TOP #19 - Game Theory And Bluffing

[wallenborn]

Game theory attempts to discover mathematically the best strategies against someone also using the best strategies. Against an opponent you think is weaker than you are, you would usually rely on your judgement rather than game theory. However, against an opponent you think is better than you or against an opponent you don't know, game theory can sometimes enable you to overcome the other's judgemental edge.


Using Game Theory to Bluff

If you never bluff, your opponent's optimal stratey is to never call your bets. If you always bluff, your opponent's optimal strategy is to always call your bets. If your opponent calls too often, or not often enough, you can adjust your strategy accordingly. But against an observant opponent who adjusts to exploit weaknesses in your strategy, you should employ a mixed strategy: you should randomly bluff with a certain frequency. At this optimum bluffing frequency, your opponent has exactly the same expectation whether he always calls, always folds ot mixes these two strategies randomly: you are indifferent to him calling or folding. The optimum bluffing frequency is when your odds against you bluffing are identical to your opponent's pot odds.

Suppose you have a 20 percent chance of making your hand, there's $100 in the pot, and the bet is $25. Your opponent is getting 5-to-1 odds if you bet. The ratio of your good hands to your bluffs should, therefore, be 5-to-1. Since you have a 20 percent chance of making your hand, you should randomly bluff 4 percent of the time.


Adjusting to Your Opponents

Game theory tells you how to bluff against an opponent who is observant and smart enough to play optimally against you. Against a superior opponent this is a good assumption. Against an unknown it is a good first approximation. But if you notice systematical mistakes in your opponent's play, you should deviate from the game theoretical optimum. Don't bluff a calling station, value bet him instead. Don't rely on value bets against a weak-tight player, bluff him often instead.

Using Game Theory to Call Possible Bluffs

Against a good opponent you can't always fold to his bets. Obviously, you cannot always call, either. Just as there is an optimum bluffing frequency, there is an optimum calling frequency, too. You figure out what odds your opponent is getting on his possible bluff, and you make the ratio of your calls to your folds exactly the same as the ratio of the pot to your opponent's bet. If your opponent bets $20 to win $100, he is getting 5-to-1 on a bluff. Therefore, you make the odds 5-to-1 against your folding. That is, you must call five times and fold once. Of course, you must randomize your calls.



Older threads:

TOP #1 - Beyond Beginning Poker
TOP #2 - Mathematical Expectation and Hourly Rate
TOP #3 - The Fundamental Theorem of Poker
TOP #4 - The Ante Structure
TOP #5 - Pot Odds
TOP #6 - Effective Odds
TOP #7 - Implied Odds and Reverse Implied Odds
TOP #8 - The Value of Deception
TOP #9 - Win the Big Pots Right Away
TOP #10 - The Free Card
TOP #11 - The Semi-Bluff
TOP #12 - Defense Against the Semi-Bluff
TOP #13 - Raising
TOP #14 - Check-Raising
TOP #15 - Slowplaying
TOP #16 - Loose and Tight Play
TOP #17 - Position
TOP #18 - Bluffing

[wallenborn]

It's odd when you try to apply this to, say, c-bets, and look at the numbers. They're surprisingly small. Assuming your c-bet is pot-sized, then you make a legitimate bet when you connected with the flop, which is about 1/3 of the time. Giving your opponent 2-1 odds, you should bluff-cbet half of that. So your opponent will see you betting the flop 50% of the time, two thirds of which are legitimate, and one third bluffs.

Obviously, most people here c-bet more often than that. So these bets, to be successful, need some extra equity. A weak-tight opponent who check-folds too often, for example. Or a value component: when you open-raise from the button with AK, and the BB calls and then checks a ragged flop, you might well have the best hand. Or some disguise: if you bet a Q high flop with AK, you know that you missed. Your opponent doesn't know that, and he is in a RIO situation here, especially if you have position on him.

Mathematically, 50% c-bet rate is "free". Everything extra you have to earn.

[Klompy]

wallenborn,

We can't really apply this to c'bets can we? The way I read the OP it sounded like it would apply more to river bluffs.

[wallenborn]

i actually tend to think it applies to all kinds of bluffs. With the classic river bluff the strategic situation becomes easy, as does the math. Maybe cbets are a little murkier than other situations. But i would think every time you represent strength there should be a mixed strategy driving your action.

[HesseJam]

I don't think you can apply this to c-bets because the "hammer of future bets" is missing here. C-bets are more threatening because the raisor is likely to ask for even more.

That being said, I think an optimum of c-betting would be between 65 and 80% for the standard better player with a 14/8 to 22/12 profile. I cannot back this up at all, of course. Just a gut feeling.

[Red_Diamond]

Well I am glad that there is at least a LITTLE chapter dedicated to Game Theory here. Unfortunately I found it quite lacking, but perhaps this is due to having Chen's textbook in my library. I also understand that they had to put some sort of limit on the pages in TOP.

Which brings up another point, who would be up for a 'Mathematics of Poker' study group? :P