A Slave to Variance

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 (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 players 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.

In his latest CardPlayer column Thomas Keller describes a key hand for both of us.

A couple of observations:

• Our memories of the hand differ. He lists the rag as a 6, and I list it as a 9. I have a bit more confidence in my memory since I was logging hands within minutes at the table after they occured, wheras I’m not so sure when Thomas Keller logged this hand.
• He was thinking about the actual play of the hand more than I was. It might be that he didn’t think too much during the hand either, but I distinctly remember thinking “Does he have KQ? No, I bet!”. I really didn’t put him on a specific hand at any point, but I was worried about an underfull with 99 or possibly QQ, and certainly thought AA/AK was a possibility.
• His perspective on my play is interesting. He seems to indicate that I was underbetting the pot by a fair amount. In general, this was my strategy. I wanted to make two kinds of bets, underbets and overbets. Most of my bets would be underbets, and my overbets would be reserved for very strong hands or very strong draws. This was a medium-strong hand and a (very) strong draw, so I didn’t really care if I was “giving him odds”. If I’m ahead or behind, I’m happy if he calls, and happy if he pushes back.
• He asked me my name after he busted, but forgot it soon afterward. Not that I’m bitter. In a lot of ways, I prefer to be anonymous, but it does illustrate that memory is fallable, and egos love stroking.
• It was a hard hand for AK no draw to play. With 100/200 blinds, our stacks are 50 BB deep. It’s hard to see what he can do to get away from the hand. But given the way I played it, it’s hard to see how it can be right to get away from it.
• Never underestimate your opponent. Not only did Thomas Keller not know who I was, but I did not know who he was. I had some idea that he had been on TV since I saw him on a monitor in the “Lifestyles” show. So even when you don’t really know what you are up against, it’s important to still play as if your opponent has a clue. I think we both did pretty good in that regard.

In the end it was a pivotal hand for both of us, and does a lot to account for both of our final results.

So, what is the point of giving myself advice, if I’m going to ignore it?

hold'em[L] $ 40/80 Paradise Poker - Malakal        Sun Aug 14
         ante:0.00 blinds: 20.00 40.00  rake:3.00 pot: 1197 [3d Ah Ac 3h 2h]
 yumyum12      2462  6 f                            0     0
 onesickchick  3185  5 f                            0     0
 Sad Max       1314  4 f                            0     0
 pfflyer       2094  3 c    kf                     40   -40
 Andrew        3948  2 c    b    bf               160  -160 [4h 5h]
 Miss Swan     1517  1 Sc   kf                     40   -40
 pokerdoll     3158  0 Bk   kc   kcr  bc          480   118 [As Th]
 rspain        3976  8 c    kc   krc  r           480   118 [Ad Qd]
 Canterbury    5500 -1 -                            0     0
 zacky         1251  7 f                            0     0

This is about the most obvious turn check through in the world: 5-high with one out to the absolute freakin’ nuts. But what I do? I bet myself off the hand instead. Of course, the only reason I post this is because the one out draw made it, and both my opponents had the obvious full house.

Double duh!