Geometric Bet Sizing: Engineering the River All-In Across Three Streets

When you hold a polarized range and want stacks in by the river, the right bet size on each street isn't a feel — it's a solvable equation. Here's the math.

You flop the nuts in a single-raised pot, 100 in the middle, 1000 behind. You want all of it in by the river. The amateur instinct is to "bet big and hope they pay." The reg instinct is to bet two-thirds, two-thirds, then jam the rest. Both leak chips. The correct answer is a specific number on each street — and it's the same fraction of the pot every time.

That fraction is geometric bet sizing, and it is the single cleanest example in poker of value extraction being an engineering problem, not a vibe.

The problem geometric sizing solves

You have a polarized range — nutted value plus some bluffs — and a clear plan: get the effective stack in by the river. The question is how to slice the remaining stack across the streets you have left.

If you bet too small early, you arrive at the river with too much behind relative to the pot, and your final bet becomes an awkward overbet that prices your opponent out of calling. If you bet too big early, you blow them off the hand before you've extracted three streets of value. Somewhere in between is a sizing that:

• keeps the maximum number of worse hands "along for the ride" across every street, and
• arrives at the river with exactly a pot-committing bet left.

Geometric sizing is the line that does both. The insight: bet the same fraction of the pot on every street. Equal-ratio bets. Because each bet grows the pot by the same multiplier, the pot compounds smoothly, and you can choose the fraction so that after your last bet the stack is exactly gone.

The math, stated correctly

Let the pot before your first bet be \(P_0\), and let \(S\) be the effective stack you have behind. Define the stack-to-pot ratio:

\[\text{SPR} = \dfrac{S}{P_0}\]

You want to be all-in after \(n\) streets of betting (typically \(n = 3\): flop, turn, river). On each street you bet a constant fraction \(f\) of the current pot, and your opponent calls.

When you bet \(f \cdot P\) into a pot of \(P\) and get called, both players put in \(f \cdot P\), so the new pot is:

\[P_{\text{new}} = P + 2(f \cdot P) = P \cdot (1 + 2f)\]

So each called street multiplies the pot by the growth factor \(g = 1 + 2f\). After \(n\) streets:

\[P_{\text{final}} = P_0 \cdot (1 + 2f)^n\]

Now, "all-in on the final street" means the total chips that went in beyond the starting pot equal both stacks: \(P_{\text{final}} = P_0 + 2 \cdot S\). Substituting \(S = \text{SPR} \cdot P_0\):

\[P_{\text{final}} = P_0 \cdot (1 + 2\,\text{SPR})\]

Set the two expressions for \(P_{\text{final}}\) equal and the \(P_0\) cancels:

\[(1 + 2f)^n = 1 + 2\,\text{SPR}\]

Solve for the per-street pot fraction:

\[f = \dfrac{(1 + 2\,\text{SPR})^{1/n} - 1}{2}\]

That's the whole thing. Plug in your SPR and the number of streets, and \(f\) is the exact fraction of the pot to bet on every street to land all-in on the last one. No street-by-street guessing — one equation, one number, applied identically each time.

A useful reference point falls straight out of this: at SPR ≈ 13 over three streets, \(f = 1\) — you bet exactly pot, pot, pot, and you're all-in. Memorize that anchor and you can eyeball everything around it.

Worked example: SPR 10, three streets

Setup: pot \(P_0 = 100\), effective stack behind \(S = 1000\), so \(\text{SPR} = 10\). We want all-in over \(n = 3\) streets.

First, the target final pot:

\[P_{\text{final}} = 100 + 2 \cdot 1000 = 2100\]

The required total growth multiplier is \(2100 / 100 = 21\). Spread geometrically over three streets:

\[g = 21^{1/3} \approx 2.759\]

So \(1 + 2f = 2.759\), giving:

\[f = \dfrac{2.759 - 1}{2} \approx 0.879\]

You bet ≈ 88% of the pot on every street. Watch it track all the way to all-in:

| Street | Pot before | Bet (88%) | Both put in | Pot after | |--------|-----------:|----------:|------------:|----------:| | Flop | 100.00 | 87.95 | 175.89 | 275.89 | | Turn | 275.89 | 242.64 | 485.28 | 761.17 | | River | 761.17 | 669.42 | 1338.83 | 2100.00 |

Check the stack: each player put in \(87.95 + 242.64 + 669.42 = 1000.0\) beyond the starting pot. The river bet of 669.42 is exactly what's left of the 1000 behind. The pot closes at 2100, both stacks in, no chips stranded, no awkward shove that's smaller or larger than the geometric line wants. The river bet is a clean pot-committing size, not a sad min-shove or a bloated overbet. That's the entire point.

Contrast the lazy alternatives at the same SPR 10:

• Three two-thirds bets: pot grows ×(1 + 2·0.667) = ×2.33 per street, so 100 → 233 → 543 → 1267. You arrive at the river with ~835 behind into a 543 pot — a forced 1.5× pot overbet jam that's far bigger than the value-maximizing size and folds out the hands you wanted to tax.
• Three pot bets: ×3 per street, 100 → 300 → 900 → 2700 of pot, but you only had 2100 of room — you're committed before the river and the sizing was too hot to keep callers along.

Geometric is the Goldilocks line, and it's not a judgment call. It's \(f = (21^{1/3} - 1)/2\).

Why equal-ratio bets maximize value with a polarized range

The "why" matters as much as the "how," because the result only holds under specific conditions.

With a polarized range — you're either nutted or bluffing, nothing in the middle — your opponent's calling range is a set of bluff-catchers. Every street, they face the same decision: defend enough to not be exploited by your bluffs (Minimum Defense Frequency), and you charge them with both your value and your air. Because your range is polar, your opponent's continuing range stays roughly the same shape relative to the bet as the hand progresses.

Equal-ratio bets exploit this. When the bet is the same fraction of the pot every street:

• The price offered to the bluff-catcher is identical on every street, so the same class of hands keeps calling. You don't blow callers off the hand early (which would forfeit later streets) and you don't underbet (which leaves value on the table).
• The total chips extracted is maximized for the constraint "all-in by the river." Geometric growth is the unique schedule where each street's bet is the largest it can be while still leaving room for an equal-ratio bet behind it. Any front-loaded schedule strands value behind; any back-loaded schedule caps your earlier streets.

Put simply: the geometric line keeps the maximum number of worse hands paying you off across all three streets, and it converts the entire stack into the pot. With a nutted polar range, that is the definition of maximum value extraction.

This is also why solvers, when you give them a clean polar range and deep-ish stacks, gravitate toward geometric-shaped sizings across streets. It's not a coincidence — it's the equilibrium answer to "extract a fixed stack from a bluff-catcher over n streets."

When geometric is NOT the answer

Geometric sizing is a tool with a narrow, correct domain. Reaching for it everywhere is how good players talk themselves into thin spots.

Merged ranges

If your value range is merged rather than polarized — top pair good kicker, second pair, the medium-strength stuff — geometric sizing is a trap. Three big equal-ratio bets fold out everything you beat and only get called by hands that beat you. You bet-fold yourself into a corner and get bluff-caught on rivers. Merged value wants smaller, often single-street or two-street value bets, and a lot of checking. The geometric line is for the nuts, not for your bread-and-butter top pair.

Overbet lines beat it sometimes

Geometric is the optimal multi-street line for a given SPR, but it implicitly assumes you want three streets. When a turn or river card massively polarizes your range — a card that smashes your nutted combos and bricks your opponent's — a single large overbet can extract more than continuing the geometric schedule, because it leverages your nut advantage on that street rather than averaging across three. Overbets also do work geometric sizing can't: maximum equity denial against draws on wet, dynamic boards, where letting a flush draw "come along for the ride" at 88% pot is actively giving up equity. If the board is screaming for one big bet, take it.

Multiway pots

Everything above assumes heads-up. Multiway, the math breaks: ranges are wider and weaker, your polar range faces multiple bluff-catchers whose combined defense is much stickier, and "all-in by the river" against two opponents requires a totally different equity threshold to even commit. Geometric sizing is a heads-up construct. Three-way, size down, value-bet more selectively, and forget the elegant all-in-by-the-river plan.

Your range simply isn't polar enough

The condition that makes geometric work is range polarity. If you don't have enough nutted combos to credibly bet three streets — if your "value" is actually capped — geometric sizing turns into a fast way to stack off when behind. Be honest about whether the river jam is the nuts-or-bluff bet the theory assumes, or whether you've just talked yourself into a big number.

The takeaway

Value sizing across multiple streets is not a feel. For a polarized range that wants stacks in by the river, the per-street pot fraction is the solution to one equation:

\[f = \dfrac{(1 + 2\,\text{SPR})^{1/n} - 1}{2}\]

Bet that fraction every street and you land all-in on the river with the maximum extractable chips, having kept the most worse hands along for the ride. The deviations — overbets on polarizing cards, smaller merged value, multiway caution — are all measured against this baseline. You can't deviate intelligently from a number you've never computed.

shadepoker's Geometric Sizing tool does exactly this: feed it any SPR and street count and it returns the per-street pot fraction, then tracks the pot and stack street by street so you can see it close out all-in — the same table we built above, for any spot you're studying. Run a few of your own deep-stacked nut-flush and set spots through it until the SPR-13-is-pot-pot-pot anchor and the shape around it become second nature. Then the river all-in stops being a guess and starts being arithmetic you do at the table.

Once that arithmetic feels automatic, pressure-test it in the Geometric Sizing Quiz.