optimal bluffing frequency
 

we know that the optimal bluffing frequency is s.t the odds against bluffing are the same as the pot odds the opponent receives. At the optimal bluffing frequency it doesn't matter if the opponent always calls or always folds. But suppose the opponent wavers somewhere in between (i.e sometimes calls, sometimes folds). Will he not be playing optimally? how does 1 prove that the opponents optimal strategy is to always call or always fold?

thanks.

Re: optimal bluffing frequency
 

If you are bluffing at the optimal rate as you describe, it makes no difference to the overall result whether your opponent always calls, always folds or mixes both in any ratio you like.
However, if you happen to know that they are more likely to call more, you should bluff less, and that if they are more likely to fold more, you should bluff more. The matching odds scenario is a hedge for where you dont know what your opponent is likely to do, so you make their call/fold decision irrelevant.

Re: optimal bluffing frequency
 

IMHO, the concept of an "optimal bluffing frequency" is more or less purely theoretical. In other words, it is not at all like pot odds or the like because you cannot say "well, I have not bluffed in x number of hands, so now is the appropriate time".

As we all know, poker is a game that is part math, part intuition - you cannot get by without both - and bluffing is almost all intuition. The SIZE of your bluff is a mathematical issue, but deciding when to bluff is much more a matter of reading your opponents and determining when the time is right for a successful bluff.

Also, IMHO, bluffing in limit games is a dangerous proposition - I very rarely bluff in limit games - but to compensate, I will semi-bluff more often, depending on the game.

Re: optimal bluffing frequency
 

Against decent opponents, I find it is often useful to vary your bluffing (or semi-bluffing) frequency. If your opponents are folding nearly every time you bet, you (semi-)bluff frequently. when they begin to call down or play back at you, you tighten up and value bet. Of course, against calling stations, you should bluff rarely, if at all, and against rocks, you should bluff a lot.

Re: optimal bluffing frequency
 

[ QUOTE ]
we know that the optimal bluffing frequency is s.t the odds against bluffing are the same as the pot odds the opponent receives. At the optimal bluffing frequency it doesn't matter if the opponent always calls or always folds. But suppose the opponent wavers somewhere in between (i.e sometimes calls, sometimes folds). Will he not be playing optimally? how does 1 prove that the opponents optimal strategy is to always call or always fold?

thanks.

[/ QUOTE ]

The opponent's optimal strategy isn't determined by your play at all, remember. He calls with some fraction of hands so that you are indifferent to bluffing at a particular threshold hand.

If either of you play a mixed strategy with any hand, then the equity of playing that hand in each way against the opponents' optimal strategy must be equal. So in a sense, if you will continue to play optimally no matter what your opponent does, he can call or fold with his threshold hand as he wishes. (In fact, there's a fairly large region of hands that are like this in most examples like this).

But if he just folds all his hands, you can exploit him. BTW for those of you who do study games, this fact is pretty useful for finding the value of games once you have the strategies. You can just pretend that one side just folds at all mixed strategies, which usually trims the game tree a lot.

Jerrod Ankenman

Re: optimal bluffing frequency
 

There is another approach to this. The old rule was, bluff after two hands that you represent as strong are not called by anyone.

Obviously you were not supposed to do this in a predictable pattern, but the idea was to keep the 2:1 ratio. Typically, you didn't bluff at all until you got some respect for your raises, once that happened twice you started looking for good opportunities to bluff. You adjusted the frequency so that the ratio was intact.

The idea of this approach is to automatically adjust your bluffing frequency to your opponents' play. With luck, it got you out of phase with them so they called your strong hands and folded to your bluffs. Of course, you risk the reverse. So it's not optimal in game theory terms, but it can play pretty well. Also, it keeps you focused on the reason for bluffing so you don't do something silly like bluff because you're mad at having lost a showdown.

Another thing to be careful about is opponent selection. You might get two strong hands when the loose players had very weak hands and folded, while the tight players folded because they're tight. This could indicate the need to run a bluff, but you want to make sure you run it against a tight player, not a loose one.

Re: optimal bluffing frequency
 

If I have taken down a couple of pots without showing, I tend to tighten up a bit, expecting that someone will think I've been stealing and will be suspicious of my bets.

I'm most likely to bet on a mediocre hand (or bluff if I miss the flop) when I have been folding a lot of hands preflop - when you don't play many hands, they're readier to believe you have a great hand.

On-line I never bluff into a crowd, but it can work in live games if you can have a good read that people are ready to muck their hands.

Re: optimal bluffing frequency
 

im approaching this purely from a game theory perspective; i'm studying a game, trying to find the optimal strategy for both players. I made the assumption that given a certain hand, the 2nd player always calls or always folds. i want to know why I can make such an assumption? it sounds like you were getting to that, Jerrod. Or can i justify this by saying that given that the first player is playing optimally, it doesnt matter if the 2nd player always calls, always folds, or wavers in between. So WLOG, assume he always calls or always folds.

Re: optimal bluffing frequency
 

[ QUOTE ]
im approaching this purely from a game theory perspective; i'm studying a game, trying to find the optimal strategy for both players. I made the assumption that given a certain hand, the 2nd player always calls or always folds. i want to know why I can make such an assumption? it sounds like you were getting to that, Jerrod. Or can i justify this by saying that given that the first player is playing optimally, it doesnt matter if the 2nd player always calls, always folds, or wavers in between. So WLOG, assume he always calls or always folds.

[/ QUOTE ]

Wait, if you're trying to find the optimal strategy for both players, then both players need to maximize their minimum value. So like if one guy bluffs 1/(p+1) as much as he value bets and the other guy folds all the time, that's not an optimal strategy pair, because the first guy can improve his equity by bluffing more often. But if the second guy calls with p/(p+1) of his in-between hands, then the first guy can't improve by bluffing more or less; he's indifferent. So that is an optimal strategy pair. (in the typical easy one street game). If either player can improve their equity unilaterally by changing their strategy, then the strategies aren't optimal.

Jerrod

Re: optimal bluffing frequency
 

Bluff if it
1. makes you money
or
2. makes you lose a bit of money but confuses opponents enough for you to make more money with your good hands.

Keep in mind opponents have to be decently intelligent for #2 to work and I wouldnt suggest assuming that if you are playing low steaks on party.

Re: optimal bluffing frequency
 

no such thing bluffing should be psycological not mathematical

Re: optimal bluffing frequency
 

its not that the 2nd player always folds regardless of his hand, its that he always folds given that he has this specific hand. In that case, if we claim 1/(p+1) to be the first guy's optimal bluffing frequency, the argument that he should increase this frequency if he knows that the 2nd player will always fold with hand X doesnt hold true.

is it not a fact of game theory that in a "one street game" that the first player's optimal bluffing frequency is such that the odds against him bluffing are the same as the pot odds the opponent receives? Also, that regardless if the 2nd player always calls or always folds with a certain hand, hand X in this situation, his EV will be the same? it's claimed to be true in the game theory section of TOP, but there's no proof, just a numerical example. if it is true can you lead me to a link with a proof?

"If either player can improve their equity unilaterally by changing their strategy, then the strategies aren't optimal."

i understand this, but is it true at the first player's optimal bluffing frequency, given hand X, where X is a specific inbetween hand, it doesnt matter what the second player does, he will have the same EV regardless?

thanks.

Re: optimal bluffing frequency
 

There's no reason for the last actor in any game to randomize his strategy. He should take the action with the best expected value. In early stages, you sometimes randomize to affect future decisions by your opponent. But never at the end. That's why it's usually easiest to solve game theory problems backward from the end.

Re: optimal bluffing frequency
 

I agree with that. The point of the "two pots" rule is that you have taken down a pot without showing, then it happened again. Sooner or later, they have to call you or you have to start bluffing. The trick is to get them out of phase, so they call when you're not bluffing and don't call when you are.

Re: optimal bluffing frequency
 

in the problem im working on, i didnt have the 2nd player randomize his strategy. In the game im working on there's only 3 possible hands, and it's a one-street game. So i made the assumption that given the 2nd best hand, the 2nd player always calls or always folds, and he will pick the higher of the two EVs, Knowing this, the first player will try to minimize that EV, which occurs when the two EVs are equal. I want to know if my original assumption that the second player always calls or always folds with the second best hand can be justified?