Game theory question--hold'em

Could someone please explain using game theory to bluff on the end in hold'em? Suppose you have a flush draw on the turn and your lone opponent has pocket aces; therefore, you have 9 wins out of the 44 unseen cards. There is $80 in the pot and the bet is $10, so your opponent is getting 9 to 1 odds to call if you bet. So, you should pick one card to bluff with to achieve optimum bluffing frequency. If your opponent calls all ten times you bet on the end, he will beat you out of $90 the one time you are bluffing, and lose $90 the 9 times I make my flush, for a total loss of $720. But he will win $80 the 34 times you don't bet for a total of $2720, so overall he makes $2000. What am I missing?

The positive number you found for the AA is because he has the best hand and surely retains the lion's share of the pot regardless what you do.


If you make the flush then he's already lost the pot; now we are struggling ONLY over your bet. If you don't make the flush and don't bluff, he's already got the pot. If you don't make the flush but bluff, then the pot and your bet are still in the air. So your bluffing frequency affects the odds on your bet and on the pot when you are bluffing. The rest is tied up only in the relative values of your hands (AA is clearly the favorite) and since you cannot do anything about that, it doesn't affect the math.


That's the correct bluffing frequency. If he CALLS each time he'll lose $10 9 times but win $90 once for a push. If he FOLDS each time he wins zero.


- Louie

Something to consider:


Hold 'em does not work like draw or stud - you can't use the last bet to randomize your hand. I the example above, the opponent knows you have a fluh draw or a pair of jacks. So if you are going to pick a card to bluff with, it had better make sense.


If you give your opponent credit for an overpair, try picking an ace. If it is likely he has an ace or aces, then pick a card that would like go with your hand if it were jacks - a ten would be a good card, as you could have JTs and made 2 pair. If he has been raising, it is unlikely a ten helps him, but it could help you. It may also be best to make your bluff a check - raise, but this is very oppponent dependent.


All this said, sometimes you should conclude that there is no good bluffing card. If you put him on aces or kings, you will probably be better off not bluffing.


The other games work differently because your last card is private, so you can more easily pick cards to bluff with. In stud, you might bluff with a card that likely makes your opponent trips, for example. You should never have to pick cards purely at random.


Good luck.


Dan Z.

Some things that may clear up this issue for you: One: when considering bluffing questions, it's important to remember that one constraint is that your opponent's hand can only beat a bluff which would rarely be the case against pocket Aces. Two: In the case of Hold'em your opponent will know that you can't have a flush because he can see the last card, so this calculation doesn't really apply. Three: Bluffing in Hold'em is more a function of what you have been representing, and what your opponent thinks you have. You should keep in mind the basic game theory principles when bluffing, but integrate them into the balance and flow of the game at that time . Typically, your feel for the situation will be a much better gauge as to whether you should bluff or not.

"All this said, sometimes you should conclude that there is no good bluffing card. If you put him on aces or kings, you will probably be better off not bluffing."


Not true. If the frequency is only 1 in 10 then that is even more reason to bluff if your opponent will fold when you bet. In fact in a case were the correct bluffing frequency is 1 in 10 and you are against an opponent that is quick to fold when you bet then you should bluff more often. Add another bluff card until he starts calling more often and then bluff correctly.


The only reason to pick a card that makes sense is if you want your opponent to call less than optimal. Otherwise if you are bluffing correctly then the card you pick doesn't mean a hill of beans.


Vince

Ed,


I wish I had read your response before I responded to Dan Z.


vince