Pushbot / Push-Fold Trainer

Nash equilibrium ranges exist; knowing them is a different skill

A Nash equilibrium shove range is a list of hands from which pushing all-in is mathematically unexploitable given a specific stack depth, position, and opponent-calling range. These ranges exist, they're computed by solver software, and most tournament players know approximately where they are for common situations. The gap between approximately knowing and actually executing under pressure is what the pushbot trainer addresses.

The model

The trainer pulls Nash equilibrium push/fold ranges derived from the Jam-or-Fold solution tables, which compute the game-theoretically optimal strategy for all-in-or-fold play. The core calculation for whether a hand is a profitable shove is the "breakeven fold equity" analysis:

EV(shove) = FE * pot + (1-FE) * [equity * (pot + hero_stack) - (1-equity) * hero_stack]

where FE is the fraction of time villain folds to the shove, pot is the current pot (blinds plus antes), equity is hero's equity against villain's calling range, and hero_stack is the stack being risked.

The Nash solution finds the (shove range, call range) pair where neither player profits by deviating. At 10bb with the SB vs. BB, this produces a shoving range of approximately 75-80% of all hands, because the value of stealing the blinds is large relative to the risk. At 5bb the range narrows slightly in theory but widens in practice because effective fold equity collapses.

How the trainer works

You're shown a stack depth (in big blinds), a position, and two hole cards. You input "push" or "fold." The trainer compares your decision to the Nash equilibrium range for that exact spot and marks it correct or incorrect. Over a session of 50-200 drills, it tracks your accuracy by stack depth, position, and hand category. The leak report at the end shows where you deviate from Nash, expressed as: "over-folding suited aces at 8-12bb in the SB" or "over-shoving offsuit connectors at 15-18bb on the BTN."

Worked example: a common leak

A5o from the SB vs. BB at 8bb. The Nash range at this stack depth includes A5o as a shove for most ante structures. Many players fold this hand because "ace-five offsuit feels weak." The feeling is wrong. Against a BB calling range that is GTO-calibrated to 8bb shoves (roughly the top 30-35% of hands), A5o has approximately 48-52% equity, and the pot odds from blinds and antes make this a mandatory shove. Folding A5o at 8bb from the SB is leaving chips on the table.

The trainer surfaces this leak explicitly: after 50 drills, if you fold A5o, A6o, and A7o at 8-10bb more than 50% of the time in the SB, the report flags "over-folding ace-x offsuit in the SB, 8-10bb range." You drill the specific hands until the pattern reverses.

What each output means

Accuracy rate is the percentage of drills where your decision matched Nash. Leak summary groups your errors by hand category and stack depth. Stack heatmap shows a 13x13 grid of all 169 starting hands at your current stack depth, colored by push (green), fold (red), and marginal (yellow). Session record tracks accuracy across sessions so you can see improvement over time.

Where the model breaks

Nash equilibrium ranges are theoretically optimal against a perfectly adaptive opponent. Your actual opponents are not perfectly adaptive. Against a BB who over-folds to shoves (calling with top 15% rather than 30%), the correct shove range widens beyond Nash. Against a BB who calls too liberally, the range tightens. The trainer builds Nash as the baseline, which is the right starting point; exploitative adjustments come from reads and population tendencies.

ICM changes the ranges materially near the bubble and at the final table. Nash chip-EV shove ranges are correct for chip-EV maximization only. In ICM-heavy spots, the correct range tightens because losing chips costs more in dollar terms than winning chips earns. The trainer's ICM mode applies this adjustment, but it requires inputting stack distributions and payout structure, which adds friction relative to the quick-drill mode.

BB-ante vs. traditional-ante structures produce meaningfully different ranges. Most modern tournaments use the BB ante, which inflates the pot by more per orbit than traditional antes at the same blind level. The trainer's ante toggle switches between structures. For the full deal on push/fold mathematics, the M-ratio calculator gives you the tournament stack context, and the Sklansky-Chubukov rankings provide the theoretical chip-EV push thresholds.