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Push/fold chart: heads-up Nash shove and call ranges

With 15 big blinds or less, shoving or folding is close to the best a small blind can do. The charts give, for every hand, the largest stack at which the equilibrium shoves it, and the largest at which the big blind calls.

Stacks are the effective stack in big blinds, heads-up, with no antes; 30+ means at every stack up to 30 bb. For 17 hands, mostly small suited ones, the equilibrium also folds at a few smaller stacks inside that span (53s shoves at 2 bb and from 4.5 to 12.5 bb); the app's chart shows every stack.

Working it out

Example: at 10 bb the small blind shoves 58.4% of hands, down to Q7o, and the big blind calls with 37.3%, down to K5o and Q9o. 76s shoves at every stack up to 30 bb but calls a shove only up to 5 bb: shoving wins the blinds whenever the big blind folds, while calling has to win at showdown.

Formula

Shove (small blind, stack S, fold probability f, equity when called e): EV = f × 1 + (1 − f) × (2S × e − S) bb; folding costs the 0.5 bb small blind. Call (big blind): call when 2S × e − S > −1, that is e > (S − 1) ÷ (2S); at 10 bb, 45%.

Heads-up only, with no antes, in chips (chip EV). At a full table the ranges get tighter the more players are left to act, antes widen them, and tournament payouts (ICM) tighten the calls.