Cash Format Transitions
how your strategy shifts when the format changes

Why this book exists

A solver trained on 6-max, 100bb, no-ante Cash NLHE does not give you the right strategy for a 3-blind bomb-pot game, a 200bb deep game, or a 9-max table with 5% rake. Those aren't small adjustments — they're different games. This book measures exactly how much strategy shifts as each parameter moves, using signal-probe data from the same solver the app runs in production.

This chapter exists because GAME-RULES.md is a researcher's reference document: field names, CUDA source citations, trained ranges. It's not a reader's introduction. Before you absorb 100+ data points across five chapters, you need to know what the five axes actually are, what the baseline represents, and what scale of change to expect.

The baseline: Cash 1.1.0

Every delta in this book is measured against a single reference point called Cash 1.1.0. This is the QuintAce solver's default 6-max cash game configuration:

When a chapter says "BTN opens 43%," that is the BTN's VPIP at Cash 1.1.0. When a chapter says "at 2bb ante, BTN opens 96%," the +53pp change is the ante's contribution.

The five axes

Axis 1 — Ante (0 to 2.5 bb)

An ante is dead money posted by every player before cards are dealt. Unlike blinds, antes are posted equally — the pot is pre-built before anyone acts. The result: entering the pot becomes cheap relative to what you can win, so ranges widen dramatically. At 2.5bb ante, CO plays 3× its baseline range.

The boundary of this book's trained range is 2.5 bb. Above 3.5 bb, the format router switches to Splash (a different format entirely). There is no claim about strategy above 2.5 bb ante.

Axis 2 — Blind structure (2-blind vs 3-blind)

2-blind is the standard structure: SB posts 0.5bb, BB posts 1bb. 3-blind adds a straddle: the UTG player posts 2bb before cards are dealt and acts last preflop. The straddle creates two simultaneous effects — more dead money (like an ante) and a live option (the straddler can re-open action). The dominant effect is on BTN, which drops from 43% to 29% VPIP because BTN now risks being squeezed by the live straddle.

This axis is a binary toggle — the solver does not model positional straddle variants (button straddle, Mississippi straddle) separately. "3-blind" in this book always means UTG straddle, seat 2.

Axis 3 — Table size (2 to 9 players)

Table size determines how many opponents you face and how far you are from the button. The biggest effect is on early position: UTG plays 28% VPIP at 4-max and 11% at 9-max — a 17pp tightening — because there are more opponents who can have you dominated. BTN stays rangebound (43–50% across 4/6/8/9-max) because its positional advantage is structural regardless of table size.

Positions that exist at 6-max (UTG, MP, CO, BTN, SB, BB) collapse at smaller tables. At 3-max, there is no "CO" — there is only BTN, SB, and BB.

Axis 4 — Stack depth (20 to 600 bb)

Stack depth controls how many streets of betting fit in a hand, which drives the value of implied odds, the danger of calling preflop, and the frequency of aggression postflop. At 50bb, a standard c-bet commits 30% of your stack. At 200bb, the same c-bet commits 7.5% — you have much more room to maneuver postflop, but you also risk a much larger amount on big hands.

Preflop, deeper stacks push BTN VPIP slightly wider (+6pp from 100bb to 600bb) because implied odds improve. Postflop, deeper stacks cause c-bet frequency to collapse (CO on K72r: 90% at 50bb → 68% at 600bb) because the SPR is too high to build in a pot you can't comfortably commit.

Axis 5 — Rake (0% to 5%)

Rake is the house's share of each pot, taken as a percentage capped at a fixed BB amount. At 3%/3bb, a 30bb pot yields 0.9bb of rake before winnings are distributed. Rake applies only when a flop is dealt (no-flop-no-drop). The key consequence: rake penalizes marginal preflop calls (you pay to see a flop that may not be profitable enough to offset the rake) and reduces postflop aggression (smaller expected value on thin c-bets means checking up is better).

The interaction with depth matters: at very deep stacks, pots quickly exceed the cap, so effective rake rate declines as stacks get deeper.

What each axis moves, at a glance

Ante is the largest single-axis effect. Table size and blind structure have the next largest preflop impact. Stack depth dominates postflop decisions. Rake is meaningful but secondary — it dampens rather than redirects.

How this book is organized

The baseline appears in every chapter. When a figure says "BTN +8pp from baseline," baseline is always Cash 1.1.0 unless the chapter header states otherwise.

What this book does not cover

Formats that are out of scope — not because they're unimportant, but because the solver routes them as distinct training families with different mechanisms:

• Splash — ante > 3.5bb triggers a separate format in the solver. Covered separately.
• Bomb Pot — no blinds, single large ante; no voluntary preflop decision-tree.
• MTT/ICM — stack depth means something different when ICM pressure applies.
• PLO / short deck — different card removal and equity distributions.
• Straddle variants (button straddle, Mississippi) — the solver collapses all straddle seats to identical strategy output.

Sources: GAME-RULES.md (ground truth), signal-probe log (c:/tmp/probe_axis_signals.out, 2026-04-12). All VPIP/frequency figures in this chapter come from the same probe run as Appendix A of GAME-RULES.md.

Cash poker is not one game. Change the ante, add a straddle, and the equilibrium reshapes from the ground up. This chapter covers two axes that share a chapter because they interact directly: the ante (how much dead money sits in the pot before anyone acts) and the blind structure (whether a third forced bet — the straddle — exists).

Every number in this chapter is measured against the Cash 1.1.0 baseline: 6-max, 2-blind, no ante, 100bb effective stacks, 3% rake with a 3bb cap.

Measurement conditions: 6-max, 2-blind (no straddle), 100bb effective, rake=3%/3cap unless noted. Ante and blind structure vary as labeled.

1.1 Open VPIP across the ante sweep

The single largest effect in this book is the ante's pull on preflop ranges. Every position widens — and the widening is massive.

Open VPIP by position and ante level. 6-max · 2-blind · 100bb effective · rake=3%/3cap · ante sweeps columns

Source: ch01-ante.md §1 — Open VPIP by position and ante

UTG goes from 17.2% to 57.2% — more than tripling. BTN goes from 43.3% to 97.4%, playing nearly every hand dealt. The widening is monotonic at every position: more ante always means more hands played.

The effect is not uniform across positions. BTN widens +54pp over the full sweep. UTG widens +40pp. The gap between them grows as ante climbs. At zero ante, the BTN–UTG spread is about 26pp. At 2.5bb ante, the spread is 40pp.

1.2 Raise sizes grow with ante

Opening wider is only half the story. The solver also opens bigger as ante climbs.

Average raise size (bb) by position and ante level. 6-max · 2-blind · 100bb effective · ante sweeps columns

Source: ch01-ante.md §1 — Raise sizes (CO avg raise bb)

CO's average open goes from 3.3bb to 11.4bb across the sweep. BTN goes from 4.8bb to 17.4bb. These are not small adjustments — they are 3–4× the baseline sizing.

1.3 Limping is equilibrium — not a leak

This is the headline finding of the ante axis. At zero ante, the solver never limps. Above 0.5bb ante, limping appears. By 2.5bb ante, it dominates the opening strategy.

Limp% by position and ante level. 6-max · 2-blind · 100bb effective · ante sweeps columns

Source: ch01-ante.md §2 — Limp% by position and ante (2-blind)

At 1.0bb ante, BTN limps half the time. At 2.0bb ante, CO limps 77% and BTN limps 95%. This is not a model artifact. This is the equilibrium response to an inflated pot.

The per-hand composition confirms the story. At 1.0bb ante, the first hands to limp from CO are speculative implied-odds holdings: T9s, 98s, 87s, small pairs like 33 and 22, suited aces like A9s and A8s. These are hands that want to see a flop cheaply and can win big pots when they connect.

At 2.0bb ante, the limp range extends to the entire range including premiums. AA, KK, QQ, JJ, AKs, AKo — all classified as "notable limpers" from CO. When the dead money pool is large enough, even premium hands gain more from entering a multiway pot than from raising and folding everyone out.

1.4 BB defense widens with ante — and shifts to 3-betting

BB defense against a CO 2.5bb open tells the same dead-money story from the defender's perspective.

BB defense vs CO 2.5bb open by ante level. 6-max · 2-blind · 100bb effective · ante sweeps columns

Source: ch01-ante.md §4 — BB defense vs CO 2.5bb open (2-blind)

At 2.5bb ante, BB defends 91.2% — nearly every hand. Fold% drops to 8.8%. Blind steals are nearly impossible.

There is a regime shift inside the defense action. At low ante (0–0.5bb), BB defends primarily by calling — 61.5% call at 0.25bb ante. At high ante (2.0–2.5bb), BB's 3-bet% climbs to 35–39%. The call% actually drops from 68% at 0.5bb to 51.9% at 2.5bb.

The explanation: with a massive pot already built, marginal calling hands become marginal 3-betting hands. The enlarged pot makes preflop squeezes profitable across a much wider range. BB stops passively calling and starts aggressively 3-betting to capture dead money before the flop.

1.5 The 3-blind structure: straddle as positional modifier

Adding a third blind — a mandatory straddle posted by the UTG seat at 2bb — changes the game asymmetrically. BTN, which was last to act preflop in a 2-blind game, now has the straddler acting after it.

Open VPIP by position: 2-blind vs 3-blind at 0bb ante. 6-max · ante=0 · 100bb effective · blind structure sweeps columns

Source: ch01-ante.md §3 — Open VPIP: 2-blind vs 3-blind at ante=0

UTG, MP, and CO all widen slightly — each by about 2.5–3.8pp. The straddle adds dead money to the pot, which makes opening slightly more profitable from early and middle positions.

BTN moves in the opposite direction. It drops 14.4pp — from 43.3% to 28.9%. This is not a small adjustment. BTN in a straddled game is tighter than CO.

The reason is positional. In a 2-blind game, BTN is the last voluntary actor preflop. It opens with full information about who has folded ahead of it. In a 3-blind game, the straddler (seat 2, traditionally UTG) posts 2bb and acts last preflop — after BTN. BTN's open-raise now faces a squeeze threat from behind.

BTN raise sizing confirms: BTN's average raise in 3-blind is 14.5bb at 0bb ante — versus 4.8bb in 2-blind. The 3× larger sizing is the cost of buying fold equity over the straddle. A small BTN open (4bb) gives the straddle excellent odds to call and squeeze on later streets. A large open (14bb+) prices out everything except the straddle's premium hands.

1.6 Ante × blind structure: opposing forces at BTN

What happens when both axes move at once? The cross-tabulation reveals a clean structural pattern.

BTN VPIP: 2-blind vs 3-blind across ante levels. 6-max · 100bb effective · BTN · ante and blind structure both vary

Source: ch01-ante.md §3 — 3-blind ante × open VPIP

Ante and the straddle pull BTN in opposite directions. Ante widens it (dead money to capture). The straddle narrows it (squeeze threat). The two effects subtract rather than compound.

At 2.5bb ante, 2-blind BTN plays 97.4% — nearly every hand. 3-blind BTN at the same ante plays 78.3%. The gap never closes. Even at the highest ante level tested, the straddle suppresses BTN by about 19pp.

For UTG, MP, and CO, the picture is different: both ante and the straddle widen them. They compound. At 2.5bb ante in a 3-blind game, CO opens 66.8% and UTG opens above 44%.

The extreme at BTN: jam or fold. At 2.5bb ante + 3-blind, 19.1% of BTN opens are all-in. BTN's average raise in this configuration reaches 19.6bb. The dead money is so large that BTN either limps speculative hands or near-jams premium ones. Mid-size raises are ineffective because the straddle can call anything that is not close to all-in.

BB defense in 3-blind is already wide at zero ante — 75.5% (versus 51.8% in 2-blind). The straddle's forced 2bb adds dead money that BB must defend against. At 2.5bb ante + 3-blind, BB defense rises to 92.7%.

How adding opponents reshapes every position

UTG tightens monotonically as the table grows; BTN barely notices — and at 9-max it actually opens wider than at 6-max.

Measurement conditions: 2-blind, no ante, 100bb effective, rake 3%/3cap. Table size sweeps rows.

UTG contracts with every player added

The model sees table size as a continuous conditioning input from 2 to 10 players. We swept six sizes — HU (2), 3-max, 4-max, 6-max, 8-max, and 9-max — and measured VPIP by position at each.

VPIP by position and table size. 2-blind · ante=0 · 100bb effective · rake=3%/3cap · table size sweeps rows

Source: ch02-table-size.md §1 VPIP by position and table size

The pattern is clean: {{num:ch02.vpip.4max.utg}} at 4-max, {{num:ch02.vpip.6max.utg}} at 6-max, {{num:ch02.vpip.8max.utg}} at 8-max, {{num:ch02.vpip.9max.utg}} at 9-max. Each step adds opponents behind UTG and tightens the set of hands that can profitably survive 3-bet pressure from that many players.

BTN is range-bound — and at 9-max it is widest

Look at the BTN column in the VPIP table above. Once you are past HU and 3-max (which are their own regimes, covered below), BTN barely moves: {{num:ch02.vpip.4max.btn}} at 4-max, {{num:ch02.vpip.6max.btn}} at 6-max, {{num:ch02.vpip.8max.btn}} at 8-max, {{num:ch02.vpip.9max.btn}} at 9-max. The range is 43–50% across four different table sizes.

The reason is structural. BTN always faces the same two opponents — SB and BB — regardless of how many players sit at the table. Adding a seventh, eighth, or ninth player behind UTG does nothing to change who BTN has to get through. BTN's fold equity target is constant.

What does change is the information BTN receives before acting. At 6-max, three players fold before BTN. At 9-max, seven fold. Those extra folds carry a signal: the remaining SB and BB are drawn from a population where seven players already declined to play. Their expected ranges weaken slightly with every prior fold. BTN can exploit this by opening wider — which is exactly what the solver does.

Nine-max BTN at {{num:ch02.vpip.9max.btn}} is wider than 6-max BTN at {{num:ch02.vpip.6max.btn}}. That runs counter to the intuition that a bigger table means tighter play for everyone. It is true for UTG. It is not true for BTN.

Heads-up: a different game entirely

HU BTN/SB plays {{num:ch02.vpip.hu.btn}} of hands. That is the positional gradient taken to its extreme — BTN faces exactly one opponent, the pot odds on a single-blind steal are enormous, and there are no other players to worry about. HU strategy diverges from full-ring in so many dimensions that it warrants its own treatment; the remaining sections focus on multi-player tables (3-max through 9-max) where the table-size gradient is the finding.

Raise sizing collapses at larger tables

The sizing story is as dramatic as the VPIP story — and moves in the opposite direction from what many players expect.

BTN and CO average raise size by table size. 2-blind · ante=0 · 100bb effective · table size sweeps rows

Source: ch02-table-size.md §1 Raise sizes shrink dramatically at larger tables

At 6-max, BTN opens to 4.8bb — a large sizing by modern standards. At 9-max, BTN opens to 2.2bb — a min-raise. CO follows the same pattern: 3.3bb at 6-max, 2.2bb at 9-max. At 6-max, BTN concentrates 88% of its raise mass at the 4.8bb sizing. At 9-max, the standard is a min-raise across all positions.

BB defense narrows with table size

BB defense vs table size. 2-blind · ante=0 · 100bb effective · table size sweeps rows

Source: ch02-table-size.md §2 BB defense by table size

BB defense drops from 75.0% at HU to 50.2% at 9-max. The big drop happens between HU and 6-max; after that, defense flattens around 50–52%.

Once the table reaches six players, the opener's surviving-range quality starts to level off. The difference between a CO open that survived four folds (6-max) and one that survived six folds (9-max) is smaller than the difference between a BTN open that survived zero folds (HU) and a CO open that survived three (6-max). That is why BB defense converges at 50–52% from 6-max onward.

C-bet peaks at 8-max, then drops

This was the finding we did not predict.

CO c-bet frequency on K72r by table size. 2-blind · ante=0 · 100bb effective · CO vs BB SRP · K72r · table size sweeps rows

Source: ch02-table-size.md §3 Flop c-bet by table size and board texture — K72r

The progression from 3-max through 8-max looks like a clean trend: 69.7% → 82.5% → 83.6% → 91.7%. Bigger tables, stronger opener ranges, more c-betting on dry boards. The "stronger range ⇒ higher c-bet" story fits.

Then 9-max drops to 81.1% — below 6-max. Adding one player to an 8-max table costs 10.6 percentage points of c-bet frequency on K72r. This reversal is not fully explained by the "stronger range advantage" narrative. At 9-max, CO has survived more folds and presumably has a stronger range. Yet c-bet drops.

One possible contributing factor from the source: at 9-max, cold callers are more common, so more hands enter the flop multiway. C-betting into potential multiway pots on K72r is less automatic than into a heads-up pot. But the source explicitly notes this is an empirical finding without a complete theoretical explanation.

HU c-bet: lower than you would expect

At HU, BTN c-bets K72r only 69.1% — well below the 83.6% at 6-max. This is counterintuitive. BTN has position and the board favors high cards. Why is the c-bet lower?

The answer is range width. BTN played 84.6% of hands preflop. BB also played close to 100% of hands (defending near-universally at HU). On K72r, BB holds top pair or better at roughly the population frequency — the selective "only strong hands call" filter is absent. The opener's range advantage on K72r is real but diluted when both players entered with everything.

HU multi-board c-bet comparison. HU BTN vs BB · ante=0 · 100bb effective · note: position conventions differ at HU

Source: ch02-table-size.md §3 Multi-board HU c-bet (Batch 02d data)

A notable pattern: at HU, T98 c-bet (66.2%) is nearly as high as K72r (69.1%). The dry/wet ordering reverses compared to 6-max. When both players have near-universal ranges, a connected wet board is not structurally worse for the opener — both players have equity everywhere. The monotone board (K94ss) remains the lowest at both formats.

The 3-max anomaly: smaller table, tighter BTN

Three-max BTN opens 43.6% — tighter than 4-max BTN at 45.9%. This is the only point in the table-size sweep where a smaller table produces a tighter result at BTN.

The resolution is in BB's 3-bet frequency.

Source: ch02-table-size.md §4 3-max vs 4-max BTN anomaly — BB defense comparison

At 3-max, BB 3-bets 24.9% — eight percentage points higher than at 4-max. The reason is information. At 3-max, BTN is the first voluntary actor — there are no prior folds. BB faces a BTN whose range has not been filtered by anyone declining to play. BB recognizes the opening range is at its widest possible and responds by 3-betting aggressively to punish it.

At 4-max, UTG has already folded before BTN acts. This fold mildly signals that UTG did not hold a premium, which weakly narrows the expected strength of the field. BTN exploits this by opening wider and sizing larger (3.3bb at 4-max vs 2.2bb at 3-max). BB, facing a BTN that has some informational advantage, 3-bets less often.

The practical takeaway is that "shorter table = looser play" does not hold for BTN when the table structure removes prior-fold information entirely. At 3-max, BTN is effectively the first player to speak — and it pays the price in BB's 3-bet aggression.

The key adjustments at a glance

Summary of key metrics across table sizes. 2-blind · ante=0 · 100bb effective · rake=3%/3cap

Source: ch02-table-size.md §§1–3

How deeper stacks reshape every street

Stack depth is the quietest format lever. Ante adds dead money. A straddle rearranges the action order. Larger tables pack in more opponents. Those changes are obvious the moment you sit down. Stack depth, by contrast, looks like the same game with bigger numbers — until you realize the solver plays it completely differently at every street.

This chapter sweeps effective stacks from 20bb to 600bb and tracks what changes. The short version: shallow stacks compress every decision into shove-or-fold; deep stacks expand postflop options so dramatically that river overbets become the default action — and the threshold where overbets "unlock" is sharper than you would expect.

Measurement conditions: 6-max, 2-blind, no ante, rake 3%/3cap. Stack depth sweeps rows.

3.1 Preflop opens by depth

BTN widens steadily as stacks get deeper. UTG and MP barely move.

Preflop VPIP by position and effective stack depth. 6-max · 2-blind · ante=0 · rake=3%/3cap · stack depth sweeps rows

Source: ch03-depth.md §1 — batches_ch03_opens_by_depth

BTN widens +18.6pp from 20bb to 600bb. UTG and MP stay within a ±2pp band across the entire sweep from 100bb to 600bb. CO shows a modest +4pp widening but flattens above 200bb.

Why does BTN widen so much more than the other positions?

The 20bb entry deserves its own discussion — see §3.6 below. At 20bb, BTN contracts to 30.6% because many hands cannot profitably call a 3-bet at that stack depth. The open-raise effectively functions as a commitment vehicle, not an information-gathering action.

3.2 BB defense shifts with depth

The way BB defends changes character as stacks get deeper: at 20bb, nearly all defense is raises (shoves). At 600bb, nearly all defense is calls.

BB defense breakdown vs CO 2.5bb open, by effective stack depth. 6-max · 2-blind · ante=0 · stack depth sweeps rows

Source: ch03-depth.md §2 — batches_ch03_defense_by_depth

Three patterns stand out.

20bb is a different game. Defense collapses to 32.6%. BB folds 67.4% of the time and raises (mostly shoves) 23.9%. Flat-calling barely exists — 8.7%. The postflop SPR after a call would be so thin that calling and then folding to a c-bet wastes chips. The solver treats 20bb as shove-or-fold territory.

The 200bb dip in call%. Call% is not monotonic — it drops to 37.3% at 200bb (below the 100bb baseline of 39.2%) before recovering at 400bb (40.8%) and climbing to 43.4% at 600bb. At 200bb, the high SPR creates room for opponents to barrel repeatedly and re-raise the turn, which makes marginal flat-calls from 100bb unsuitable. At 400bb+, implied odds become dominant and wide calls return.

Raise% trends toward zero at deep stacks. At 600bb, raise% drops to 8.6%. 3-betting at 600bb forces an enormous pot with high commitment risk. The solver prefers calling to preserve postflop flexibility.

3.3 C-bet frequency drops with depth

This is the most consistent pattern in the chapter. On every board texture, CO c-bets less often as stacks get deeper. The magnitude depends on the board.

CO c-bet frequency across four board textures and six stack depths, single-raised pot. 6-max · 2-blind · ante=0 · CO vs BB SRP · stack depth sweeps rows

Source: ch03-depth.md §3 — batches_ch03_cbet_by_depth

K72r (dry K-high): drops −15.3pp from 100bb to 600bb. At 50bb it actually peaks at 89.6% — the near-commitment dynamic at shallow stacks pushes frequency upward before the standard SPR attenuation takes over.

A94r (dry A-high): the dramatic outlier at 20bb. The solver c-bets 98.4% of the time at 20bb — near-mandatory. At that stack depth, calling preflop on an ace-high board has essentially committed BB's stack. The solver recognizes this and fires almost always. From 100bb onward, frequency stabilizes in the 58–65% range.

T98 (connected wet): the most predictable curve. Monotonic from 65.0% down to 46.4%. Wet connected boards are dangerous at deep stacks because opponents hold more drawing equity, and deeper SPR lets those draws realize their equity over multiple streets.

K94ss (monotone): uniformly low (22–46%) and declining. On a monotone board, every caller has a credible flush draw. The opener's c-bet gains less fold equity at any depth, and the problem compounds as stacks get deeper.

C-bet sizing grows only marginally with depth — on K72r, average bet moves from 1.8bb at 20bb to 2.6bb at 600bb. The action is in frequency, not sizing.

3.4 3-bet pots: frequency is flat, but texture flips the depth direction

In single-raised pots, c-bet frequency drops with depth on every board (§3.3). In 3-bet pots, the story is more complicated — and on some boards, it reverses entirely.

The K72r baseline: 100% at every depth

CO opens 2.5bb, BTN 3-bets to 8bb, CO calls. Flop Kd7s2h. CO checks (OOP), BTN to act.

BTN c-bet frequency and sizing in a 3-bet pot on K72r. 6-max · 2-blind · ante=0 · CO vs BTN 3BP · K72r · stack depth sweeps rows

Source: ch03-depth.md §4 — batches_ch03_3bp_cbet_by_depth

Frequency is saturated at 100% regardless of depth. On K72r, BTN's 3-betting range has such a strong range advantage that checking is never correct at any SPR. The variation shows up only in sizing: 6.1bb at 100bb, rising to 7.5bb at 300bb, then settling at 7.0bb at 600bb.

This makes K72r the wrong board to study 3BP depth dynamics. It is an extreme case.

The texture reversal: A94r grows, T98 shrinks

The full texture sweep tells a different story.

BTN c-bet frequency in 3-bet pots across four boards and six stack depths. 6-max · 2-blind · ante=0 · CO vs BTN 3BP · board named per row · stack depth sweeps columns

Source: ch03-depth.md §4b — batches_ft_phase3_3bp_textures

The direction reverses between boards. On A94r and K94ss, the 3-bettor c-bets more as stacks get deeper — from 4.5% at 20bb all the way to 87.1% at 600bb on A94r. On T98, the solver goes from 74.8% at 20bb down to 5.1% at 600bb. That is a near-complete reversal of the single-raised-pot pattern, where all boards dropped with depth.

Pre-condition: On dry A-high and monotone boards, BTN's 3-bet range is polarized toward value (big pairs, nut draws). Deeper stacks let BTN commit with that value over multiple streets — c-bet frequency grows because the range can sustain the deep-stack play.

Post-condition: On connected wet boards (T98), the caller's range smashes the flop. At shallow stacks, geometric commitment forces BTN to fire even with a range disadvantage. At deep stacks, BTN defers to check-back — the caller's equity realization improves with SPR.

Why it splits: the caller's flop equity relative to the 3-bettor's range. On A94r, the caller rarely connects. On T98, the caller connects hard.

This finding was verified across three different position pairs (CO-BTN, UTG-CO, MP-BTN). The direction is perfectly position-invariant — the same texture-split pattern appears regardless of who opens and who 3-bets.

3.5 The river overbet unlock

This is the chapter's headline finding. At deep stacks, river average bet size does not grow gradually — it jumps.

The step function on K72r → 3d → 7c

CO opens, BB calls, flop c-bet/call, turn bet/call, river BB checks, CO to act. Runout: Kd7s2h → 3d → 7c.

Multi-street sizing and frequency across stack depths, CO vs BB on K72r → 3d → 7c. 6-max · 2-blind · ante=0 · CO vs BB · K72r→3d→7c · stack depth sweeps rows

Source: ch03-depth.md §5 — batches_ch03_river_by_depth + batch v11e (500bb river-only)

Two distinct phases.

Phase 1 (100–400bb): frequency rises, size stays near 25bb. River bet% climbs from 60.3% to 74.4% while average bet stays in the 24–30bb range (roughly pot-sized). More of the range polarizes — medium-strength hands increasingly prefer to check as SPR grows, which concentrates the betting range in strong value and bluffs.

Phase 2 (400bb onset → 600bb): the size jump. Average bet is 29.8bb at 400bb, jumps to 38.7bb at 500bb (river-only batch), and reaches 54.4bb at 600bb. The overbet "unlock" begins in the 400–500bb window — not as a gradual curve but as a step. Below 400bb, bet sizing is constrained to pot-sized territory. Above 400bb, sizing breaks free into 2× pot territory.

Turn barrel frequency is essentially flat at 55–56% from 200bb onward. The strategic variation concentrates on the river.

What drives the step: all-in scaling, not new hands betting bigger

Per-combo data clarifies the mechanism. The average bet rise at 600bb is not driven by medium-value hands switching to overbets. Premium hands (QQ, JJ, TT) shove all-in at every depth — but at 400bb they shove 387.5bb, and at 600bb they shove 587.5bb. The larger all-in size pulls the average up mechanically.

Medium-value hands (77) actually bet smaller at 600bb than at 400bb. The "overbet unlock" is an all-in-scaling phenomenon: premiums always commit; the commit size grows with the stack.

Cross-runout comparison

The K72r step function is not the universal pattern. Four runouts spanning different texture classes show different characters:

River average bet size at each depth across four runouts. 6-max · 2-blind · ante=0 · CO vs BB · stack depth sweeps rows

Source: ch03-depth.md §5b — batches_v11_river_step (K72r), batches_ft_phase2_second_runout (A94r), batches_ft_phase6_more_runouts (T98, K94ss)

The step function is K72r-specific. The other three runouts scale continuously (geometrically) with depth. But the direction is universal: bigger average bet at deeper stacks, on every tested runout.

The magnitude varies enormously. At 600bb, average bet ranges from 54.4bb (K72r dry) to 363.7bb (K94ss monotone) — a spread of nearly 7×. K94ss is the polarization extreme: the river betting range on a monotone K-high board is nearly flush-or-air, so shoves dominate the value range. At 600bb, average bet approaches stack size.

3.6 Shallow stacks: the 20bb regime

At 20bb, the game compresses into a near-commitment structure. Every preflop decision is a decision about whether to put your stack in.

BTN tightens to 30.6%. At 100bb, BTN opens 43.3%. The 12.7pp contraction at 20bb reflects the commitment reality: many hands cannot profitably call a 3-bet when calling costs a third of the remaining stack. BTN opens only hands it would be comfortable committing with.

BB defense collapses. Only 32.6% defense, with 67.4% folds and 23.9% raises (near-shoves). Call% is 8.7% — almost no flat-calling. The solver treats the preflop decision as binary: fold or shove.

A94r c-bet spikes to 98.4%. On a dry ace-high board at 20bb, calling preflop has essentially committed both players. The solver responds by c-betting near-universally. This is the shallowest SPR regime where top-pair equivalent hands should never check back.

K72r at 20bb is slightly lower than at 50bb (84.4% vs 89.6%). This may reflect the near-commitment dynamic reducing mixed bet/check strategies — at 20bb, the solver either commits or does not, producing slightly different mixing than the 50bb sweet spot where c-betting is a real strategic choice.

Rake changes everything about how BB defends — and almost nothing about how anyone opens.

That asymmetry is the headline finding. Across a sweep from 0% to 5% rake, preflop open frequencies barely move. But BB's calling range shrinks dramatically, flop c-bet strategy diverges by board texture, and the river does something nobody predicted.

Measurement conditions: 6-max, 2-blind, no ante, 100bb effective. Rake rate sweeps columns. Note: 2%/2cap rows show anomalous behavior documented in Research notes.

4.1 Preflop opens are rake-insensitive

Open VPIP — the rate at which each position voluntarily enters the pot — barely reacts to rake.

Preflop open VPIP by position and rake level. 6-max · 2-blind · ante=0 · 100bb effective · rake sweeps columns

Source: ch04-rake.md §1 Preflop opens × rake

The largest movement is CO at 5% rake — 6-max context aside, the shift is only +1.3pp. UTG moves +0.3pp. BTN doesn't move at all.

Why? Rake is charged when a flop is dealt (the no-flop-no-drop rule). When you open-raise from CO and everyone folds, no flop happens and no rake is taken. The open-raiser's EV calculation doesn't include rake — rake only enters the picture after BB decides to call and a flop is dealt.

What this means in practice: When you transition to a higher-rake game, don't change your opening ranges. The adjustment belongs to the defenders, not the openers.

4.2 BB defense drops sharply at 5% rake

While opens stay flat, BB's response changes dramatically.

BB defense breakdown vs CO 2.5bb open, by rake level. 6-max · 2-blind · ante=0 · 100bb effective · rake sweeps columns

Source: ch04-rake.md §2 BB defense × rake

At 5% rake, BB's overall defense drops from 51.8% to 43.9% — a 7.9pp decline. The components tell the story of how that defense contracts:

• Call% drops 10.2pp (39.2% → 29.0%). This is the primary driver. Marginal calling hands — the ones near the break-even threshold at 0% rake — become losing plays at 5%.
• 3-bet% rises 2.4pp (12.6% → 15.0%). Some hands that were flat-calls at 0% rake now 3-bet instead. The 3-bet resolves the hand preflop; if CO folds, no flop is dealt and no rake is paid.
• Fold% rises 7.9pp (48.2% → 56.1%). The rest of the marginal hands simply fold.

The 0% and 3% rows are identical. This is expected — the 3%/3cap configuration is the Cash 1.1.0 baseline, and the 0% query uses the same trained model at its default parameter. See §4.6 for the technical explanation.

What this means in practice: At 5% rake, tighten your BB flatting range by roughly 10pp. Hands at the margin — suited gappers, weak suited aces, dominated broadways — should shift to either fold or 3-bet. The 3-bet is a rake-avoidance play: resolve preflop, skip the rake.

4.3 C-bet frequency diverges by board texture

Rake doesn't suppress c-bets uniformly. On dry boards it pushes frequency down. On a connected wet board, it pushes frequency up.

CO c-bet frequency across four flop textures, by rake level. 6-max · 2-blind · ante=0 · 100bb effective · CO vs BB SRP · rake sweeps columns

Source: ch04-rake.md §3 Flop c-bet × rake

On the dry boards:

• K72r drops 6.5pp (83.6% → 77.1%)
• A94r drops 7.4pp (64.9% → 57.5%)
• K94ss (monotone) drops 3.8pp (32.2% → 28.4%)

These follow the intuitive logic. Thin value bets — hands where the c-bet EV is near zero — become losers at 5% rake. The opener checks back more with marginal holdings and keeps a cleaner, value-heavy c-bet range.

Then there is T98. It rises 5.6pp (59.5% → 65.1%). The direction reverses.

What this means in practice: At 5% rake on dry boards, check back more marginal c-bets — second pair, weak top pair. On connected wet boards, c-bet more frequently. Your opponent is folding at a higher rate on those boards because of rake pressure, so your thin bets become profitable, not unprofitable. This is counterintuitive: high rake makes you more aggressive on wet boards, not less.

4.4 The river does the opposite of what you would expect

Before we ran the data, the prediction was straightforward: higher rake → less river betting. Thin value becomes unprofitable, so the solver should check more on the river.

The data disagreed.

River betting on K72r → 3d → 7c, CO to act, by rake level. 6-max · 2-blind · ante=0 · 100bb effective · CO vs BB SRP · K72r → 3d → 7c · rake sweeps rows

Source: ch04-rake.md §4 River thin-value × rake

At 5% rake, river bet frequency jumps from 60.3% to 91.3% — a 31pp increase. Average bet size drops from 25.5bb to 21.1bb. The solver bets more often but smaller.

This was flagged as a wrong-direction prediction during the research. The initial hypothesis was "thin river value bets should drop with rake." The data shows the opposite. Two effects appear to operate simultaneously:

Effect 1 pulls average bet down. Effect 2 pushes bet frequency up. Net result: CO bets the river 91.3% of the time, but for a smaller amount.

This finding is directionally confirmed across three additional runouts tested in later phases — A94r → Js → 2d, T98 → Jc → 2d, and the rainbow control Ks5d2h → 8c → 7h all show frequency rising and average size falling at 5% rake. The mechanism is not runout-specific. (See Research notes for a monotone-board caveat.)

What this means in practice: At 5% rake, do not assume river thin value bets are worse. They may be better — your opponent reaches the river less often, so when you do bet, they fold at a higher rate and you need less sizing to get called by the hands they do hold. Bet more often on the river at high rake, but size down slightly.

4.5 The 2%/2cap anomaly

One rake configuration does not behave like the others.

At 2%/2cap, BB defense widens to 56.1% — above both 0% (51.8%) and 3% (51.8%). BB call% rises to 44.9% (vs 39.2% at 0%). K72r c-bet jumps to 88.7% (vs 83.6% at 0%). The 2% row is consistently wider than both bounding configurations.

This is not a measurement error. All four rake settings produce distinct cache keys (confirmed via direct investigation). The behavior is genuine, but the reason 2%/2cap specifically produces a call-heavy equilibrium — when both 0% and 3% produce identical results — remains unexplained.

2%/2cap data is excluded from all analysis in this chapter. The 0%, 3%, and 5% rows form a coherent, monotonic pattern (flat from 0% to 3%, then compressing at 5%). The 2% row breaks that pattern in a way that cannot be explained by any tested mechanism. Until the anomaly is resolved, treat 2%/2cap numbers as unreliable reference points.

4.6 Why the 0% and 3% rows are identical

The Cash 1.1.0 baseline model was trained and queried at 3%/3cap rake. When the research queried the same model at 0%/0cap, the returned values match the 3% baseline exactly — they are the same model state.

This is expected behavior, not a measurement error. The two rows represent the same trained equilibrium. The meaningful comparison is between this shared baseline and the 5%/5cap configuration, which produces distinctly different strategy output. All claims in this chapter derive from the 0%/3% baseline versus 5% delta.

Single-axis findings are clean. You vary ante and hold everything else fixed; you vary depth and hold everything else fixed. The patterns are neat, monotonic, and easy to teach.

Real poker games are not neat. A Friday night 9-max game with a 2bb ante is not the 6-max zero-ante baseline plus a single adjustment. The ante interacts with the table size. Depth interacts with rake. The straddle interacts with depth. Each compound can amplify, cancel, or even reverse the effects you would predict from single-axis analysis alone.

This chapter covers five directly tested cross-axis interactions. Each section starts with a prediction, shows what actually happened, and explains the mechanism. The chapter ends with a summary matrix and three compound-format scenarios you are likely to encounter.

Measurement conditions: compound format combinations — see each table caption for the specific axes in play. Baseline where not varied: 6-max, 2-blind, 100bb effective, rake=3%/3cap.

5.1 Ante × blind structure: the straddle brake holds even at high antes

The prediction seemed obvious: at a large enough ante, the dead-money pool should be so massive that the straddle's squeeze threat becomes irrelevant. BTN in a 3-blind game at 2.5bb ante should approach BTN in a 2-blind game at 2.5bb ante — the squeeze threat drowns in dead money.

The prediction was partially confirmed and partially wrong.

BTN VPIP by ante level and blind structure. 6-max · 100bb effective · rake=3%/3cap · ante and blind structure both vary · BTN

Source: ch05-transitions.md §X-1 (re-analysis of ch01 batch 01a)

The widest cell in the entire table is 2-blind BTN at 2.5bb ante (97.4%), not 3-blind BTN at any ante level. Ante widens BTN; the straddle narrows BTN. These two forces pull in opposite directions, and the straddle wins — even at the highest ante tested, 3-blind BTN (78.3%) is still ~19pp below its 2-blind counterpart.

The gap does not close. It actually widens slightly in absolute terms as ante grows, from 14.4pp at zero ante to ~19pp at 2.5bb ante.

The compound runs differently by position

For UTG, MP, and CO, ante and blind structure pull in the same direction — both widen. The source notes that UTG goes from 17.2% to 44.5% in a 2-blind game at 2.5bb ante, and achieves a similar 44.5% in a 3-blind game at 2.5bb ante. The positive compound for early positions is real: ante widens their ranges, and the straddle's dead money adds a further incentive to open.

BTN is the exception because the straddle sits directly behind it. The straddle's squeeze threat discounts BTN's raise EV at every ante level — that discount is structural, not pot-size-dependent.

The all-in dynamic at 2.5bb ante + 3-blind

At 2.5bb ante in a 3-blind game, 19.1% of BTN's opens are all-in. The combination of the straddle squeeze threat plus the large ante pot creates near-jam-or-fold dynamics. BTN's choices in this format are: limp in, raise to near-jam sizes, or fold. There is no comfortable mid-size isolation raise.

5.2 Depth × rake: deeper stacks are slightly more rake-sensitive

The prediction: BB's call% drop at 5% rake should be smaller at 600bb than at 100bb. The logic seemed solid — at 600bb the fixed rake cap (5bb) is a smaller fraction of larger pots, so rake should matter less.

The prediction was wrong.

BB call% at 0% and 5% rake by stack depth. 6-max · 2-blind · ante=0 · table size=6-max · depth and rake both vary · BB vs CO

Source: ch05-transitions.md §X-2 (Batch 05b)

The 600bb game shows more rake sensitivity (−11.4pp) than 100bb (−10.2pp). The cap-fraction argument is correct in isolation but misses the composition of BB's calling range at deep stacks.

5.3 Table size × depth: 9-max shows a slightly larger depth premium

The question: does BTN's depth-driven widening (100bb → 600bb) look the same at 9-max as at 6-max?

BTN VPIP by table size and depth. 2-blind · ante=0 · rake=3%/3cap · table size and depth both vary · BTN

Source: ch05-transitions.md §X-3 (Batch 05c)

The finding is confirmed but modest: 9-max BTN gains 7.5pp from depth versus 6-max BTN's 5.9pp. A 1.6pp difference.

At 9-max, BTN's range is already wider at 100bb (49.8% vs 43.3%) because seven prior folds weaken the remaining blind ranges. Going deeper at 9-max adds implied-odds hands on top of an already-loose baseline, producing a slightly larger absolute depth premium.

Raise sizes stay put. 9-max BTN opens to 2.2bb at both 100bb and 600bb. The implied-odds widening does not translate into larger sizing — the min-raise convention at larger tables holds regardless of depth.

5.4 Ante × table size: the most dramatic interaction in the book

This is where compound effects stop being modest.

The prediction: ante widens all positions, and the widening should be roughly proportional regardless of table size. Table size and ante operate on different dimensions — prior-fold count and dead money.

The prediction underestimated what happens when both dimensions are extreme simultaneously.

CO VPIP

CO VPIP by table size and ante. 2-blind · 100bb effective · rake=3%/3cap · table size and ante both vary · CO

Source: ch05-transitions.md §X-4 (Batch 05d)

At zero ante, 9-max CO is only 2.6pp wider than 6-max CO. At 2bb ante, the gap widens to 5.4pp. The ante effect is amplified at 9-max.

BTN raise sizes — the headline number

BTN average raise by table size and ante. 2-blind · 100bb effective · rake=3%/3cap · table size and ante both vary · BTN

Source: ch05-transitions.md §X-4 (Batch 05d)

6-max BTN goes from 4.8bb to 13.8bb at 2bb ante — a 2.9× increase. 9-max BTN goes from 2.2bb to 45.7bb — a 20.8× increase.

This is not a quirk. The mechanism is arithmetic.

CO VPIP at 9-max + 2bb ante reaches 85.4% — nearly identical to BTN at 6-max + 2bb ante (96%). At a 9-max table with a 2bb ante, CO benefits from 7 prior folds AND the ante pool. The positional edge compounds with dead money at a larger table.

5.5 Rake × table size: larger tables are more rake-sensitive

The prediction: rake sensitivity should be similar across table sizes. Rake operates on individual hand EV, not table-level dynamics. A 5% rake is 5% whether six players or nine players sit at the table.

The prediction was wrong.

BB call% at 0% and 5% rake by table size. 2-blind · ante=0 · 100bb effective · table size and rake both vary · BB vs CO

Source: ch05-transitions.md §X-5 (Batch 05e)

9-max BB shows 36% more rake sensitivity than 6-max BB — a −13.9pp drop versus −10.2pp. Table size does amplify rake sensitivity.

How this connects to the depth interaction

The §5.3 finding showed 9-max BTN gains 7.5pp from depth (100bb → 600bb). The §5.5 finding shows 9-max BB loses 13.9pp from 5% rake. At 9-max + 600bb + 5% rake, the rake contraction nearly cancels the depth premium. BB's call% at 9-max + 600bb + 5% rake is approximately 24.7% — close to the 9-max 100bb baseline without depth widening.

5.6 The cross-axis picture at a glance

The five interactions above, plus the compound-format implications, form a single reference table. Interaction type classifies whether the two axes reinforce each other, oppose each other, or produce a non-linear compound.

Cross-axis interaction summary — tested compound effects. All interactions measured within the parametric Cash NLHE sweep. Baseline: 6-max, 2-blind, 100bb effective, 3%/3cap rake.

Source: ch05-transitions.md §Cross-axis interaction summary

The dominant pattern: axes that involve dead money (ante, rake, table size) amplify each other. Axes that involve positional mechanics (straddle) oppose dead-money axes at BTN specifically while compounding at other positions.

5.7 Compound format scenarios

Single-axis charts underpredict what happens in real-world formats because real formats combine multiple axes simultaneously. Here are three compound scenarios you are likely to encounter, with the specific adjustments the data supports.

"Big game" — high ante + straddle

Setup: 2bb+ ante, 3-blind structure (straddle at UTG), 100bb effective.

This is common in live high-stakes cash games. The combined effects from §5.1 and the single-axis chapters:

1. UTG is dramatically wider. UTG goes from 17.2% (baseline) to 44.5% at 2.5bb ante + 3-blind. The ante overcomes the 3-blind's minor BTN-narrowing effect at early positions.
2. BTN is wide but constrained. BTN reaches 78.3% at 2.5bb ante + 3-blind — wide, but still ~19pp below the 2-blind equivalent (97.4%). The straddle consistently acts as a brake on BTN width.
3. BTN raise sizes are large. 19.6bb average at 2.5bb ante + 3-blind. Near-jams become standard BTN opens in this format (19.1% AI%).
4. BB almost never folds. 92.7% defense at 2.5bb ante + 3-blind. Blind steals are ineffective.

The adjustment: Widen all preflop opens except BTN. Expect near-jam sizing from BTN and CO. BB defends nearly everything. Postflop is where value is realized — c-bet near 100% on dry boards with the pot already bloated by antes.

Deep stack + high rake — 5% rake, 600bb

The two effects partially counteract:

• Depth widens BTN and BB's call% (implied odds)
• 5% rake narrows BB's call% (rake-sensitive marginal calls)

From §5.2, the rake sensitivity is larger at 600bb (−11.4pp) than at 100bb (−10.2pp). The implied-odds widening is outweighed by rake's damage to exactly those implied-odds hands.

The adjustment:

1. BB calling range is tighter than the 100bb/3% baseline. From the source data: 32.0% calls at 600bb/5% rake, versus 39.2% at the standard baseline.
2. BTN widens to ~49%+ VPIP at 600bb regardless of rake.
3. River play stays aggressive — 5% rake does not suppress river betting. Expect overbets on dry boards at 600bb.
4. Flop c-bet on dry boards drops from the 100bb baseline (depth effect from Ch 4) and drops further under 5% rake. Expect roughly 62% dry-board c-bet frequency at 600bb/5% rake.

Short-handed + deep — 4-max, 300bb+

• 4-max UTG plays 28% (effectively MP in a 6-max game)
• BTN approaches 55%+ at 600bb deep in a 4-max game
• High SPR means multi-street builds and river overbets

The adjustment: UTG has more range flexibility than in any other table format. BTN opens extremely wide. BB calls very wide (close to HU-level defense at 4-max, widened further by implied odds). Postflop is SPR-controlled — flop bets are small relative to stacks; plan for river overbets starting around 500bb effective.