No-Limit Texas Hold’em is a complex game with many different variables, including your cards, the cards on board, the number of chips that you and your opponents have, the number of chips in the pot, and more. With all of these elements changing every hand, it may seem that there is no equilibrium strategy in hold’em simply because there are too many different strategies. In fact, in many cases there isn’t, as a previous post discusses. However, in certain very specific situations, there are Nash Equilibria in which sticking with one strategy is optimal for both players, and deviating from the strategy in any way is detrimental to the player who deviates.

Consider the situation at the end of a poker tournament with two players remaining (known as heads up play), when the blinds (forced bets before the hand starts) are large compared to the amount of chips that the shorter stack (person with less chips) has, and the small blind (player first to act) chooses either to move all-in (bet all of his/her chips) or fold. This situation is in fact very common in many online tournaments. Because it is profitable to play optimally, this situation has been analyzed by many experts and it turns out that a Nash Equilibrium exists. The SAGE system, created by Lee Jones, a poker author/poker room manager and James Kittock, a math professor, defines how each player should act in this situation to be playing optimally (note that in practice, if one player is not playing optimally, then SAGE is merely a winning strategy for the other player; both players must be using the SAGE system for it to be an equilibrium).

The SAGE system works by defining a range of hands that you should move all-in with if you are first to act, or call with if you are second to act. Here is the reference table:

First, you compute R, which is the ratio of the short stack to the big blind. If you are the small blind, you should be moving all-in if you have the “SB cutoff hand” corresponding to the current R value as shown in the table, or if you have a better hand. Otherwise, you should fold. If you are the big blind, you should be calling an all-in if you have the “BB cutoff hand” or a better hand. This is a Nash Equilibrium; if both players are employing this strategy, then if either player changes strategy, they lose expected value (they lose money in the long run).

Employing the SAGE system will guarantee you at least breaking even in the long run in these specific situations. Once you learn the SAGE system, when playing against an opponent who is not playing optimally, you gain an advantage estimated between 5 and 40 percent, which is huge in poker. SAGE illustrates that there are indeed known optimal plays under very specific conditions at a poker table, and those who use SAGE can’t lose money in these situations in the long run. Note that SAGE won’t make you a winning poker player automatically, as you will need to play well in a tournament before the conditions necessary to use SAGE are met. However, it can certainly help a player of any skill level play a better heads up game, and therefore make more money (or lose less) in the long run.

References:

http://www.cardplayer.com/magazine/article/15250

http://klopzi.blogspot.com/2007/09/sage-sit-and-go-endgame-system.html