Powerful poker tools and mathematical models have have enabled poker players to develop and implement game-theory based unexploitable and optimal poker strategies and plays in specific, key situations.

One concept which is highly applicable to poker tournaments and SNGs is "Nash Equilibrium".

For heads-up play (e.g. blind vs blind hands) it's possible to calculate game-theory optimal (GTO) shove and calls ranges which are profitable in the long run, based on your stack size, and cards / those of your opponents.

Poker Pushbot charts can answer important questions quickly- for heads-up situations, for example:

• Can I go all in with this hand profitably?
• Which hands can I profitably call a shove with vs my opponents stack size, when they go all in for X big-blinds?

Equilibrium rankings are meant as a guide to help you to develop an idea of which hands and hand ranges are good enough to warrant an all-in and which hands are good enough to call an all-in vs an aggressive opponent.

It folds to you in the small-blind. You have 12 big-blinds and Queen ten offsuit. What's your play? Can you profitably and unexploitably go all in, or do you wait for a better sport?

You can Shove (go "all in") profitably if:

• If you are in the small blind,
• everyone before you has folded,
• your effective stack (the smallest of the stack sizes of you vs your opponent in blinds blinds) is smaller than the number given in the table below for your specific hand

in this spot when you have 12 big-blinds.. From the pushbot chart below, we can see that QTo is a profitable SHOVE (as is Q9, and Q8, but NOT Q7o)

*Pro Tip: the push chart can also be used when you’re short-stacked and interested in jamming "all in" from the button. In this case account for the extra player that you’ll be shoving into (the Small Blind) by dividing all the stack size numbers in the chart by 2. Similarly you could divide by 4 for stack size for a protiable cutoff jam.

#### Nash Equilibrium pushbot chart for going all-in from the small-blind (SB)

 Suited Cards Offsuit Cards AA50 AKs50 AQs50 AJs50 ATs50 A9s50 A8s50 A7s50 A6s50 A5s50 A4s50 A3s50 A2s48 AKo50 KK50 KQs50 KJs50 KTs50 K9s50 K8s50 K7s49 K6s36 K5s32 K4s26 K3s20 K2s19 AQo50 KQo50 QQ50 QJs50 QTs50 Q9s50 Q8s50 Q7s20 Q6s29 Q5s24 Q4s16 Q3s14 Q2s13 AJo50 KJo50 QJo50 JJ50 JTs50 J9s50 J8s50 J7s32 J6s19 J5s16 J4s14 J3s11 J2s8.8 ATo50 KTo50 QTo45 JTo46 TT50 T9s50 T8s50 T7s36 T6s25 T5s12 T4s11 T3s7.7 T2s6.5 A9o45 K9o24 Q9o24 J9o29 T9o32 9950 98s50 97s36 96s27 95s14 94s6.9 93s4.9 92s3.7 A8o43 K8o19 Q8o13 J8o14 T8o18 98o21 8850 87s43 86s31 85s19 84s10 83s2.7 82s2.5 A7o41 K7o16 Q7o10 J7o8.5 T7o9.9 97o11 87o16 7750 76s36 75s24 74s14 73s2.5 72s2.1 A6o35 K6o15 Q6o9.8 J6o6.5 T6o5.7 96o5.2 86o7.1 76o11 6650 65s29 64s16 63s7.1 62s2 A5o37 K5o14 Q5o8.9 J5o6 T5o4.1 95o3.5 85o3 75o2.6 65o2.4 5550 54s24 53s13 52s2 A4o35 K4o13 Q4o8.3 J4o5.4 T4o3.8 94o2.7 84o2.3 74o2.1 64o2 54o2.1 4450 43s10 42s1.8 A3o32 K3o13 Q3o7.5 J3o5 T3o3.4 93o2.5 83o1.9 73o1.8 63o1.7 53o1.8 43o1.6 3350 32s1.7 A2o29 K2o12 Q2o7 J2o4.6 T2o3 92o2.2 82o1.8 72o1.6 62o1.5 52o1.5 42o1.4 32o1.4 2250

#### Nash Equilibrium chart for calling an all-in in the big blind

The action folds to the small-blind who goes "all in" for 15 BBs - half of your 30bb Stack. You look down at King Queen Suited.

Can you call the all-in profitably?

• If you are in the big blind (BB)
• everyone before the small blind has folded,
• The SB moved all-in,
• and your effective stack (in BBs) is smaller than the number specified in the table above.

Looking at the chart we can see that we can profitably call with QKO, as well as KJo, KTo and K9o

 Suited Cards Offsuit Cards AA50 AKs50 AQs50 AJs50 ATs50 A9s47 A8s41 A7s36 A6s31 A5s30 A4s26 A3s25 A2s23 AKo50 KK50 KQs50 KJs45 KTs32 K9s24 K8s18 K7s15 K6s14 K5s13 K4s12 K3s11 K2s11 AQo50 KQo46 QQ50 QJs29 QTs24 Q9s16 Q8s13 Q7s11 Q6s10 Q5s8.9 Q4s8.5 Q3s7.8 Q2s7.2 AJo50 KJo27 QJo20 JJ50 JTs18 J9s14 J8s11 J7s8.8 J6s7.1 J5s6.9 J4s6.2 J3s5.8 J2s5.6 ATo50 KTo24 QTo16 JTo13 TT50 T9s12 T8s9.3 T7s7.4 T6s6.3 T5s5.2 T4s5.2 T3s4.8 T2s4.5 A9o40 K9o18 Q9o12 J9o9.9 T9o8.5 9950 98s8.3 97s7 96s5.8 95s5 94s4.3 93s4.1 92s3.9 A8o35 K8o14 Q8o9.8 J8o7.7 T8o6.7 98o6.1 8850 87s6.5 86s5.6 85s4.8 84s4.1 83s3.6 82s3.5 A7o29 K7o13 Q7o8 J7o6.4 T7o5.5 97o5 87o4.7 7750 76s5.4 75s4.8 74s4.1 73s3.6 72s3.3 A6o22 K6o11 Q6o7.4 J6o5.4 T6o4.7 96o4.2 86o4.1 76o4 6650 65s4.9 64s4.3 63s3.8 62s3.3 A5o21 K5o10 Q5o6.8 J5o5.1 T5o4 95o3.7 85o3.6 75o3.6 65o3.7 5543 54s4.6 53s4 52s3.6 A4o19 K4o9.2 Q4o6.3 J4o4.8 T4o3.8 94o3.3 84o3.2 74o3.2 64o3.3 54o3.5 4432 43s3.8 42s3.4 A3o17 K3o8.8 Q3o5.9 J3o4.5 T3o3.6 93o3.1 83o2.9 73o2.9 63o3 53o3.1 43o3 3322 32s3.3 A2o16 K2o8.3 Q2o5.6 J2o4.2 T2o3.5 92o3 82o2.8 72o2.6 62o2.7 52o2.8 42o2.7 32o2.6 2215

* The maximum stack size considered in this model is 50 BBs

Data Source: Mathematics of Poker (2006) by Bill Chen