Nash Equilibrium and SAGE

by Cocky Fish on July 9, 2009

There are two pieces of paper that sit next to me when I play poker.  They’re old, crumbled, covered in grease stains and dotted with bitter tear stains from past bad beats.  These tattered pieces of paper are the Nash Push/Call charts and I’d never play a Sit ‘N Go without them.

Most SNG players know about the Independent Chip Model (ICM).  ICM is the theory that determines when shoving/calling a pre-flop shove is unexploitable.  Don’t get the words “unexploitable” and “optimal” confused.  A move that is unexploitable isn’t always optimal.

For example, shoving aces from late position in an unopened pot when the blinds are 15/30 is an unexploitable move but it’s hardly optimal.  That said, Nash equilibrium tells you when shoving or calling a shove is unexploitable heads-up.

Here are the Nash Equilibrium charts:

nash pusher Nash Equilibrium and SAGE

nash caller Nash Equilibrium and SAGE


The numbers represent the number of effective big blinds needed to make the shove unexploitable.

Personally, I like to make standard raises and play poker until the small stack has about 13 or 14 big blinds.  But if you’re seriously outmatched or your opponent is a complete lunatic, you can use the Nash charts to negate any edge they have.  The only way they can adjust to your play is to start playing Nash too.  If they don’t, you’ll have the edge.

Nash is great if you’re playing online, but you won’t be able to take your charts to the table with you if you’re playing live.  If you’re playing live, there’s a little trick you can use to play close to Nash called SAGE.  It’s not perfect, but it’s close enough.  Here’s how you do it:

SAGE uses two numbers: the power index and the ratio of the small stack to the big blind.

Finding the ratio of the small stack to the big blind is easy.  Just take the small stack and divide it by the big blind.  So if the small stack is t1000 and the big blind is t100, the ratio is 10.

The power index is a bit more complicated.  You assign a value to your hand based on the cards you hold and whether or not they’re suited or a pocket pair.  Here’s how the scoring system works:

Ace = 15

King = 13

Queen = 12

Jack = 11

Other cards = face value

Your hand also gets bonus points if it’s suited or a pocket pair.

Suited = +2

Pocket Pairs = +22

The way you figure out your power index is to double the score of your highest card, then add the score of your other card and any bonus points.

So a hand like J9 offsuit would have a score of 31 (11*2+9) whereas J9 suited would be 33 (11*2+9+2).  The most powerful hand you can hold is pocket Aces (15*2+15+22=67) and the least powerful hand you can hold is 32 offsuit (3*2+2=8).

Now here’s what you do with that information:

sage Nash Equilibrium and SAGE


The chart tells you what your power index needs to be in order to shove or call a shove when the effective stack size is 7 big blinds or less.

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Obviously you won’t win every heads-up situation using Nash Equilibrium or SAGE, but you’ll win more often than you’ll lose.  Obviously these charts aren’t a substitute for learning solid heads-up play, but they’ll definitely help you take down more tournaments.

Ready to try out your new poker skills? Play for real or free at:

 Nash Equilibrium and SAGE

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