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Nash Equilibrium Push/Fold Calculator

Master short-stack tournament poker with Nash equilibrium push/fold charts. This calculator provides mathematically optimal shove and call ranges based on your stack depth, position, and number of opponents. Named after Nobel Prize-winning mathematician John Nash, these equilibrium strategies are unexploitable when executed correctly.

Whether you're navigating a tournament bubble, facing short-stack situations, or studying tournament poker strategy, understanding Nash ranges transforms your decision-making from guesswork into precise game theory application.

Select Chart Type

Shove Range: Button, 10 BB

Shove
Marginal / Mixed
Fold

Button Shove Range

42.5%
Range Percentage
42.5%
of starting hands
Total Combos
564
hand combinations
Unique Hands
72
distinct holdings

Strategy Notes

From the button with 10 BB, Nash equilibrium suggests a wide shoving range. You have fold equity against the blinds and position advantage. At this stack depth, standard raise sizing loses too much fold equity, making all-in the optimal play with your entire opening range.

Quick Hand Lookup

SHOVE

Understanding Nash Equilibrium in Poker

Nash equilibrium, named after mathematician John Nash whose work on game theory earned the 1994 Nobel Prize in Economics, represents a state where no player can improve their expected value by unilaterally changing their strategy. In poker terms, a Nash equilibrium push/fold strategy cannot be exploited by opponents playing perfectly.

When your tournament stack falls below approximately 15 big blinds, traditional raise sizing becomes mathematically suboptimal. At these stack depths, the correct approach is often all-in or fold, maximizing your fold equity while eliminating difficult post-flop decisions. This is where Nash charts become essential tools for tournament players.

The Power of Unexploitability

When you execute Nash ranges perfectly, no opponent strategy can outperform breaking even against you in the long run. While exploitative adjustments can increase EV against weaker players, Nash provides a baseline guarantee - a mathematically sound floor that ensures you're never making fundamental errors in push/fold situations.

When to Use Push/Fold Strategy

Nash push/fold charts are most valuable in specific tournament situations:

  • Short Stack Play (under 15 BB): The mathematics strongly favor all-in plays when standard raise sizing would commit too much of your stack
  • Bubble Situations: Understanding both shove and call ranges becomes critical when survival equity matters most
  • Final Table Play: ICM considerations often require tighter adjustments to Nash ranges
  • Sit-and-Go Tournaments: SNGs frequently reach push/fold territory, making these charts essential
  • Turbo and Hyper-Turbo Events: Fast blind structures mean more decisions at short stack depths

Learn more about tournament-specific applications in our Complete Tournament Strategy Guide and M-Ratio Calculator.

Shove Ranges vs Call Ranges

A common misconception is that shove and call ranges should be identical. In reality, they serve different purposes and require different constructions:

Aspect Shove Ranges Call Ranges
Primary Goal Maximize fold equity + showdown value Profitable calls vs opponent's shove range
Typical Width Wider (includes bluffs) Narrower (pure value)
Position Impact Widens significantly from late position Adjusts based on opponent position
Stack Depth Effect Wider with shallower stacks Depends on pot odds offered
Example (10 BB BTN) ~42% of hands ~25-30% vs tight shove

ICM Adjustments Required

These Nash charts assume chip EV (cEV), where all chips have equal value. In real tournaments, especially near the bubble or at final tables, ICM pressure requires tighter adjustments. Big stacks can afford wider ranges while short stacks must often play tighter than Nash suggests. Use our ICM Calculator to understand bubble factors.

Position and Range Construction

Position dramatically affects Nash ranges because it determines how many opponents remain to act and thus your expected fold equity:

Early Position (UTG, MP)

With many players behind, fold equity is limited and you face higher risk of being called. Nash ranges here are tight, focusing on premium hands that play well even when called. A 10 BB UTG range might be only 15-20% of hands.

Late Position (CO, BTN)

Fewer players to act means higher fold equity and wider profitable shove ranges. From the button with 10 BB, Nash suggests shoving approximately 40%+ of hands because you only need the blinds to fold. Study position dynamics further in our Complete Position Guide.

The Blinds (SB, BB)

Small blind has the widest shove range when it folds to you - you only need to get through one opponent. Big blind calling ranges depend on pot odds and the opponent's position-based shoving range.

Stack Depth Considerations

Your stack size profoundly impacts Nash ranges. Understanding the mathematical thresholds helps you adjust on the fly:

Stack Depth Recommended Approach Range Width (BTN)
1-5 BB Pure push/fold; any playable hand is a shove 50-100%
6-10 BB Standard push/fold territory 35-50%
11-15 BB Push/fold or min-raise depending on table 30-40%
16-20 BB Can employ some standard raises 25-35%
20+ BB Standard tournament poker applies Varies widely

For a deeper understanding of stack health in tournaments, explore our M-Ratio Calculator and the strategic zones developed by Dan Harrington.

Practical Application Tips

Knowing the theory is one thing; executing correctly under pressure is another. Here are practical tips for applying Nash ranges:

  • Memorize Key Thresholds: You don't need to memorize every hand - learn the boundaries (e.g., "I shove any Ace at 8 BB from the button")
  • Account for Antes: Antes increase the pot and widen correct shoving ranges significantly
  • Adjust for Player Tendencies: Against very tight players, shove wider. Against calling stations, tighten up
  • Consider ICM Pressure: Near bubbles or pay jumps, adjust ranges based on relative stack sizes
  • Study Showdowns: Review your push/fold hands to verify you're executing close to Nash

Practice Makes Perfect

Use our Preflop Trainer to build muscle memory for hand classifications, then apply that knowledge to push/fold scenarios. The better you know hand rankings and relative strengths, the faster you can reference Nash ranges in real-time.

The Mathematics Behind Nash Ranges

Nash equilibrium calculations for poker push/fold scenarios consider several mathematical factors. According to game theory research documented by ScienceDirect, the equilibrium point balances risk and reward across all possible opponent responses.

The calculation involves:

  • Fold Equity: The probability opponents fold multiplied by the pot you win
  • Showdown Equity: Your expected value when called, based on your hand vs calling range
  • Risk/Reward Ratio: Stack size relative to blinds and antes
  • Position Adjustments: Number of players yet to act affects fold probability

Learn more about the mathematical foundations in our Complete Poker Probability Guide and Expected Value Calculator.

Frequently Asked Questions

What is Nash Equilibrium in poker?

Nash Equilibrium in poker refers to a mathematically optimal strategy where neither player can improve their expected value by unilaterally changing their strategy. Named after mathematician John Nash, this concept provides unexploitable push/fold ranges for short-stacked tournament play, ensuring you cannot be outplayed when following these charts.

When should I use push/fold charts?

Push/fold charts are most applicable when your stack is 15 big blinds or fewer in tournament poker. At these stack depths, the mathematics favor an all-in or fold approach rather than standard raise sizing. The exact ranges depend on your position, stack depth, and number of players yet to act behind you.

What is the difference between shove and call ranges?

Shove (push) ranges define which hands you should go all-in with when you are the first to act. Call ranges define which hands you should call with when facing an all-in from another player. Call ranges are typically tighter because you need to beat an opponent's shove range rather than just have positive EV in isolation.

How does ICM affect push/fold decisions?

ICM (Independent Chip Model) affects push/fold decisions by adjusting the value of chips based on tournament payout structure. On the bubble or at final tables, ICM pressure makes survival more valuable, requiring tighter push ranges and even tighter call ranges. Standard Nash charts assume chip EV, so real tournament play often requires ICM adjustments. Our ICM Calculator helps quantify these adjustments.

More Learning Resources

Expand your tournament poker knowledge with these related tools and guides:

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