Poker complexity

Posted March 21st, 2009 by LuckyMe

Poker is a very complex game obviously. Because poker is a game with a great deal of combinatorial complexity, most poker analysis is done by making a variety of simplifying assumptions.

In the case of the Jam or Fold problem, the game is restricted to a heads-up action in the preflop round. The action of the first opponent is limited to to either folding or raising all-in. Likewise, the second opponent is only given the option of either calling the all-in raise or folding.

Under these restrictions an exact solution to the problem can be formed. The solution consists of an attack matrix for the first opponent, and a defend matrix for the second opponent. The usual formulation of the Jam or Fold problem is as follows:

  • The first player is in the small blind
  • The second player is in the big blind
  • The blinds are 1/2
  • The raise is expressed in terms of big blinds
  • Both players may choose to play any hand

The solution to this problem can either be solved directly using minimization techniques like the simplex method, or iteratively where updates are done in a way to ensure that both matrices eventually converge to a stable solution.

So an exact solution to the Jam or Fold problem exists. However, that solution is still an approximation. If any of the actual game parameters deviate from the assumed parameters optimal strategy will differ to a lesser or greater degree. Even if everyone folds to you in the small blind, the distribution of cards has been skewed significantly. The presence of a player who has entered the pot voluntarily will make a Jam or Fold solution inapplicable.

If the two players have over 30 big blinds to play out the hand with, the solution again becomes impractical. You are forced to open with a very narrow range of hands, which denies you many good opportunities to play hands which are clearly profitable.

In practice, Jam or Fold can be used as an effective opening strategy when a player’s stack shrinks to a level between 10-15 big blinds or less. For example, if you have 10 big blinds and can open with a Jam from late position, you should consider using a solution to the Jam or Fold problem as a guide in choosing your action. Likewise, if you are in the big blind and facing a Jam from a late position short stack, you might want to consider whether or not to call based on a Jam or Fold solution.

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