simple game theory question

[SNOWBALL]

Playing heads up limit hold em - how many bets max can it be optimal to go without the nuts on the river?

Same game - How many bets max can it be correct to go with a pure bluff on the river?

[_D&L_]

I do not believe there is a theoretical cap.

I'll give u a no-limit exampe first:

Pretend you have AdKs and 3 out of 5 cards on the board have diamonds on it, with no straight flush possibilities. If your Ks was any diamond card, u would have the absolute nuts, but you don't - and pretend u got nothing else (no pair, etc.)

If the pot was $100, it would be correct to go all in with a deepstack here, even if both you and your opponent had trillions in chips. You know your opponent cannot hold the nuts - you know he must fold. And as long as you play your value hand the same way, and you don't bluff more than 49.999% of your hands in this situation, your opponent would be a fool to call you.

Now applied to limit, the same theory I think holds - but just barely. In game theory, every value bet has to be coupled with a potential bluff. Since we would never stop betting the nuts, in theory, we would never totally stop bluffing when we are representing the nuts.

Now for the VERY BIG game-theory exception to this rule. Unike no-limit, the pot is capable of growing very large in relation to any future re-raise. As the pot grows, our opponents pot-commitment grows. Thus, we don't need to bluff him as often, to force him to call us down.

This means, that every time he raises us back, we should drop a certain percentage of our bluffs, re-raise our nut hands and the remaining percentage of our bluffs. Our bluff percent can never hit zero, because we never stop raising with the nuts here. What i mean by our bluffs never hits zero - is that our bluff percentage is asymptotic to zero. Meaning it gets infinitely close to zero, but never quite equals it.

Ok - before I get flamed by a bunch of people who purport to understand game theory, let me say this. This was ONLY a game theory analysis. Obviously a human would feel pot committed at a crtain point even with a second best hand and would always call a limit-reraise. A computer programmed to play poker optimially though would be capable of in-human like laydowns.

P.s. The above examples i used where there was no theoretical limit only hold where your bluff hand rules out the possibility of your opponent holding the nuts when you bluff. If you can't rule that out, gametheory would impose a theoretical cap.

----_Dirty & Litigious_----

[PhlegmWad]

wow that's a good question - Im not sure that it's possible to define that ... my best guess is that it's between 1 and 4!

[jay_shark]

This is not a simple game theory question at all .
I don't even think any of the Putnam contestants can solve this problem in the alloted time .

This type of problem is already challenging when we restrict the game without any raises .

[Paxinor]

D&L is correct...

in game theory with no more cards to come there are basicly just pure bluffs and valuebets.

you will pick your worst cards to bluff at the pot and check behind medium strength hands...

basicly if the definition of pure bluff is that no worse card will call you then there is never a cap because you will always be bluffing at least a small portion.

if you define "pure bluff" to a special hand strengh meaning no pair or whatever the question is not really answerable because it depends on the flop and the action before...

[Yepitis]

Once you are reraised in a limit game why would you take it any further with no chance of winning?
If he has called and raised a couple of times why would he ever fold instead of just calling?
Unless you are saying he is on a bluff also and you both believe this and are both just trying to get each other to fold so you don't have to split the pot since neither of you can beat the broad.
I play low limit so I would never raise on the river and fold to one more bet no matter what I had. I would just use that fact I couldn't beat his 4 high as table image...heh.

[jstill]

depends on the player... some players some boards going 5 bets would be -EV other players or boards not so

shoot some examples Id say, itd probably be a more productive discussion

3 bet bluffing the river is somethin i rarely rarely do and probably never in HU where no one raise folds the river and everyone raise calls (depends I guess) 4betting the river on a bluff is not somethin ive ever done

the example theoretically has come up before of huge pot both players know eadh has nothing... the answer comes back if both are good they will call with the range of K Q hi hands instead of bluff reraising so going off for tons of bets in this scenarios would be sub optimal by both players

[_D&L_]

[ QUOTE ]
Once you are reraised in a limit game why would you take it any further with no chance of winning? If he has called and raised a couple of times why would he ever fold instead of just calling?


[/ QUOTE ]

You are correct that most every human player would have entered "call down" mode at this point. But from the point of game theory, call down mode is not an optimal strategy. It acts as an overdeterrent to bluffing, and at the same time pays off big hands.

Game theory is about how you would play against an opponent playing the optimal strategy. Thus, u are correct, u wouldn't use these ideas versus a human.

A game-theory optimal player is balancing at least two considerations here. Minmaxing the EV of his opponent hands that beat him, and Minmaxing the EV of his opponents bluff hands. The more he folds, the less EV he gives to hands that beat him, and the more he calls/re-raises, the less EV he gives to bluffs.

An optimal player who is not holding the nuts would not enter call down or no-fold mode. He would always fold his hand some percentage of the time. He doesn't need to call with his hand 100% of the time to deter bluffing.

In fact, he can even make bluffing the prefered choice for a few of our really trashy hands (hands with virtually no call value), and at the same time deter us from bluffing with moderate hands (hands that get higher EV from calling, than bluffing).

Thus, against an optimal player we do continue to bluff some percentage of the time (in some cases, near zero), but never quite zero percent of the time.

Against a human player, this strategy of always coupling value bets with bluffs may not be needed on big pots. But it wouldn't keep us from stealing the human players money, we would just be less efficient at it.

Think of it as trading mistakes - we bluff at a person who always calls, but we have more value hands than bluff hands, thus his mistake is bigger than ours.

----_Dirty&Litigious_----

[jogsxyz]

[ QUOTE ]


Game theory is about how you would play against an opponent playing the optimal strategy. Thus, u are correct, u wouldn't use these ideas versus a human.


[/ QUOTE ]

If that statement were correct, you wouldn't be able to use game theory against a player who didn't know how to play.

[_D&L_]

[ QUOTE ]
[ QUOTE ]


Game theory is about how you would play against an opponent playing the optimal strategy. Thus, u are correct, u wouldn't use these ideas versus a human.


[/ QUOTE ]

If that statement were correct, you wouldn't be able to use game theory against a player who didn't know how to play.

[/ QUOTE ]

Yes and no. If your opponent doesn't now how to play chess, you can use the three-move checkmate to beat him. But that doesn't say anything about Kasparov's strategy other than a more efficient strategy might exist for this opponent.

A player that doesn't know how to play will do worse than a player using an optimal counter-strategy. So we still win versus bad player using game theory.

----_Dirty&Litigious_----