Rake is a flat shift, not a tax

Bob forwarded a forum thread where a guy had "corrected" his own SALSA output by running it a second time at "rake-adjusted skill." Tau's reply was four words long.

Tau was already halfway through a taco when Bob sat down. The waiter started toward the booth with the combo menu.

Uncle Tau: We'll talk about el combo another time.

The waiter nodded and receded.

Bob: So I sent you that thread.

Uncle Tau: You sent me that thread. The guy is wrong.

Bob: Which guy?

Uncle Tau: All of them, but especially the one who ran SALSA twice. Let me just get this out so we can move on.

Rake is linear.

Bob: OK, go.

Uncle Tau: When we say a player has X percent skill, we mean their skill in a rake-zero world. The pure edge over a field, before the house takes anything. Our convention:

$$\text{net ROI} = \text{skill} - \text{rake percent}$$

Skill in, rake subtracted, net comes out. One subtraction. A flat shift. That's it. If your skill is plus thirty and the rake is ten, your net is plus twenty. If your skill is plus five and the rake is ten, your net is minus five and you should find a different tournament.

Bob: So what's the double-counting version?

Uncle Tau: The double-counting version is people running SALSA once to get skill, then running SALSA again with the rake folded into the payouts to "see how the rake tax compounds against their edge."

Bob: And that's wrong because...

Uncle Tau: Because rake doesn't compound. It's not a tax on returns. It's a slice off the top of the prize pool before a single chip is dealt. The payout vector itself already accounts for it. Look at the definition the SALSA paper uses: the payouts sum to N times one-minus-r. The rake has already been subtracted before the distribution over finishing positions is even computed.

So when SALSA gives you an ROI number, that number is already computed against the post-rake prize pool. The answer you got is your net ROI. If you then subtract rake again, or re-run the simulator with rake-adjusted payouts, you're taxing yourself twice.

Bob: So the claim on forums that rake compounds against your edge...

Uncle Tau: Is people confusing two things. One is how rake interacts with a pot in a cash game — where the rake comes out of each hand and yes, there's a compounding-ish effect over many hands in that specific setting. The other is how rake interacts with a tournament, which is one event, one buy-in, one payout distribution drawn from a pool that's already had the rake removed. Totally different math. People read cash-game rake articles and apply the framing to tournaments and get themselves into knots.

Bob: So the single subtraction handles the whole thing.

Uncle Tau: The single subtraction handles the whole thing. In a tournament, you pay the buy-in, the house keeps the rake fraction, the remainder becomes the prize pool, the prize pool gets distributed according to the payout vector, and your finishing distribution — the SALSA distribution — tells you how that prize pool flows back to you in expectation. That expectation is your net ROI. Your skill number, before rake, is what we use to describe the shape of your finishing distribution against a zero-rake benchmark so we can compare players across formats. Once you run SALSA on the actual payout vector — which already has rake removed — you're looking at net. Done.

Bob: So what went wrong in the thread?

Uncle Tau: The guy had a SALSA output saying his net ROI in the €22 was plus eight percent. Then he got nervous and said "but rake is eight percent, so my real edge is zero." And he ran SALSA again with payouts scaled down by another eight percent to "confirm." Came back with minus eight. Posted: "see, rake eats my whole edge."

Bob: And the error is —

Uncle Tau: The error is that the plus eight was already post-rake. Subtracting another eight made it minus. He didn't discover that rake was killing him. He manufactured a second subtraction. He was playing a winning game and he talked himself out of it with bad bookkeeping.

Bob: Couldn't you argue skill moves with rake? Like, higher rake means weaker fields?

Uncle Tau: Fields can change with rake. Rake can affect who shows up. If a €22 tournament has nine percent rake and the €50 has five, the two fields are not the same population and your skill parameter for each might genuinely be different. That's real. That lives in the prior. That's why we have archetypes and 180 cells — different cells, different priors. Fine.

But given a specific tournament with a specific rake, your skill number is a property of you-versus-that-field in that-cell, and your net ROI is that number minus the rake fraction. One subtraction. Not a second simulator run. Not a compounding tax. A subtraction.

Bob: OK, one more. What about live games with a cap on rake?

Uncle Tau: Same math, different r. If the tournament caps rake at a lower number, the r in the formula is smaller, your net is higher, you're welcome. Still one subtraction. Still linear. Still not a tax that compounds.

Bob: And if the rake is a weird time-based thing like in some live games?

Uncle Tau: Then you convert it to a percentage of the buy-in over the expected duration and subtract that. Still linear. Still one subtraction. There is no nonlinear rake under this framework. None. You want nonlinear rake, you're in a different game — cash, not tournaments — and you should read a different wiki page.

The one-line math, for the doubters

Bob: Give me the formula so I can paste it.

Uncle Tau: The payout vector in a tournament satisfies

$$\sum_{k=1}^{K} w_k = N(1 - r)$$

which is Definition 1 in the SALSA paper. The sum of all prize pool shares is the number of entrants times the after-rake fraction. Your expected gross return from the SALSA distribution is

$$\mathbb{E}[w_k] = \sum_k w_k \cdot P^*(k)$$

and your net ROI is that minus one — minus one because the buy-in goes in first. The rake is already baked into the w's. One subtraction — from the gross return, not a second one on the net — and you're done.

There is nowhere in that chain for rake to compound with itself. It's one scalar multiplier on the pool. The pool gets distributed. You get your share. That's the whole story.

Bob: So the practical version is —

Uncle Tau: The practical version is: if the app tells you your net ROI in a game is plus X, and the rake in that game is Y, your skill number — if you want to compare yourself across rakes — is X plus Y. You don't subtract twice. You don't simulate twice. You don't compound. You add Y once when you want skill-in-a-rake-zero-world. You leave net alone when you want to decide whether to play.

Bob: This really was a short one.

Uncle Tau: It's short because it's one idea. Rake is linear. Stop turning it into a story. Stop running your simulator a second time to confirm a mistake. Stop posting on forums that rake eats your edge when the number the app gave you already ate it once. Everybody's a hypocrite and everybody's also doing middle-school arithmetic wrong.

Bob: Thanks, Uncle Tau.

Uncle Tau: Go estimate your shapes, kid. And if you run SALSA twice on the same event, I will find out, and we will be back in this booth until you stop.

What's next

• SALSA internals (paid) — what the two-parameter distribution looks like from the inside, how cash frequency and HU winrate get turned into a finishing vector, and why the rake lives in exactly one place in that pipeline.
• Drift (paid) — why your skill — not your rake — does change over time, and how Phase 2 of the model handles it.
• Reading the Strategy tab — applying linear-rake reasoning to a real event decision.

Further reading

• Felix Dahlke, Information-Theoretic Bounds on Achievable Tournament Poker Winrates, Definition 1. The explicit statement that the payout vector sums to N(1 − r).
• Bob and Uncle Tau: The ROI Ceiling Nobody Knows About on the muchomota Substack — the ceiling tables are computed with rake already baked in, exactly the way this lesson describes.