Thoughts on Game Theory: Part 6

The most frequent application of game theory to poker occurs when trying to determine optimal frequencies for bluffing and value betting. These also lead to optimal frequencies for calling when you can only beat a bluff.

The simplest cases to analyze revolve around heads-up river play because there are no more cards to come. The player with the best hand will win the pot unless his opponent bets and he folds. There is one more simplification we need to make in order to do game theoretic calculations. Value bets always win when called, but bluffs never win when called.

Obviously, in real life, these situations are relatively rare. A typical example might be a hand where you have the A and another card. Your opponent has K 4. On the river, the board is Q J 8 7 2. If your other card is a spade you will win. If it isn’t, you won’t win unless your opponent folds.

Your opponent checks, and you have to decide what to do. Clearly you always want to bet when your other card is a spade – your value bets. You also want to bet some of the time when it isn’t – your bluffs. There is 100 in the pot. You have 200 behind, and your opponent has you covered. If you decide to bet, either for value or as a bluff, you also have to decide what amount to bet.

When you are the player facing a bet, and you have a marginal hand or a bluff catcher, you have to call some of the time. If you don’t call often enough, your opponent can bet with any two cards and show a profit. This is often referred to as your minimum defensive frequency.

I will present a chart with all this information a little later, but first let’s examine one situation to see how these numbers can be calculated.

You decide to make a half pot sized bet on the river with a polarized hand. You either have the nuts or a bluff. According to game theory you should bluff with a frequency that makes you indifferent to what your opponent does. If he always calls, always folds, or choses a mixture of the two, you will have the same result.

Your bluffing percentage can be calculated by a simple formula. Bet size divided by the sum of the pot size plus twice your bet size. If the pot is 100, the formula is 50/(100+2x50) = .25. If you plan to make a half-sized bet, you should bluff 25 percent of the time and value bet 75 percent of the time. Now how often must a half pot bet be called to avoid the bettor showing a profit by betting any two cards? The minimum defensive frequency is determined by subtracting the bet divided by the pot plus the bet from one. In this case the MDF is 1-(50/150) = 2/3 or 67 percent. Don’t worry about doing the calculations, here is the chart I promised you.

Bet Size
As % of Pot Value Bet % Bluff % MDF %
25% ¼ pot 83% 17% 80%
33 1/3 pot 80 20 75
50 ½ pot 75 25 67
67 2/3 pot 72 28 60
100 pot size 67 33 50
150 1 1/2 pot 62 38 40
200 twice pot 60 40 33

It is probably a good idea to memorize these numbers, but it is even more important to get a feel for what they mean. When you are making small bets, you should be making mostly value bets. This is why you should seldom bluff on the river in limit poker games. Likewise, when you are making big bets, you can bluff a lot more.

If you are thinking of calling with a marginal hand or bluff catcher, call small bets very frequently and large bets only when the situation seems clear. I frequently hear commentators get this wrong. They’ll say something along the lines of it being right to bet small, when you have a wide range. Against good opponents this is the opposite of what game theory says you should do.

Versus weak opponents, however, using an exploitative strategy is fine. If they fold 40 percent of the time to your bet of one-quarter of the pot, you’ll show an immediate profit betting any two cards. The chart shows that you must call at least 80 percent of the time to prevent this play from being profitable. ♠

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.