ABC Tools

Push/Fold Nash Charts, Explained

After reading this you will know why heads-up push/fold has an exact solution, how to read the jam and call ranges from 1 to 25 big blinds, and how to check any single hand's expected value against a perfect opponent.

What the charts are

When two players are left and the small blind has a short stack, the game shrinks to something you can solve exactly. The small blind either shoves all in or folds. The big blind either calls or folds. There is no flop strategy to worry about, no bet sizing, no bluffing on the river. With only those choices, game theory gives one equilibrium: a jam range for the small blind and a call range for the big blind that neither player can beat by changing their own play.

The tool draws both ranges on the standard 13×13 grid of starting hands. Drag the stack from 1 to 25 big blinds and the shaded cells update. Suited hands sit above the diagonal, offsuit hands below, pairs on the diagonal.

One hook to fix the idea: at 10 bb the small blind should jam A7s every time, but the big blind should only call it with a fairly narrow range. That single hand, at that stack, is the demo you can reproduce below.

When to use these charts, and when not

Use them for heads-up no-limit hold'em when the effective stack is short, roughly 12 bb or less, and when the small blind's realistic choice really is shove or fold. That covers the end of most heads-up matches and the final two players of a sit-and-go.

Do not treat the charts as gospel at deep stacks. Once you carry 15 bb or more, a smaller raise usually beats jamming, because you keep the option to fold to a re-raise and you win more when your strong hands get action. The equilibrium here assumes jam-or-fold only, so read the 15 bb+ charts as a worst-case baseline, not as the best available line.

They also assume no antes and blinds of 0.5/1 bb. Antes shift the math toward wider jams because there is more dead money to win. If your game has antes, these ranges are slightly too tight.

These are equilibrium ranges, not exploit ranges. They are built to be unbeatable, not to punish a specific opponent. If the big blind calls far too tight, you can profitably jam any two cards; if the big blind calls far too loose, you should jam tighter. Nash is your default when you have no read.

The math behind a single jam

The small blind's decision reduces to a comparison of two expected values in big blinds. Folding surrenders the small blind of 0.5 bb, so the fold EV is fixed. Jamming risks the whole stack to win the pot. The equilibrium call range for the big blind is exactly the set of hands where calling beats folding.

EV_{\text{jam}} = f \cdot 1 + (1 - f)\left[2S \cdot e - S\right]

Here S is the effective stack in big blinds, f is the fraction of the time the big blind folds, and e is your hand's equity when called. When the big blind folds you win the pot of 1 bb (the two blinds already in). When called, you invest S and get back 2S times your equity. Compare that number against the fold EV of -0.5 bb and jam when it is larger.

The big blind faces the mirror comparison. Facing a jam, folding costs the 1 bb already posted, leaving a stack of S - 1. Calling risks the rest to contest a pot of 2S. Calling is correct when

2S \cdot e \gt S

which simplifies to e \gt 0.5 against that exact jamming range. In practice the required equity is a touch above 50% because of the dead blind, and the software finds the fixed point where both ranges are best responses to each other.

A worked example: A7s at 10 bb

Reproducing the demo

Set the stack to 10, position to small blind, and the hand to A7s. This is the demo button's sample data.

  1. Against the big blind's equilibrium calling range at 10 bb, A7s gets called only by the equilibrium calling range, which folds about 65% of the time, so f = 0.65.
  2. When called, A7s holds about 53% equity, so e = 0.53. The calling range is wide enough that A7s is ahead of much of it.
  3. Compute the called branch: 2 \cdot 10 \cdot 0.53 - 10 = 10.6 - 10 = +0.6 bb.
  4. Blend the two branches: 0.65 \cdot 1 + 0.35 \cdot 0.6 = 0.65 + 0.21 = +0.86 bb from the point of jamming.
  5. Fold EV is -0.5 bb. Jamming at about +0.83 bb (the tool’s exact figure) beats folding by more than 1.3 bb, so A7s is a clear jam.

The tool reports the jam EV for the hand as a single number in big blinds. Small differences from the hand-computed value above come from using exact enumerated equities against the full mixed range rather than the round figures used here.

How the range widens as the stack shrinks

The shorter your stack, the wider you jam, because the pot you can win (1 bb) is larger relative to what you risk. At 1 bb effective the small blind is already all in after posting, so every hand jams. At 2 bb the jam range is close to 90% of hands. By 10 bb it is about 58%. By 20 bb it drops to about 40% in the pure jam-or-fold game.

The small blind jams almost everything at 1 to 2 bb and tightens smoothly as the stack grows. The marker sits at the 10 bb demo stack, around 58% of hands.

The big blind's calling range stays well behind the jam range until the stacks get very short. At 10 bb the big blind calls a jam with roughly the top 37% of hands. Only at very short stacks does calling become automatic: at 2 bb the big blind is priced in with every hand, but already by 5 bb the call range has tightened to about 62%, well behind the jam range.

At 10 bb, jamming A7s is worth about +0.83 bb while folding is worth -0.5 bb, so the jam gains over 1.3 bb. A7s stays a jam at every stack up to 25 bb: its jam EV slides from about +0.88 bb at 8 bb to +0.17 bb at 25 bb, approaching but never crossing the fold line. A junk hand like 72o sits below the fold line at every stack beyond 1 bb and never jams.

Reading and interpreting the grid

A fully shaded cell means the equilibrium plays that hand 100% of the time (jam for the small blind, call for the big blind). A blank cell means never. A partly shaded cell is a mixed strategy: the equilibrium jams or calls that hand only part of the time.

Do not agonize over mixed cells at the table. A cell is mixed precisely because its jam EV and fold EV are almost equal, so choosing either one costs you almost nothing. If a hand is 40% jam, jamming it every time or folding it every time each loses a tiny fraction of a big blind versus the mix.

Click any cell to read its EV in big blinds. Compare that against the fold baseline of -0.5 bb for the small blind, or the fold baseline of S - 1 bb for the big blind, to see how much the correct action gains.

Common mistakes

Calling as wide as you jam
The small blind jams 58% at 10 bb, but the big blind calls only about 37%. Calling that wide bleeds chips because you are putting the last of your stack in as roughly a coin flip or worse.
Ignoring antes
These charts have no antes. Antes add dead money, which widens both ranges. Treat the charts as slightly tight if your game has an ante.
Using them deep
At 20 bb the pure jam-or-fold solution jams around 40% of hands, but in a real game a min-raise beats jamming with most of that range. Deep charts are a floor, not a plan.
Overvaluing offsuit gappers
Hands like J7o or 96o look playable but have poor equity when called and low fold-out value. They sit outside the jam range until the stack is very short.

Related tools

The push/fold charts tell you the equilibrium action. To check the equity numbers behind them, use the Texas Hold'em Odds Calculator for a single matchup, or the Hold'em Hand vs Range Equity tool to run one hand against a whole jam or call range at once. For post-flop spots where you are weighing a call, the Poker Pot Odds & Outs tool converts outs into the equity you need. If you play other games, the Omaha Odds Calculator, Seven-Card Stud Odds Calculator, Razz Odds Calculator, Stud Hi/Lo Odds Calculator and Omaha Hi/Lo Odds Calculator cover their own equities.

Frequently asked questions

Why does the small blind jam so many hands?

Because the pot is 1 bb and the fold EV is only -0.5 bb. Any time the big blind folds, the small blind pockets a net 0.5 bb. That fold-out value makes even weak hands profitable jams when the big blind cannot call wide.

What equity does the big blind need to call?

Just over 50% against the small blind's exact jamming range. From 2S \cdot e \gt S the break-even equity is 50%, nudged slightly higher by the blind already invested. That is why the call range is much tighter than the jam range.

Are the numbers exact?

The equilibrium is computed by fictitious play over all 169 hand classes with exact enumerated preflop equities and card removal. Residual exploitability is under 0.001 bb, so for practical purposes the ranges are exact.

Do the charts work with antes?

Not directly. They assume 0.5/1 bb blinds and no antes. Antes add dead money that widens both the jam and call ranges, so treat these charts as a slightly tight baseline in an ante game.

Can I use these three-handed or in a full ring?

No. The solution is specific to heads-up, where the small blind acts first and only one caller remains. With more players behind you the fold-out math and the call ranges both change.

Try it yourself in the tool

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