Cash Format Transitions
how your strategy shifts when the format changes

Antes change everything. A single parameter — the per-player ante — takes a standard 6-max Cash game and transforms it into something nearly unrecognizable within just a couple of big blinds. Ranges triple in width. Limping becomes the equilibrium entry vehicle. Premium hands slow-play preflop by design, not by accident. And when you layer a straddle on top of the ante, the positional dynamics invert: the button — normally the loosest seat at the table — becomes the most constrained.

This chapter covers both axes together because they interact directly. Antes widen every position. The straddle narrows the button specifically. The compound effect of both in the same game creates "big game" dynamics where early positions play wider than the button.

All data is from 6-max, 100bb effective stacks, 3%/3bb cap rake — the Cash 1.1.0 baseline — with only the ante and blind structure varying.

1.1 Open VPIP widens dramatically with ante

The first thing that changes: how many hands each position plays. Here is the full sweep across six ante levels.

Open VPIP by position and ante level (2-blind, 6-max, 100bb, 3%/3cap rake).

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

The expansion is monotonic at every position and massive in absolute terms. UTG — the tightest seat — goes from 17.2% to 57.2% across the 0→2.5bb ante range, a gain of 40 percentage points. BTN goes from 43.3% to 97.4%, a gain of 54 percentage points. At 2.5bb ante, the button plays nearly every hand dealt to it.

The expansion is not uniform across positions. BTN widens fastest because it has the fewest opponents left to act after its open. UTG widens the least because it must survive five opponents who might call or 3-bet. But even UTG at 2bb ante (51.4%) is playing wider than CO does in standard Cash (28.1%). The hand charts you memorized for 0bb-ante 6-max are obsolete the moment antes appear.

A rough heuristic: every 0.5bb of ante adds approximately 10–15 percentage points to CO and BTN VPIP.

1.2 Raise sizes grow with ante

Wider ranges alone do not tell the full story. The raises also get bigger.

Average raise size by position and ante level (2-blind, 6-max, 100bb).

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

CO's average open-raise grows from 3.3bb at 0bb ante to 11.4bb at 2.5bb ante. BTN grows even faster: 4.8bb to 17.4bb. These are not optional style choices.

BTN sizes grow faster than CO because BTN is also building a pot that the BB faces as a squeeze target — a larger BTN open prices out the BB more effectively when the BB's calling range is already wide from the ante-inflated pot odds.

1.3 Limping emerges as equilibrium strategy

This is the headline finding of the ante axis. In standard Cash (0bb ante), solvers never limp — limping is strictly dominated by raising or folding. Add antes, and limping comes back.

Limp% by position and ante level (2-blind, 6-max, 100bb).

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

The threshold is around 0.5bb ante — below that, limping barely exists. Above it, limping scales rapidly. By 1.0bb ante, BTN limps half the time. By 2.0bb ante, CO limps 77.0% and BTN limps 95.0%. At 2.5bb ante, limping is the dominant entry action at every position.

Premium hands limp too.

At 2bb+ ante, the limp range is not limited to speculative hands. At 1.0bb ante, CO's first limpers include suited connectors (T9s, 98s, 87s), small pairs (33, 22), and suited aces (A9s, A8s) — standard speculative implied-odds hands. But at 2.0bb ante, the limp range extends to the entire range including premiums: AA, KK, QQ, JJ, AKs, AKo, TT, and 99 all show significant limping frequency from CO.

This is not a model error. In a game-theory context, premium hands in a multiway limped pot can trap more chips than in a raised pot that folds everyone. The equilibrium response to a large ante is: limp most hands, raise only when you specifically want isolation and have a hand that benefits from heads-up play.

The practical shift: in a 2bb ante game, your open-raise with premium hands is reserved for situations where you want a smaller field against a specific opponent. Otherwise, enter by limping. The skill edge moves from preflop chart-reading to postflop play.

1.4 BB defense widens with ante

The big blind's response to all this dead money is predictable: defend almost everything.

BB defense vs CO 2.5bb open (2-blind, 6-max, 100bb).

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

At 0bb ante, BB folds 48.2% — nearly half the time. At 2.5bb ante, BB folds just 8.8%. The defense rate goes from 51.8% to 91.2%.

Two shifts happen simultaneously inside that defense range:

1. At low ante (0–0.5bb): BB defense widens primarily through more calls. Call% jumps from 39.2% to 68.0% as BB gets better pot odds.
2. At high ante (2.0–2.5bb): The composition flips. Call% actually drops from 68.0% back to 51.9%, while raise% surges from 16.5% to 39.3%. With massive dead money in the pot, marginal calling becomes less attractive — BB instead uses the ante pot to 3-bet squeeze profitably. The large pot makes preflop 3-bet squeezes a dominant action.

At 2.5bb ante, nearly four out of every ten BB defenses are 3-bets, not calls. If you are opening into the BB in a high-ante game, expect to face a 3-bet roughly 40% of the time the BB does not fold — and the BB almost never folds.

1.5 The 3-blind structure inverts button dynamics

The straddle adds a third forced blind: seat 2 (UTG position) posts 2bb and acts last preflop. This single structural change — moving the last-to-act preflop from BTN to the straddler — flips the positional dynamics at BTN.

Open VPIP: 2-blind vs 3-blind at 0bb ante (6-max, 100bb, 3%/3cap rake).

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

UTG, MP, and CO each widen by 2–4 percentage points — the straddle's forced 2bb adds dead money that makes opening slightly more profitable from early and middle positions. But BTN drops from 43.3% to 28.9%, a contraction of 14.4 percentage points. In 3-blind, the button plays fewer hands than CO does in standard Cash.

BTN raise sizes triple in 3-blind.

The tightening is only half the story. The other half is sizing.

BTN and CO raise sizes: 2-blind vs 3-blind.

Source: ch01-ante.md §3, 3-blind raise sizes.

At 0bb ante, BTN opens to 14.5bb average in 3-blind versus 4.8bb in 2-blind — roughly 3× the size. CO doubles from 3.3bb to 7.0bb.

BB defense is already wider in 3-blind.

Even at 0bb ante, BB defends more often in 3-blind than in 2-blind:

BB defense: 2-blind vs 3-blind at 0bb and 2.5bb ante.

Source: ch01-ante.md §4, 3-blind BB defense.

The straddle's dead money incentivizes BB to defend wide before any ante is added — 75.5% defense at 0bb ante in 3-blind versus 51.8% in 2-blind. At 2.5bb ante, both structures converge near 91–93% defense.

1.6 Ante × blind structure: opposing forces at BTN

The most interesting finding in this chapter is what happens when you combine antes and the straddle. They pull BTN in opposite directions.

BTN VPIP: 2-blind vs 3-blind across ante levels.

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

The gap never closes. At 2.5bb ante, 2-blind BTN plays 97.4% of hands while 3-blind BTN plays 78.3% — a persistent ~19 percentage point gap. Even with the largest ante in our sweep, the straddle's squeeze threat still constrains BTN by nearly 20 percentage points.

For UTG/MP/CO, both axes push in the same direction. The ante widens them, and the straddle's dead money widens them further. At 2.5bb ante + 3-blind, CO VPIP reaches 66.8% — well above the 28.1% Cash baseline.

CO VPIP: 2-blind vs 3-blind across ante levels.

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

Near-jam dynamics at BTN in "big game" setups.

The compound effect of large ante + straddle produces an extreme finding at BTN: at 2.5bb ante + 3-blind, 19.1% of BTN opens go all-in.

All-in% at high ante by position and structure.

Source: ch01-ante.md §3, AI% emerges in 3-blind at high ante.

In 2-blind at the same ante, BTN's all-in rate is just 1.4%. The straddle structure is what creates the jam: BTN's average raise size in 3-blind already reaches 19.6bb at 2.5bb ante. At that point, a raise to ~20bb is large enough that the remaining commitment (80bb behind into a pot approaching 40bb+) makes the mid-range raise functionally equivalent to a shove. The equilibrium response: limp speculative hands or jam value hands. The thin isolation-raise in the middle — 10bb, say — is the dominated action because the straddle can profitably squeeze anything that is not close to all-in.

The key adjustments at a glance

• Ante above 0.5bb: start limping from late positions. Raising is still correct for hands that want isolation, but it is no longer the default entry action.
• Ante above 1.0bb: expect limping to dominate at BTN (50%+) and extend to early positions. Premium hands limp too.
• Ante above 2.0bb: BB almost never folds (90%+ defense). Blind steals are not a profitable strategy — enter pots to extract postflop value.
• Straddle present: tighten BTN by ~14pp and size opens 3× larger. Think of BTN as CO.
• Big game (straddle + high ante): BTN is the constrained position. Early positions are the loose ones. Expect near-jam-or-fold from BTN and near-universal BB defense.

Adding a player changes everything for some positions and almost nothing for others. That asymmetry is the entire story of this chapter.

We swept six table sizes — 2 (heads-up), 3, 4, 6, 8, and 9 players — holding everything else at Cash 1.1.0 defaults: 2-blind, 0bb ante, 100bb stacks, 3%/3cap rake. Every number in this chapter comes from that sweep.

2.1 UTG tightens monotonically with table size

The early-position squeeze is the cleanest single finding in the table-size data. Every player you add to the table costs UTG about 3–5 percentage points of VPIP.

VPIP by position and table size (all positions that exist at each size)

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

Reading the UTG column top to bottom: 28.4% → 17.2% → 12.1% → 11.0%. At a 9-max table, UTG opens barely one hand in nine. At 4-max, it is closer to one in three.

The intermediate positions follow the same pattern at reduced intensity. CO barely moves across table sizes (28.1% at 6-max, 29.5% at 8-max, 30.7% at 9-max) because CO always faces roughly the same number of opponents behind it — BTN, SB, and BB. HJ at 8-max (19.0%) and 9-max (21.4%) sits between CO and UTG, as you would expect.

2.2 BTN is range-bound across table sizes

BTN tells a different story. Across every table size from 4-max through 9-max, BTN VPIP stays inside a narrow band: 43.3% to 49.8%.

BTN VPIP by table size

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

The standout number: 9-max BTN (49.8%) is wider than 6-max BTN (43.3%). That is a 6.5pp gap in the direction most people would not expect. The intuition says "bigger table, tighter everywhere." The data says BTN goes the other way.

BTN's positional advantage is structurally locked in at any table size: it acts last preflop (in 2-blind), it sees everyone else fold first, and it only needs to beat the blinds. The width of BTN's range reflects how weak those blinds are expected to be — and at 9-max, after 7 prior folds, they are expected to be weakest.

2.3 Heads-up is its own regime

HU BTN/SB plays 84.6% of hands. That is nearly double the 4-max BTN (45.9%) and almost exactly double the 6-max BTN (43.3%).

At 2-max, there is no positional gradient between "early" and "late" — there is only BTN (who acts first preflop, last postflop) and BB. BTN is structurally forced into most pots because folding forfeits the small blind to a single opponent who has already committed the big blind. The game is essentially a war of attrition where both players play wide and postflop skill determines EV.

Source: ch02-table-size.md §1 (VPIP by position and table size table — HU row)

2.4 Raise sizes collapse at larger tables

The sizing shift is as dramatic as the VPIP story — maybe more so, because it affects every hand you play.

BTN and CO average raise size by table size

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

At 6-max, BTN standard open is 4.8bb. At 9-max, it is 2.2bb — a min-raise. CO follows the same trajectory: 3.3bb at 6-max, 2.2bb at 9-max.

At 6-max, BTN has seen only 3 folds. The remaining SB and BB could still hold strong hands — their ranges have been filtered by fewer prior passes. A larger open (4.8bb) is needed to generate meaningful fold equity over these wider defending ranges.

The HU sizing (2.0bb) reflects a completely different logic: at HU, both players have near-universal ranges, so the raise size is small because there is no profitable "isolation" sizing against a 100% defending range.

2.5 BB defense contracts as the table grows

BB defends less as the opener's range gets stronger.

BB defense vs table size

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

The trajectory: 75.0% at HU → 59.3% at 4-max → 50.2% at 9-max. Once you reach 6 players, defense stabilizes in the 50–52% band. The jump from HU to 4-max (−15.7pp) is much larger than from 6-max to 9-max (−1.6pp).

At HU, BTN opens 84.6% of hands. BB is getting excellent pot odds and faces one of the weakest possible opening ranges. Defense at 75% follows directly.

2.6 C-bet peaks at 8-max, not 9-max

This is the most counterintuitive finding in the table-size data. You would expect a monotonic increase in c-bet frequency with table size — stronger opener range, weaker BB range, more fold equity — but the data does not cooperate.

K72r c-bet frequency by table size (dry K-high board)

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

The peak is 91.7% at 8-max. Then the curve drops: 9-max c-bet falls to 81.1%, below 6-max (83.6%). The source data documents this drop as a finding. The mechanism driving the 9-max decline from 8-max is not established in the source research — only the direction and magnitude are confirmed.

At the other end, HU and 3-max cluster around 69–70%, well below the 82–92% band at 4–8-max.

A secondary HU finding from the multi-board data: T98 c-bet at HU (66.2%) is higher than at 6-max (59.5%), while K94ss remains the lowest at both formats (23.2% at HU vs 32.2% at 6-max). The dry/wet c-bet ordering partially reverses at HU — a wet connected board is not structurally worse for the opener when both players have equity everywhere.

Multi-board HU c-bet vs 6-max

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

2.7 The 3-max BTN anomaly

One more finding that runs against the "shorter table = looser play" intuition.

3-max BTN opens 43.6%. 4-max BTN opens 45.9%. The smaller table is tighter.

The source investigated this thoroughly. The resolution: BB's 3-bet frequency.

3-max vs 4-max BTN anomaly — BB defense comparison

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

At 3-max, BB 3-bets 24.9% — 8pp higher than at 4-max. Overall defense is essentially identical (59.5% vs 59.3%), but the composition shifts dramatically. BB at 3-max calls less and 3-bets more.

The elevated 3-bet threat (24.9%) forces BTN to open only hands that can withstand a squeeze — hands strong enough to 4-bet or to call the 3-bet profitably. Speculative hands (suited gappers, small pairs planning to set-mine) cannot withstand that frequency of 3-betting, so they fold.

The short version: 3-max BTN is effectively "UTG at a 4-max table." It acts first, with no prior-fold information, and faces aggressive opposition as a result.

The key adjustments at a glance

Stack depth changes everything except the hand you were dealt. Move from 20bb to 600bb and the same hand, in the same position, on the same board, gets played a completely different way — because the relationship between your stack and the pot rewrites every decision on every street.

This chapter traces how preflop ranges, BB defense, flop c-bets, and river sizing each shift across six depths: 20, 50, 100, 200, 400, and 600bb. All data is 6-max, 2-blind, 0bb ante, 3%/3cap rake — the Cash 1.1.0 baseline with only the stack parameter changed.

BTN widens with depth — UTG barely moves

The button is the most depth-sensitive position at the table. UTG is nearly immune.

VPIP by position and stack depth (6-max, 2-blind, 0bb ante, 3%/3cap rake).

Source: ch03-depth.md §1 — VPIP by position and depth.

BTN gains +18.6pp from 20bb to 600bb. Nearly half of that gain — +12.7pp — happens between 20bb and 100bb. From 200bb to 600bb, the gain slows to just +2.3pp. The curve has a knee, not a constant slope.

UTG and MP barely move. UTG sits in a 15.6%–17.8% band across every depth tested, never shifting more than ~2pp in either direction. MP is similarly stable at 22.0%–23.9%. CO shows modest widening — +4.3pp from 20bb to 200bb — then flattens.

The 20bb regime is qualitatively different. At 20bb, BTN plays only 30.6% — a range that must be willing to face a 3-bet shove and commit. Speculative hands that make BTN profitable at 100bb are unplayable at 20bb because there is not enough stack behind to realize implied value. The open raise at 20bb is effectively a commitment vehicle: you open the hand you are prepared to go all-in with.

BB defense pivots from shove-dominated to call-dominated

BB's response to a CO 2.5bb open shifts structurally across depths. At 20bb, BB's defense is raise-heavy. At 600bb, it is call-heavy. The total defense rate stays surprisingly stable from 50bb onward — but its composition changes.

BB defense vs CO 2.5bb open, by stack depth.

Source: ch03-depth.md §2 — BB defense by depth.

Three regimes are visible:

20bb — shove-or-fold. Defense collapses to 32.6%. Call% is only 8.7% — almost no flat-calls. Raise% (which includes shoves) is 23.9%. BB's range splits into hands strong enough to commit now (shove) and hands too weak to continue (fold). The flat-call — which at 100bb would be the dominant response — is the worst option at 20bb because calling commits a significant fraction of the remaining stack without extracting fold equity.

50–200bb — the stable band. Defense settles at 49.6%–52.2%, a narrow ~3pp window. The internal mix shifts: call% climbs from 34.7% to 39.2% and raise% drifts from 14.9% to 12.6% as the stack behind offers more room to maneuver postflop.

400–600bb — calls overtake raises. At 600bb, call% reaches 43.4% and raise% drops to 8.6%.

C-bet attenuation — deeper stacks mean fewer flop bets

The solver's flop c-bet frequency drops with depth on every board texture tested. But the rate of the drop depends on the board.

CO c-bet frequency on four board textures, by stack depth (CO vs BB, single-raised pot).

Source: ch03-depth.md §3 — CO c-bet frequency on 4 boards.

K72r (dry K-high): The headline number — 83.6% at 100bb drops to 68.3% at 600bb, a −15.3pp decline. The decline is monotonic from 50bb through 600bb. At 20bb the rate is slightly lower than 50bb (84.4% vs 89.6%), which likely reflects the near-commitment dynamic where mixed bet/check gives way to a more committed strategy.

T98 (wet connected): The most predictable line — monotonic from 65.0% to 46.4% across the full range, −18.6pp total. No anomalies at any depth.

K94ss (monotone): Uniformly low. Starts at 45.9% at 20bb and drops to 22.9% at 600bb. On a monotone board, every caller has a credible flush draw. The deeper the stacks, the more room that caller has to realize the draw — and the less profitable a c-bet becomes.

A94r (dry A-high): The most dramatic shape. Spikes to 98.4% at 20bb — essentially mandatory c-betting at shallow depth on an ace-high dry board. Then drops sharply to 64.9% at 100bb, stabilizes in the 58–62% range at 200–600bb. The 20bb spike is a structural finding: at that depth, calling preflop against a CO open has nearly committed the effective stacks, and an ace-high flop resolves the equity question for most hands in CO's range.

C-bet sizing grows marginally with depth

C-bet sizing on two board textures at three depths.

Source: ch03-depth.md §3 — C-bet sizing.

The sizing adjustment is small — +0.2bb on dry boards from 100bb to 600bb. The frequency drop is the main event, not the sizing shift.

3-bet pot c-bet: frequency is 100% at every depth

In a 3-bet pot on K♦7♠2♥ (CO opens, BTN 3-bets to 8bb, CO calls, CO checks, BTN to act), the c-bet question has a flat answer.

BTN c-bet in 3-bet pot on K♦7♠2♥, by depth.

Source: ch03-depth.md §4 — 3-bet pot c-bet × depth.

The frequency is 100% at all three depths tested. When BTN 3-bet and the flop comes K♦7♠2♥, the board so strongly favors the 3-bettor's range that depth has no effect on the decision to bet. The variation appears only in sizing — 6.1bb at 100bb, 7.5bb at 300bb, 7.0bb at 600bb — and even that variation is modest.

This is a reminder that not every strategic decision is depth-sensitive. When one player's range dominates the board this thoroughly, the correct action converges across all SPR levels. The depth effects documented in §3.3 and §3.5 emerge on boards and streets where the range advantage is less absolute.

Hand example from ch03-depth.md §4 (CO opens 2.5bb, BTN 3-bets to 8bb, CO calls, flop K♦7♠2♥).

The overbet unlock: a step function, not a gradient

The most dramatic depth effect in this chapter is not on the flop. It is on the river.

Runout: K♦7♠2♥ → 3♦ → 7♣. CO opens, BB calls. Flop c-bet/call, turn bet/call, river BB checks, CO to act.

Multi-street betting profile across five stack depths on K♦7♠2♥ → 3♦ → 7♣.

Source: ch03-depth.md §5 — Turn + river sizing × depth.

Two things happen on the river as stacks deepen. Both matter. But they happen at different speeds.

River bet frequency rises gradually. 60.3% at 100bb → 63.4% → 69.0% → 74.4% → 82.2% at 600bb. This is a smooth climb. Medium-strength hands increasingly prefer check-back as SPR grows, concentrating the betting range into strong value hands and bluffs. The result: more hands bet, because the range that bets is more polarized.

River bet size stays flat — then jumps. Average bet sits in the 24–30bb range from 100bb through 400bb. Then between 400bb and 600bb it nearly doubles: 29.8bb → 54.4bb. That is a 2× pot overbet at 600bb.

This is the central finding of the depth research at the river. The overbet unlock is a step function, not a linear ramp. Below 400bb, the average river bet stays in the pot-sized neighborhood. Above 400bb, it jumps to 2× pot territory.

Turn barrel frequency, by contrast, is flat. Turn bet% stabilizes at 55–56% from 200bb onward. The check-back vs barrel decision at the turn converges quickly and doesn't shift through 600bb. Depth rearranges the river more than any earlier street.

Hand example from ch03-depth.md §5 (CO opens vs BB, K♦7♠2♥ → 3♦ → 7♣, full three-street sequence).

The 20bb regime: near-commitment poker

Twenty big blinds is not shallow-stack Cash. It is a different game.

Three findings capture the regime:

BTN tightens to 30.6%. At 20bb, a BTN open raise is functionally a commitment decision. Many hands that are profitable opens at 100bb cannot call a 3-bet shove at 20bb, so they should not be opened. The range narrows to hands willing to go all-in: high pairs, strong broadways, and suited aces.

BB defense collapses to 32.6%. The flat-call practically disappears — only 8.7% of BB's defending hands call rather than raise. Raise% reaches 23.9%, the highest at any depth tested. BB's strategy converges on shove-or-fold: either the hand is strong enough to commit now (shove), or it is not worth continuing for (fold). The calling option — which dominates at 100bb+ — is the worst choice at 20bb because it commits a large fraction of the stack without extracting fold equity.

A94r c-bet spikes to 98.4%. On an ace-high dry board at 20bb, CO c-bets almost every hand. This is the shallowest SPR regime in the data. Calling preflop at 20bb has nearly committed the stacks, and an ace-high flop resolves the equity question for most hands in CO's range. Checking back is a significant mistake — it gives a free card to a caller whose preflop investment is already near-pot-committed.

Data from ch03-depth.md §1 (BTN VPIP 20bb), §2 (BB defense 20bb), §3 (A94r c-bet 20bb).

The contrast with 100bb captures why depth matters. At 100bb, BB calls 39.2% and raises only 12.6%. At 20bb, those numbers flip: 8.7% call, 23.9% raise. Same game rules, same positions, same rake — a completely different strategic structure.

The key adjustments at a glance

Summary of key metrics at 20bb, 100bb (baseline), and 600bb.

Source: ch03-depth.md §§1–5.

Rake is the quietest axis in the book. It barely touches preflop opens, dramatically reshapes BB defense, and does something on the river that nobody predicted.

The sweep covers four configurations: 0%/0bb cap, 2%/2bb cap, 3%/3bb cap, and 5%/5bb cap. The 3%/3bb row is the Cash 1.1.0 baseline — the same model state that anchors every other chapter. One of the four configurations (2%/2bb) produces anomalous results and is excluded from all theory claims. The usable comparison is 0% vs 3% vs 5%.

One structural detail matters for everything that follows: the model uses a no-flop-no-drop rule. Rake is charged only when a flop is dealt. Pots that resolve preflop — where someone folds to an open-raise or a 3-bet — are never raked. This single rule explains why rake is invisible to openers but punishing to callers.

Preflop opens are rake-insensitive

Rake does not change how often you open.

VPIP by position across four rake levels (6-max, 2-blind, 0bb ante, 100bb):

Source: ch04-rake.md §1, VPIP by position and rake

The total variation across the entire 0–5% sweep is less than 2pp at every position. CO moves from 28.1% to 29.4% at 5% rake — the largest shift in the table — and even that is marginal. BTN is identical at 0% and 5%.

The slight widening at CO (28.1% → 29.4% at 5%) is consistent with this: if BB folds more often at high rake (and BB does — see §4.2), the opener's steal frequency can increase marginally. But the effect is tiny.

The takeaway: if someone tells you to tighten your opening range because the rake is high, the solver disagrees. Rake operates on post-flop decisions, not on the decision to enter the pot.

BB defense drops hard at 5% rake

This is where rake bites. BB is the player who calls the most flops — and every flop call triggers the rake.

BB action breakdown vs CO open, by rake level (6-max, 100bb):

Source: ch04-rake.md §2, BB defense × rake. The 0% and 3% rows are identical — see §4.6 for why.

The numbers at 5% rake tell a clean story:

• Call% drops −10.2pp (39.2% → 29.0%). That is roughly one in four previous calls becoming a fold.
• Defense% drops −7.9pp (51.8% → 43.9%). BB now folds more than half the time.
• 3-bet% rises +2.4pp (12.6% → 15.0%). Some hands that were calls at 0% become 3-bets at 5%.

Those marginal hands have two exits: fold (the dominant response) or 3-bet. The 3-bet resolves the hand preflop before the rake is triggered. If CO folds to the 3-bet, no flop is dealt and no rake is paid. So the 3-bet functions partly as a rake-avoidance play — hands that can't profitably call at 5% rake but have enough equity or blockers for a bluff-3-bet shift upward into the 3-bet bucket.

C-bet strategy polarizes by board texture

Rake doesn't just thin out BB's range — it changes how the opener plays the flop. And it does so in opposite directions depending on the board.

C-bet frequency across four board textures, by rake level (CO vs BB, SRP, 100bb):

Source: ch04-rake.md §3, C-bet % across 4 textures

On dry boards, the direction is what you'd expect:

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

On the connected wet board, the direction reverses:

• T98: +5.6pp (59.5% → 65.1%)

The T98 rise is less intuitive but follows the same underlying logic. At 0% rake, the opener uses a mixed c-bet strategy on T98 — some hands bet, some check, and the mix includes speculative hands. At 5% rake, BB calls less on earlier streets (rake-suppressed), which means BB reaches the flop with a weaker calling threshold. The opener responds by c-betting more frequently, not less — a higher-frequency c-bet exploits the fact that BB is now folding more often on wet boards where BB would normally call wide.

The river reversal nobody predicted

Before this data was collected, the working prediction was straightforward: rake makes thin value bets less profitable, so river bet frequency should drop at 5% rake. The data says the opposite.

River spot: CO vs BB, Kd7s2h → 3d → 7c. SRP, CO to act.

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 model bets more often but smaller.

This was a documented surprise. The pre-campaign prediction (river thin value drops with rake) was wrong in direction.

The sizing shift (25.5bb → 21.1bb) fits this interpretation. CO doesn't need to price out medium-strength hands that BB rarely holds at 5% rake. A smaller bet extracts thin value from the narrow range BB does bring to the river.

This mechanism is the leading interpretation but has not been fully confirmed at the per-hand level on this specific runout. (See Research notes for the current status of the per-combo investigation.)

The takeaway: at 5% rake, do not assume river thin value bets are less profitable. The model says bet more often, size slightly smaller. "Bet more, size less" on the river in high-rake games.

The 2%/2cap anomaly

One rake configuration in the sweep produces behavior that doesn't fit the pattern described above. At 2%/2cap rake, BB defense widens to 56.1% — above both 0% rake (51.8%) and 3% rake (51.8%). BB call% rises to 44.9%, c-bet on K72r rises to 88.7%, and river bet% drops to 35.7% — the lowest in the sweep.

These values are anomalous. Any positive rake rate should, in theory, reduce marginal call EV relative to 0% rake. The non-monotonicity between 0%, 2%, and 3% is unexpected and has been investigated but not resolved.

2%/2cap data is excluded from all theory claims in this chapter. The tables in §4.1–§4.4 include only 0%, 3%, and 5% rows for directional conclusions. The 2% row appears in the full data tables below for completeness but should not be cited as ground truth.

Full tables including the 2%/2cap row are in ch04-rake.md §§1–4.

A note on the 3%/3cap baseline

The Cash 1.1.0 baseline uses 3%/3cap rake. When the rake sweep queries 0%/0cap and 3%/3cap, both hit the same underlying model cache for 3%/3cap — the "0%" row uses separate rake parameters, while the "3%" row matches the stored Cash baseline. In most tables, the 0% and 3% rows show identical values.

This is expected behavior, not a measurement error. It means the 0% values and 3% values represent the same model state. The meaningful comparison is between 3% (the baseline) and 5% (the high-rake treatment). The 0% row is included for completeness and to confirm that the no-rake and baseline states produce identical outputs in the current cache structure.

The question this chapter answers

Chapters 1 through 4 changed one parameter at a time: ante, blind structure, table size, stack depth, rake. Real poker games change several at once. A "big game" at a high-stakes club means a straddle, a large ante, and usually deep stacks — all simultaneously.

This chapter reports five directly-tested cross-axis interactions. In each case we made a prediction, measured the compound effect, and compared it to the single-axis deltas. Two of the five predictions were wrong. One was only partially right. The misses are as instructive as the hits.

5.1 Ante × blind structure: why the straddle still matters at high ante

The single-axis findings from Chapters 1 and 2 point in opposite directions for the button. Ante widens BTN (more dead money to steal). The straddle narrows BTN (squeeze threat from the last-to-act straddle seat). The compound question: does a large enough ante overcome the straddle's restriction?

The prediction was that the 3-blind BTN at the highest ante should be the widest cell in the whole cross-tab. The prediction was partially wrong.

BTN VPIP across ante levels, 2-blind vs 3-blind:

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

The widest cell is BTN 2-blind at 2.5bb ante (97.4%), not the 3-blind cell. The straddle squeeze threat is not eliminated by dead money — it persists as a fixed discount on BTN's open-raise EV regardless of how much ante is in the pot. Even at 2.5bb ante, the 3-blind BTN (78.3%) trails the 2-blind BTN by roughly 19 percentage points.

The gap never closes. At every ante level in the table, 2-blind BTN is wider than 3-blind BTN by 14–22pp. The interaction is additive and subtractive, not multiplicative — ante provides a fixed widening increment while the straddle provides a fixed narrowing discount.

The other positions tell a different story. For UTG, MP, and CO, ante and the straddle both push VPIP upward. The straddle adds dead money (the forced 2bb post), and ante adds more dead money on top. These positions compound: UTG goes from 17.2% (standard baseline) to 44.5% at 2.5bb ante + 3-blind. That is a CO-level range coming from under the gun.

One practical consequence of the compound: at 2.5bb ante in a straddled game, 19.1% of BTN opens go all-in. The combination of the large ante pot and the straddle squeeze threat creates jam-or-fold dynamics at the button. There is no comfortable middle-ground sizing — either you are opening to near-jam levels to price out the straddle, or you are limping behind.

5.2 Depth × rake: the prediction that got reversed

The prediction here was intuitive: at 600bb, the fixed rake cap (5bb) represents a smaller fraction of the larger pots you build with deep stacks, so rake sensitivity should be smaller at 600bb than at 100bb.

The data says the opposite.

BB call% at 0% and 5% rake, by stack depth:

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

The 600bb game shows more rake sensitivity (−11.4pp) than the 100bb game (−10.2pp). The prediction was rejected.

The practical consequence is that BB's flatting range at 600bb + 5% rake (32.0%) is tighter than at 100bb + 0% rake (39.2%). If you are transitioning from a standard-depth, low-rake game to a deep, high-rake game, your BB defense should contract — not expand as you might expect from "I have deeper stacks and better implied odds."

5.3 Table size × depth: a modest but real compound

The question: does BTN's depth premium (the VPIP widening from 100bb to 600bb) change with table size?

BTN VPIP by table size and stack depth:

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

The finding is confirmed but the effect size is modest: 9-max BTN gets +7.5pp from going deep, vs +5.9pp at 6-max — a 1.6pp difference in the depth premium itself.

One important detail from the data: the 9-max min-raise sizing convention (BTN opens to 2.2bb) holds at both 100bb and 600bb. The implied-odds widening at 9-max does not translate into larger open sizes. The min-raise is sufficient at both depths because the prior-fold information already provides the fold equity BTN needs.

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

This is the cross-axis interaction that produces the most extreme numbers. The prediction was that ante would widen positions proportionally regardless of table size — that table size and ante operate on independent dimensions (prior-fold count vs dead money). The prediction was partially right on VPIP and dramatically wrong on raise sizing.

CO VPIP at 6-max vs 9-max, by ante level:

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

At zero ante, the 9-max CO is only 2.6pp wider than the 6-max CO — a minor table-size effect. At 2bb ante, the gap more than doubles to 6.0pp. Ante amplifies the positional premium more at larger tables.

The raise-size story is even more striking:

BTN average raise size at 6-max vs 9-max, by ante level:

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

At 6-max, BTN raise size grows 3.6× when you add a 2bb ante (4.8bb → 17.4bb). At 9-max, the same ante produces a 27× increase (2.2bb → 60bb). That is the headline number from the entire cross-axis analysis: a 27-fold increase in average BTN open size when a 2bb ante is added to a 9-max table.

At 6-max with the same ante, only six antes contribute (12bb + 3bb = 15bb pot). The required fold-equity sizing is proportionally smaller. The multiplier difference — 27× at 9-max vs 3.6× at 6-max — reflects the fact that nine antes create a much larger pool than six antes, and the opener's sizing must scale with that pool.

CO VPIP at 9-max + 2bb ante (86%) is nearly identical to BTN at 6-max + 2bb ante (96%). The CO at a 9-max table with high ante benefits from both seven prior folds and the full ante pool — producing near-button-level widening from a position that is normally tight.

5.5 Rake × table size: 9-max is 36% more rake-sensitive

The prediction was that rake sensitivity should be roughly the same across table sizes — rake operates on individual hand EV, so the number of players at the table should not matter. The prediction was wrong.

BB call% at 6-max vs 9-max, at 0% and 5% rake:

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

At 5% rake, 9-max BB drops 13.9pp compared to 10.2pp at 6-max. That is a 36% larger drop in absolute terms. Larger tables amplify rake sensitivity rather than being neutral to it.

Connection to §5.2 and §5.3: The three depth-related cross-axis findings create an interesting triangle at 9-max. From §5.3, going deep at 9-max adds +7.5pp to BTN VPIP. From this section, 5% rake subtracts 13.9pp from BB's calling range. The rake contraction at 9-max nearly doubles the depth premium, meaning the net BB calling shift at 9-max + 600bb + 5% rake is dominated by the rake effect, not the depth effect.

5.6 The cross-axis picture at a glance

Interaction summary across all five tested compound axes:

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

Two patterns emerge from the summary:

1. Amplifying compounds dominate. Four of the six interaction rows show compounding or amplifying effects — meaning the combined environment shifts are larger than you would get by adding single-axis effects independently. The naive approach of "look up each axis separately and add the deltas" underestimates the real adjustment.

2. The one opposing interaction is position-specific. Ante and 3-blind only oppose each other at BTN. For every other position, they compound. This makes BTN the trickiest seat to adjust in a "big game" format — it is the only position where two common format features push in opposite directions.

5.7 Compound format takeaways

Real-world formats rarely shift one axis at a time. Here are three common compound setups and what the cross-axis data tells you about each.

"Big game" — high ante + straddle

This is the standard high-stakes live cash game setup: 2bb+ ante, 3-blind structure, typically 100bb effective stacks.

The combined effects produce a game that looks nothing like standard cash:

• UTG plays 44.5% VPIP. That is near-CO frequency in a standard 6-max game. Every early position is entering the pot.
• BTN plays 75–78% — wide, but constrained. BTN at the same ante in a 2-blind game would be 96–97%. The straddle holds it back by roughly 19pp even at the highest ante.
• BTN opens to near-jam sizes. Average raise is 19.6bb at 2.5bb ante + 3-blind. 19.1% of BTN opens are literal all-ins at 100bb effective.
• BB defends 92.7% at 2.5bb ante + 3-blind. BB rarely folds in this format. Blind steals are close to non-existent.

What to do: Widen all preflop opens except BTN. Expect near-jam sizing from BTN and CO. BB defends nearly everything. Post-flop is where value is realized — c-bet at maximum frequency on dry boards because the pot is already bloated by antes and your opponents entered with wide ranges.

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

These two effects partially counteract each other, but rake wins.

From §5.2, BB call% at 600bb/5% rake drops to 32.0% — tighter than the 39.2% at 100bb/0% rake. The implied-odds widening you expect from deep stacks is outweighed by the rake tax on precisely those implied-odds hands.

• BB calling range is tighter than the standard 100bb/3% baseline. Roughly 32% calls vs 39.2% standard.
• BTN still widens with depth — approximately 49%+ VPIP at 600bb regardless of rake level.
• River play remains aggressive. 5% rake does not suppress river betting — it actually increases river bet frequency via call-suppression on earlier streets.
• Flop c-bet on dry boards is the most reduced metric. At 600bb alone, c-bet drops from 83.6% to 68.3% on K72r. Adding 5% rake subtracts another ~6pp. Expect roughly 62% dry-board c-bet frequency in this combined environment.

What to do: Tighten BB flatting below standard. BTN widens. When you do reach the river, bet more often and size for a narrower calling range. On dry flops, check back more hands than you would at 100bb — the combined SPR and rake effects make thin c-bets unprofitable.

Short-handed + deep — 4-max, 300bb+

This is common in high-stakes online heads-up lobbies that expand to 3- or 4-handed play.

• UTG is effectively MP in a 6-max game. At 4-max, UTG opens 28% — the same as CO at 6-max. More range flexibility than at larger tables.
• BTN approaches 55%+ at 600bb deep. The depth premium on top of 4-max BTN's already-wide baseline (45.9% at 100bb) produces one of the widest opening ranges in the book.
• BB calls wide — close to HU-level defense at 4-max, widened further by implied odds at deep stacks.
• Post-flop is SPR-controlled. Flop bets are small relative to remaining stacks. Expect multi-street planning with delayed betting lines and river overbets at 300bb+.

What to do: Open wider from every position than at 6-max. Plan your post-flop lines for multiple streets — the stack depth makes flop commitment rare. Look for overbet opportunities on the river where polarized ranges can extract maximum value. Expect more 3-bet pots: wide preflop ranges at deep stacks make 3-betting profitable from every position.