The player next to you gets the same deal you get, sized to their own posterior. Everybody's geometric growth beats their solo counterfactual, because uncorrelated pooling beats solo exposure in log-space. Markowitz 1952 for the aggregation math, Kelly 1956 for the growth rate, Shapley 1953 for the fair division. Nothing new. We just plug the pieces together.

Bob: And if the stable were badly run.

Uncle Tau: Then you lose the diversification surplus to friction — bad leverage sizing, dishonest chop, correlated player pools that don't actually diversify. Which is why the axioms matter. A stable that violates Shapley, or sets the makeup-leverage line at the wrong number, or pools correlated players under one roof — that stable does extract from you. A stable that does the math right doesn't have to.

Deleveraging over time

Bob: What happens as my own bankroll grows.

Uncle Tau: You deleverage. That's the other half of why the structure works. Early on, when your personal bankroll is small, you lean on the pool heavily — you sell a big fraction of each buy-in, the pool absorbs most of your variance. Over a month, a quarter, a year, your chop shares and your own pieces of retained edge grow your personal bankroll. Kelly goes up. The tournaments you can afford solo climb up with it. So you sell less into the pool. Same tournaments, smaller pool share.

Bob: And eventually I could take the whole thing solo.

Uncle Tau: Eventually you could, but you probably won't, and that's the point. Even when your personal bankroll is large enough to play the buy-in without leverage, keeping some piece of every tournament in the pool is cheap variance insurance. You pay a small share of edge for a disproportionate reduction in the geometric drag from variance. That trade stays positive across almost the whole curve, because the log punishes variance more than it rewards small additions of point EV. You end up with your own book, a trimmed-down pool share, and still-low-friction variance sharing with the rest of the stable.

Bob: So the stable isn't forever 50/50.

Uncle Tau: Nothing in this is forever. 50/50 is where you start when your Kelly is small relative to the room you want to play in. As your bankroll grows, the optimal pool share shrinks. The math does the adjusting — you don't renegotiate a contract, you just sell a smaller slice next time. Same structure, different dial.

Bob: The Accountant would want a fixed 50/50 split for life.

Uncle Tau: The Accountant treats relationships like payroll. He'd also charge you rent for the desk you're not sitting at.

What a Shapley-chopping stable actually looks like

Bob: OK so walk me through the month. Five-player pool. How does it go.

Uncle Tau: Every player in the pool has a posterior. The posterior tells you $\mu$ and $\sigma^2$ — expected edge and variance per tournament. The app holds those posteriors live, updated every time the player books a result.

At the end of the chop period — month, quarter, whatever the pool agreed — you pull each player's posterior, compute $\mu^2/(2\sigma^2)$, and that's their raw share. Then you normalize: divide each player's raw share by the sum of raw shares across the pool. Now you have percentages that sum to 100. Multiply those percentages by the pool's total profit. Done.

Bob: And a player who ran cold but has good fundamentals?

Uncle Tau: Their posterior reflects that. If they're a +15% ROI player who just had a rough month, their posterior hasn't collapsed — it's barely moved. Their Shapley share is roughly what it was before the month started. The bad month cost the pool dollars, but it didn't change that player's allocation share very much, because Shapley reads the posterior, not the sample.

Bob: And a player who booked a big score?

Uncle Tau: Same logic in reverse. If they had a good month but their posterior was already pretty confident before — tight curve, long sample — then the one big score barely moves their posterior, barely moves their Shapley share. The stable collected the profit. The allocation of the profit is done by how much each player was contributing in expectation that month, not by who happened to hit.

Bob: So the player who spiked the big score doesn't just walk away with all the money.

Uncle Tau: The pool is a pool. The spike happened in the pool. The spike belongs to the pool. Who contributed to the pool's growth that month — and therefore who deserves what share of the spike — is a posterior question, not a result question. Otherwise, every pool is a lottery where the prize goes to whoever got lucky. That's not a stable. That's a raffle.

Bob: So Shapley is also a defense against variance.

Uncle Tau: It's a defense against confusing variance with skill at payout time. Anyone in the pool who got lucky doesn't get overpaid. Anyone who ran bad doesn't get underpaid. Both get paid for their contribution, which is their posterior, which is what they actually bring. Everybody is insured against a rough month by the pool. Everybody accepts that a spike isn't personal winnings — it's pool income.

Why this is different from everything else

Bob: So Shapley-chopping stables are different from markup-pooling or equal-share stables how?

Uncle Tau: In markup-pooling, each player sells their action at some markup, profits get collected, and the split is whatever was agreed when the markups were set. That's a contract, not an allocation rule. If your markup was set wrong — too high, too low, stale — your share of the pool profit is wrong forever. The math under the chop doesn't correct for that.

In equal-split, you ignore contribution entirely. That's a friend group, not a stable. Great at Thanksgiving, bad at month-end.

In Shapley, the math reads each player's posterior and splits by actual variance-adjusted growth contribution. It's self-correcting — as posteriors tighten, shares refine. It's axiomatic — you can't reject Shapley without rejecting one of the four axioms and having to explain which.

And it's the one that holds up when somebody new joins or somebody leaves or somebody takes a month off. You don't re-negotiate. You just recompute. The formula doesn't care.

Bob: The Accountant would hate this.

Uncle Tau: The Accountant can't add up the pool's growth rate correctly and he's going to lecture Shapley about fairness. Yeah.

Bob: Thanks, Uncle Tau.

Uncle Tau: Go estimate your shapes, kid. And when your pool proposes a split rule, point at the four axioms and ask which one it violates. The silence after that question is the shape of the answer.

What's next

Bond — makeup and chop work when honest. What catches dishonesty.
Priors and Posteriors — the upstream object Shapley reads.