Push/Fold Strategy — Nash Equilibrium for Short Stacks

❓ Why Push/Fold?

When your stack drops below approximately 15BB in a tournament, the standard open-raise/call/3-bet game breaks down. With a low Stack-to-Pot Ratio (SPR), you effectively have only two viable options: push all-in or fold.

$$\text{SPR} = \frac{\text{Effective Stack}}{\text{Pot after preflop action}}$$

Consider a 10BB stack that open-raises to 2.5BB. The pot is now ~4BB and you have 7.5BB behind. Your SPR is approximately 1.9. At this SPR, virtually any flop bet commits you to the pot — there is no room for postflop maneuvering.

When \(\text{SPR} < 4\), postflop play becomes trivial because most decisions are effectively pre-committed. Push/fold simplifies the entire game into a single preflop decision: go all-in or surrender your hand.

This simplification is not a limitation — it is mathematically optimal. With shallow stacks, the EV difference between open-raising and pushing is negligible, and pushing has the added benefit of maximizing fold equity.

📈 Push EV Calculation

When you push all-in from any position, your expected value consists of two components: the times everyone folds (you win the blinds/antes) and the times you get called (you play for stacks).

$$EV_{\text{push}} = P(\text{fold}) \times (P + B) + P(\text{call}) \times \left[P(\text{win|call}) \times (S_v + P + B) - (1 - P(\text{win|call})) \times S_h\right]$$

Where:

• \(P\) = pot from antes (dead money)
• \(B\) = blinds already posted
• \(S_v\) = villain's calling stack
• \(S_h\) = hero's stack (amount risked)
• \(P(\text{fold})\) = probability all opponents fold
• \(P(\text{win|call})\) = our equity when called

Simplified Heads-Up Formula (SB vs BB)

In the simplest scenario — SB pushes, BB decides — the formula reduces to:

$$EV_{\text{push}} = f \times 1.5\text{BB} + (1-f) \times [e \times (2S + 1.5) - (1-e) \times S]$$

where \(f\) = BB's fold frequency, \(e\) = our equity when called, and \(S\) = our stack in BB (assuming we cover).

Worked Example

Hero is in the SB with \(8\text{BB}\) and holds A5o. BB folds 60% of the time and calls with top 40% of hands. Our equity vs BB's calling range is approximately 42%.

$$EV_{\text{push}} = 0.60 \times 1.5 + 0.40 \times [0.42 \times (16 + 1.5) - 0.58 \times 8]$$ $$= 0.90 + 0.40 \times [0.42 \times 17.5 - 0.58 \times 8]$$ $$= 0.90 + 0.40 \times [7.35 - 4.64]$$ $$= 0.90 + 0.40 \times 2.71 = 0.90 + 1.08 = +1.98\text{BB}$$

Pushing A5o from the SB with 8BB is profitable — we gain nearly 2BB in expectation.

📊 Call EV Calculation

When facing an all-in, the EV of calling is:

$$EV_{\text{call}} = P(\text{win}) \times (S_{\text{push}} + P + B) - P(\text{lose}) \times S_{\text{call}}$$

Setting \(EV_{\text{call}} = 0\) gives us the breakeven equity — the minimum equity needed to call profitably:

$$\text{Required Equity} = \frac{S_{\text{call}}}{S_{\text{call}} + S_{\text{push}} + P + B}$$

Example

Villain pushes 10BB from the SB into your BB. You have already posted 1BB, so you need to call 9BB more. The total pot after calling would be 10 + 1 + 0.5 + 9 = 20.5BB (villain's 10BB + your 1BB posted + SB's 0.5BB posted + your 9BB call).

$$\text{Required Equity} = \frac{9}{9 + 10 + 1 + 0.5} = \frac{9}{20.5} = 43.9\%$$

You need at least 43.9% equity against villain's pushing range to profitably call. With a hand like K9o (~45% vs a wide SB push range), this would be a marginal call.

🎯 Nash Equilibrium

A Nash equilibrium in push/fold is a strategy pair (push range + call range) where neither player can improve their EV by unilaterally changing their strategy.

At equilibrium:

• Pusher's range makes the caller indifferent between calling and folding with their marginal hands
• Caller's range makes the pusher indifferent between pushing and folding with their marginal hands

The equilibrium is found by iteratively adjusting both ranges until neither player wants to deviate. In practice, this is computed algorithmically because the calculation involves comparing equity of every hand against every possible range.

The key insight is that Nash push/fold ranges are unexploitable. Even if your opponent knows your exact strategy, they cannot beat it. This makes Nash charts the foundation of tournament endgame play.

📋 Nash Push/Fold Ranges by Stack Size

The following tables show approximate Nash equilibrium push and call ranges at various stack depths, assuming no antes.

BTN Push Ranges (no antes)

SB Push Ranges (vs BB, no antes)

BB Call Ranges (vs SB push, no antes)

💰 The Effect of Antes

Antes dramatically change push/fold dynamics by increasing the dead money in the pot before any action occurs.

Dead Money Comparison

Without antes, the pot before action is simply the blinds:

$$\text{Pot}_{\text{no antes}} = 1.5\text{BB} \quad (\text{SB} + \text{BB})$$

With a typical 0.1BB ante at a 9-handed table:

$$\text{Pot}_{\text{with antes}} = 1.5 + 9 \times 0.1 = 2.4\text{BB}$$

That's 60% more dead money in the pot! This has a direct impact on push EV because the fold-equity component increases:

$$\Delta EV = P(\text{fold}) \times \text{Antes}$$

If opponents fold 70% of the time, the extra 0.9BB in antes adds \(0.70 \times 0.9 = 0.63\text{BB}\) to your push EV — a significant boost.

Consequences

• Push ranges expand significantly — the extra dead money makes stealing more profitable
• Calling ranges also expand slightly — callers are getting better pot odds
• Short stacks become even more desperate — antes eat into your stack every orbit, so pushing before being blinded out becomes critical

M-Ratio

The M-ratio (also called the Magriel number) gives a more accurate picture of your stack depth when antes are in play:

$$\text{M-ratio} = \frac{\text{Stack}}{\text{SB} + \text{BB} + \text{Total Antes}}$$

For example, with a 20BB stack at a 9-handed table with 0.1BB antes:

$$M = \frac{20}{0.5 + 1.0 + 0.9} = \frac{20}{2.4} = 8.3$$

Even though you have 20BB, your M-ratio is only 8.3 — meaning you can only survive about 8 orbits. This is much more urgent than the raw BB count suggests.

When \(M < 5\): Push/fold mode is mandatory. You do not have enough chips to play a standard raising game. Every orbit costs you a significant fraction of your stack.