Thoughts On Game Theory: Part 10

The last few columns have discussed in some detail what you should do heads up on the river in position after your opponent has checked. Basically, we divided this situation into three subdivisions or buckets. In general, the procedure of subdividing complex situations makes impossibly complicated analysis more manageable.

1. Hands to bet for value
2. Hands to bluff (assuming a pot sized bet, there should be about half as many hands as 1.)
3. Hands to check back (hands in this bucket range from hands that might win a showdown, but aren’t worth a value bet down to garbage in spots where you don’t think your opponent will fold often enough to make a bluff profitable.)

Now let’s see what game theory tells about a slightly more complicated situation. This is when you are heads up on the river and out of position. Once again, we will divide the hands into three buckets:

1. Hands to bet for value
2. Hands to bluff (Assuming a pot-sized bet, there should be about half as many hands as 1.)
3. Hands to check (The third bucket has now become more complicated. More on this will follow.)

Before we go into more detail on bucket 3, let’s examine value betting a little more closely. When you were last to act, you only wanted to bet hands that would win more than they’d lose if he called.

For example, you feel that both your hand and your opponent’s hand are fairly weak so there is no danger of a raise if you bet. You think that of all the possible hands he might have, he will call with the best nine. Of those nine, if you can beat five or more, you should bet. If you lose to five or more, you should just show down your hand.

When you are first to act, this changes. When you check, you don’t get an automatic showdown. He has the option to bet. It is sometimes right to bet even when you will lose more often than you will win.

Let’s look at the case where he will call with nine hands, five of which beat you, but if you check he will only bet the best eight of these hands. If you bet, you lose five bets and win four. Your net result is a one-bet loss over nine hands. If instead, you check and call, you still lose five bets to the hands that beat you. Now you only win three bets when you have the best hand, since he doesn’t bet the worst hand with which he would have called. So, if you check and call, your net result is losing two bets. Obviously, folding is out of the question since you don’t want to give up the pot to save a fraction of a bet.

It is also possible, that he has a lot of really weak hands that would never call a bet. But since he suspects you are weak, he will bluff a lot more often than he should. Remember game theory taught us that he should have about half as many pot-sized bluffs as value bets. If you think he will bluff about half the time, which is way more than he should, you should check and call.

By now I have really entered into the discussion of the third bucket mentioned above, which is hands that check out of position. We can divide hands that check into three sub-categories:

1. Check with the intention of folding
2. Check with the intention of calling
3. Check with the intention of raising

As you know by now, game theory always recommends balancing your strategy, so your opponent can’t exploit you. In small-stakes games, weak players will frequently check-fold too often.

In an earlier column we discussed the concept of minimum defensive frequency (MDF). If your opponent makes a pot-sized bet on the river, you must defend at least one-third of the time or he can make a guaranteed profit betting every hand, even if he never wins a showdown.

Here is a simple example: The pot is 100 and your opponent bets 100. Let’s say you only defend 20 percent of the time, but you win every time you defend. He is risking 100 to win 100 with his worst hands. He wins 80 percent of 100 minus 20 percent of 100, for a net profit of 60.

When looking at MDF, there is one important caveat. Your opponent must also be playing a relatively balanced strategy with an appropriate mix of value bets and bluffs. If your opponent never (or seldom) bluffs, then it can be correct to fold almost every hand.

If you play against weak opponents, who seldom bluff, you are much better off abandoning game theory and following the exploitative strategy of folding most of the time. As a strong player, a lot of your profit will come from exploitation of weak players. They don’t bluff as often as game theory says they should, so you can fold more. They are used to playing against players who don’t bluff enough, so they fold too often. Exploit them by bluffing more often than game theory says you should.

In the next column, we will go into more detail on creating a balanced checking range when out of position, and also discuss pros and cons of game theoretic versus exploitative strategies. ♠

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.