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Game Theory - taught by a catGame Theory in Poker

Though it's easy to see why something called "game theory" might be applicable to something like poker, the term is most often attributed to the fields of political science, economics, psychology, computer science and biology.

Still, other than biology, each of those areas of study are highly applicable to poker strategy itself, which in turn makes game theory one of the most powerful weapons advanced players can add to their holdem arsenals.

Game Theory and Poker Probability Explained

By looking at tournament poker, particularly Sit and Go games, through the lens of game theory, players can develop individual play styles to be remarkably effective. Other important aspects of poker probability that can be derived from advanced game theory include the independent chip model (ICM) and Nash equilibrium, both of which are more applicable to SNGs than any other form of Texas Holdem.

Applying Game Theory to SNGs

With statistics alone, poker players can determine their odds of winning a hand or having their preferred outs become community cards. Still, without the addition of the human element and using game theory, there is nothing that dictates what a player should do. The charts in the following sections will demonstrate what we mean.  

ICM

In a nutshell, the independent chip model (ICM) tells tournament and SNG poker players how much actual money their current stack of chips are worth. The figures used to determine ICM are the total number of chips in play as compared to the prize structure. Once play gets near the bubble, ICM becomes more applicable and is useful during all-in raising, calls, and any deals (chops) made by the players.

ICM chop at final tables of MTTs or HU in SNGs at Full Tilt Poker.

Nash Equilibrium

Whereas ICM is used during the bubble stages of a poker tournament, Nash equilibrium can theoretically be used at any stage of a tournament. However, Nash is most applicable during heads-up play when the blinds are huge and the players get to a push-or-fold situation. Although using Nash equilibrium during heads-up is where easy-to-use and optimal strategy align, its applicability is rare.

The rules for using optimal Nash in poker are as follows:

  1. Opponent is using Nash equilibrium ranges
  2. Play is currently limited to push-or-fold play

The following charts explore which hands should be used to go all-in or to call an all-in raise:

Nash Shoving Chart: Heads-Up SNGs
  A K Q J T 9 8 7 6 5 4 3 2
A 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ * * 20+
K 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 19.9 19.3
Q 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 16.3 13.5 12.7
J 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 18.6 14.7 13.5 10.6 8.5
T 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 11.9 10.5 7.7 6.5
9 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 14.4 6.9 4.9 3.7
8 20+ 18 13 13.3 17.5 20+ 20+ 20+ 20+ 18.8 10 2.7 2.5
7 20+ 16.1 10.3 8,.5 9 10.8 14.7 20+ 20+ 20+ 13.9 2.5 2.1
6 20+ 15.1 9.6 6.5 5.7 5.2 7.0 10.7 20+ 20+ 16.3 * 2
5 20+ 14.2 8.9 6 4.1 3.5 3 2.6 2.4 20+ 20+ * 2
4 20+ 13.1 7.9 5.4 3.8 2.7 2.3 2.1 2 2.1 20+ * 1.8
3 20+ 12.2 7.5 5 3.4 2.5 1.9 1.8 1.7 1.8 1.6 20+ 1.7
2 20+ 11.6 7 4.6 2.9 2.2 1.8 1.6 1.5 1.5 1.4 1.4 20+
Nash Shoving Chart: Heads-Up SNGs
  A K Q J T 9 8 7 6 5 4 3 2
A 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+ 20+
K 20+ 20+ 20+ 20+ 20+ 20+ 17.6 15.2 14.3 13.2 12.1 11.4 10.8
Q 20+ 20+ 20+ 20+ 20+ 16.1 13 10.5 9.9 8.9 8.4 7.8 7.2
J 20+ 20+ 19.5 20+ 18 13.4 10.6 8.8 7 6.9 6.1 5.8 5.6
T 20+ 20+ 20+ 20+ 20+ 11.5 9.3 7.4 6.3 5.2 5.2 4.8 4.5
9 20+ 20+ 20+ 20+ 20+ 20+ 8.2 7 5.8 5 4.3 4.1 3.9
8 20+ 18 13 13.3 17.5 20+ 20+ 6.5 5.6 4.8 4.1 3.6 3.5
7 20+ 16.1 10.3 8,.5 9 10.8 14.7 20+ 5.4 4.8 4.1 3.6 3.3
6 20+ 15.1 9.6 6.5 5.7 5.2 7 10.7 20+ 4.9 4.3 3.8 3.3
5 20+ 14.2 8.9 6 4.1 3.5 3 2.6 2.4 20+ 4.6 4 3.6
4 20+ 13.1 7.9;8.3 5.4 3.8 2.7 2.3 2.1 2 2.1 20+ 3.8 3.4
3 20+ 12.2 7.5 5 3.4 2.5 1.9 1.8 1.7 1.8 1.6 20+ 3.3
2 20+ 11.6 7 4.6 2.9 2.2 1.8 1.6 1.5 1.5 1.4 1.4 20+

Though you may have come across these charts before, there's a good chance that the numbers in them weren't properly explained.

They stand for the number of big blinds the player needs to make the play unexploitable.

For instance, if the blinds are 500/1000 and you have 15,000 chips, anything on the Nash equilibrium heads-up push-or-fold charts that says anything 15 or higher is unexploitable when pushing all-in.

To clarify, any all-in push is unexploitable if your chip stack is smaller than the chart's numerical value for that specific hand. Entries that say "20+" simply mean you can comfortably go all-in with any chip stack when yielding such hole cards.

The Gap Concept

Another important aspect of game theory, the Gap Concept states that hands a player calls calls with should be stronger than hands raised with. Though notably simple, using and knowing when other players are using the gap concept is paramount in changing up your personal strategy during the middle of a game.

Against Good Opponents

However good ICM and the Nash equilibrium are for novice and amateur players, poker gets really interesting when players break away from rigid strategies and begin using advanced game theory (also called 'leveling').

This leads to bluffing, re-raising, check-raising, strategic calling and any number of the other crazy plays you've seen your favorites commit to during high stakes games.  Using you knowledge to understand what systems other players are using while randomizing your play can result in increasing your stack substantially.

Optimal Bluffing

To get the most out of bluffing, especially if you believe another player has caught on to your use of ICM, Nash or the gap concept, the consistency of your bluffs should be a specific percentage to your legitimate raises. This mix is called your 'range'.

Most players choose this to be ~25% which is enough to gain positive EV from stealing, while not getting caught too often. 

Randomizing

One of the most difficult parts of poker for beginners to understand is randomization. While experienced players will quickly have a feel for which hands to play randomly and how effective bluffs can be. Especially when playing against more talented players than yourself, the ability to randomize can get you closer to equal footing as it will make you look more like a wild card ¬than someone who simply follows rigid strategies.

To break the habit of rigid playing, you can take a route similar to optimal bluffing and play certain hole cards as though they were spectacular hands. For instance, Q♣5♥ might not be a spectacular, but if you randomize 10% of the time and play it as if it's A♣ A♦, you'll definitely keep your opponents on their toes. 

Game Theory Doesn't Imply Optimal Poker Strategy

It should be noted that using game theory strategies doesn't mean optimal plays are being made.

If we look at poker like paper, rock, scissors, simply choosing scissors every time against random opponents should yield a 66.6 percent rate of win or draw. However, when used against the same opponent time and again, your win-or-draw percentage will quickly fall to zero percent as soon as your strategy is discovered.

Though using the strategies discussed above will still more often than not lead to winning scenarios, sticking to them can lead to situations where other players exploit your play or you won't be winning as large of pots as possible. As difficult as they are for new players to implement properly, bluffing, randomizing and strategic small betting will lead players into winning situations and a much more consistent career in SNG and large-tournament play.

Practice optimal play and randomizing your bluffs at all-anonymous tables on Bovada Poker.