Thoughts on Game Theory: Part 11

A strategy is the game theoretic term for making a decision. (If you only care about poker, skip down to the middle of this column.) When someone flips a coin, you have two strategies available – heads and tails. When you are under-the-gun in hold’em, you have three strategies available – fold, call and raise. If you are playing limit hold’em, the amount you can raise is fixed. If you are playing no-limit the strategy of raising can be divided into all the possible amounts you could raise, which range from a one big blind min-raise up to a full-stack shove.

There are a number of ways that game theorists use to try to evaluate which strategy is best. They generally begin with a simple first step, which is to eliminate all dominated strategies. What is domination? Sadly, it has nothing to do with women in leather lingerie. It has to do with the results of one strategy compared with another.

If Strategy A always produces a result that is better than Strategy B, then Strategy A dominates Strategy B. If you have only two strategies and one is dominant always choose that one, and never choose the dominated one. Let’s suppose I have my two strategies A and B. My opponent also has two strategies – C and D. Usually these are shown as a matrix: with results in the middle. To keep it simple, assume my result is whatever is shown in that box.

If he chooses C, I am best to choose A (and be +4 instead of +2.) If instead he chooses D, I am still better off choosing A (lose 2 instead of losing 4.) This game can be easily solved. We have just seen that A is my dominant strategy. We can easily show that his choice of D dominates C. After eliminating the dominated strategies B and C, we know that I will pick A and he will pick D – thus my result should always be a loss of 2.

A strategy that always performs better is a strictly dominant one. A strategy that will always do as well or better is a weakly dominant one. In poker: no rational person should ever select a dominated strategy, yet I frequently see players choose a dominated strategy at the table.

In a recent $5-$10 no-limit game at Bellagio, the under-the-gun player raised to $30, and was called by the button and both blinds, creating a pot of $120. The flop was A-K-Q rainbow. The small blind checked, and the big started thinking and thinking. (I’m not sure if he knew it was his turn.) The original UTG raiser got impatient and folded. Eventually everyone checked. They all checked again on the turn of another ace and on the river of a king. The board now was two pair aces and kings with a queen kicker. None of the three remaining players showed anything of interest, and the pot was split three ways. The UTG’s players hasty fold cost him his original raise.

If your strategic choices are fold or check, then the immediate fold is weakly dominated by checking. Why weakly dominated? Most of the time, both checking and folding lead to the same result-losing the pot. As this example shows, however, checking occasionally does better than folding, and it can never do worse. Players are sometimes so eager too fold that they even fold out of turn. Folding when there has been no bet, especially when done out of turn, is also a breach of poker etiquette. Your fold may help or hurt one of the remaining players.

Folding when there has been no bet is a dominated strategy and a breach of poker etiquette. And we haven’t even reached the main reason why you should never fold when you could check. Poker is a battle to find out information about the other players’ hands. If you know what they have, you can definitely use it your advantage. When you fold in these spots you give away the information your hand is very weak. It might not mean much for you to give this fact away on hands where you can’t win anyway. The big information leak occurs when you have a reasonable hand and choose to check instead of fold. Now everyone knows you are less likely to have nothing and more likely to have something. ♠

Steve ‘Zee’ Zolotow, aka The Bald Eagle, is a successful gamesplayer. He has been a full-time gambler for over 35 years. With two WSOP bracelets and few million in tournament cashes, he is easing into retirement. He currently devotes most of his time to poker. He can be found at some major tournaments and playing in cash games in Vegas. When escaping from poker, he hangs out in his bars on Avenue A in New York City -The Library near Houston and Doc Holliday’s on 9th St. are his favorites.