simple game theory question
 

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?

Re: simple game theory question
 

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_----

Re: simple game theory question
 

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!

Re: simple game theory question
 

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 .

Re: simple game theory question
 

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...

Re: simple game theory question
 

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.

Re: simple game theory question
 

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

Re: simple game theory question
 

[ 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_----

Re: simple game theory question
 

[ 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.

Re: simple game theory question
 

[ 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_----

Re: simple game theory question
 

I mean your statement is FALSE.

Re: simple game theory question
 

[ QUOTE ]
I mean your statement is FALSE.

[/ QUOTE ]

A nash eq is defined where each player plays with the best response to his opponents strategy. Each player thus plays with an optimal strategy, because no player can gain by deviating.

A bad player is one who deviates, and therefore loses.

Hence, my statements have been correct.

Re: simple game theory question
 

[ QUOTE ]


Game theory is about how you would play against an opponent playing the optimal strategy. ....


[/ QUOTE ]

This is the garbage they teach in school.

If opponents were able to play anywhere near optimal strategy, this game is unbeatable. The rake is outrageously high.
Game theory is also about using best exploitive strategy against bad opponents. These bad opponents are oblivious to your strategy and will fail to exploit you.

Chess is a deterministic game. You don't need game theory for chess. In chess a good move against a good player is a good move against a weak player. You can't make this same statement in poker. In exactly the same situation it may be correct to call player A, fold to player B, and randomize against player C.

Re: simple game theory question
 

[ QUOTE ]
[ QUOTE ]


Game theory is about how you would play against an opponent playing the optimal strategy. ....


[/ QUOTE ]

This is the garbage they teach in school.

If opponents were able to play anywhere near optimal strategy, this game is unbeatable. The rake is outrageously high.
Game theory is also about using best exploitive strategy against bad opponents. These bad opponents are oblivious to your strategy and will fail to exploit you.

Chess is a deterministic game. You don't need game theory for chess. In chess a good move against a good player is a good move against a weak player. You can't make this same statement in poker. In exactly the same situation it may be correct to call player A, fold to player B, and randomize against player C.

[/ QUOTE ]

Your original argument was that game theory (as I defined it - a Nash Strategy) can't be used profitably against a bad player. I showed that was false.

You come back at with some silly argument about how rake would make it unprofitable to play against a perfect player. Obviously we wouldn't be playing poker if such players existed (in any significant number). A better argument would have been that a Nash strategy is unprofitable to play (given rake) against a bad player. That, i know you can't prove and i'm certain is not the case. A nash poker strategy is quite complex, and gives opponents plenty of room for mistakes.

There are many potential uses for a nash strategy. Besides being profitable in its own right, it can be used as a baseline for creating exploitative strategies, and as a default strategy while collecting data-points - which all exploitative strategies need because they rely on opponent modeling.

But unfortunately, even when begining my statements by saying that an exploitative strategy is better than a nash strategy against a flawed opponent, somone feels compelled to restate what I just said as if they were reinventing the wheel or something. And using that to argue that all discussion of Nash strategies are worthless.

Finally, if you want to call exploitative strategies game theory - go ahead. I try to avoid semantics arguments, as they are ultimately pointless. Its just in any gametheory , p1's strategy is generally a function of p2's strategy. Here, you made p1 exploitable because he doesn't adapt anymore. Because p1 doesn't adapt anymore, maximizing your payouts against him is a simple math problem that was generally understood and solved for before "game theory" as a field ever came about. Hence, when economists speak of game theory, they really aren't referring to these problems. They are referring to games with rationally adaptive opponents.

----_Dirty&Litigious_----

Re: simple game theory question
 

[ QUOTE ]


You come back at with some silly argument about how rake would make it unprofitable to play against a perfect player. Obviously we wouldn't be playing poker if such players existed (in any significant number). A better argument would have been that a Nash strategy is unprofitable to play (given rake) against a bad player. That, i know you can't prove and i'm certain is not the case. A nash poker strategy is quite complex, and gives opponents plenty of room for mistakes.



[/ QUOTE ]

You haven't a clue what's Nash equilibrium for hold'em. No one knows it. But we do know many of the most successful players deviate from it. You don't really think Ivey, Negreanu, or Hansen are using optimal strategy, do you?

In poker it's about using the best exploitive strategy. This strategy will blow up if opponents' knew what you were doing and exploited you. Still it's the best EV strategy.

Re: simple game theory question
 

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



[/ QUOTE ]

Change this a little. Fixed-limit lowball. A two-way pot, the other players have folded. After the draw. The wheel(5432A) is the best hand in lowball. Assume both players know how to play and both knows the other knows how to play. 3rd raise after the draw. You must have the wheel. With less than the wheel the 2nd raise is it. When a player puts in the 4th raise, the other players get mad at him for slowing the game down.

Re: simple game theory question
 

[ QUOTE ]
Game theory is often described as a branch of applied mathematics and economics that studies situations where players choose different actions in an attempt to maximize their returns. ...

[/ QUOTE ]

This is a much better definition of game theory, especially as it applies to hold'em.

Re: simple game theory question
 

[ QUOTE ]
Game theory is a branch of applied mathematics that is often used in the context of economics. It studies strategic interactions between agents. In strategic games, agents choose strategies which will maximize their return, given the strategies the other agents choose.

[/ QUOTE ]


http://en.wikipedia.org/wiki/Game_theory

words are just boxes to put ideas. this is the last response i'm making to any arguments over what "is" game theory. As wiki indicates, its where a players strategy is rationally adaptative to that of his opponent.

If u want it to have a broader meaning, i don't mind. I'm not on here to debate definitions. I can see u like to argue for sport, but maybe u can pick something more substantive than definitions. And even if u do stick that field, at least use accepted definitions....


----_Dirty&Litigious_----

Re: simple game theory question
 

[ QUOTE ]
You haven't a clue what's Nash equilibrium for hold'em. No one knows it.

[/ QUOTE ]
That's true. But we still have an idea of what it would look like. In this situation we know that the optimal strategy will have the probability of a player's hand being a bluff asymototically approach 0. Most of us have learned, from experience that players tend to deviate from this equilibrium by actually bringing that probability to 0 after 3 or 4 bets; and by doing this they are only deviating slightly, and the edge one could gain from this deviation is extremely small.

OP's question was about the optimal strategy, not the maximal strategy. The question you keep answering pertains to the maximal strategy.

Re: simple game theory question
 

[ QUOTE ]
That's true. But we still have an idea of what it would look like.

[/ QUOTE ]

Van nostrin is right. We can't solve for it, but we have a pretty good idea of what it looks like. This informs us of where opponents can make mistakes, even what the magnitude of those mistakes might be.

----_Dirty&Litigious_----

Re: simple game theory question
 

[ QUOTE ]

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

[/ QUOTE ]

this is a game theory question, so I am assuming no opponent modelling, which is to say we are playing unexploitably/optimally b/c we assume opponent is doing the same

Re: simple game theory question
 

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

[/ QUOTE ]

There's no one size fits all answer. It depends on the board on the river.
---
You: KsQs
Board: QdJs8s, 4c, 2s

You would be willing to maybe put in the 2nd or 3rd raise.
---
You: AhKs
Board: KdJs8s, 4s, 2s

Doubt that this hand is worth any raises.
---
You: AhJs
Board: QsJdJc, 4d, Qh

Opponent bets into you after being passive. This
is a crying call.

Re: simple game theory question
 

[ QUOTE ]
A. Game theory is about how you would play against an opponent playing the optimal strategy. ....

[/ QUOTE ]

[ QUOTE ]
B. Game theory is a branch of applied mathematics that is often used in the context of economics. It studies strategic interactions between agents. In strategic games, agents choose strategies which will maximize their return, given the strategies the other agents choose.

[/ QUOTE ]

Quote A and quote B are not the same. Quote A is wrong.
Game theory is finding strategies to maximize returns, not restricted to finding NE. Often the best strategy is NE, but not always.

Re: simple game theory question
 

the discussion about the maximal strategy is at least as theoretical and unpractical as talking about the optimal strategy

everybody talks about "blabla max strategy blabla equilibrium against donkey no good bla"

but often the strategy which is max exploiting is usually very extreme and players adjust to it the next couple hands.

(folding to 3bets more than you should and therefore making it plus ev to 3bet alone so you should raise every hand)

because you will play more than one hand against an opponent he is able to change strategies that go against your favor. this strategy is hidden and therefore it bears threat...

thats why in pracitce keeping "the match under control" is important, therfore not exploiting every single mistake...

this is somewhat like the optimal play.

of course no one has calculated a sequential nash equilibrium... but in theory there exist one and that is a pretty important thing to know how the game really works...

Re: simple game theory question
 

[ QUOTE ]
You haven't a clue what's Nash equilibrium for hold'em. No one knows it.

[/ QUOTE ]

This is something that can be solved by computational game theory (at least within some epsilon). However, computational resources are not at the stage yet in which the game of heads-up limit hold'em can be solved (a game with about 10^18 states). However, the most recent work in this area has solved an abstraction of the game with 10^12 states and then uses the solutions from that game as approximate solutions to the full-scale game.

Regret Minimization in Games with Incomplete Information

Check out the work done by the University of Alberta Computer Poker Research Group. I just started my Master's there in Computer Science. Their most recent pseudo-optimal heads-up bots are playable at Poker Academy and recently competed against Phil Laak and Ali Eslami in a Man-Machine competition.

Re: simple game theory question
 

[ QUOTE ]

Regret Minimization in Games with Incomplete Information


[/ QUOTE ]

Do you know these profs?

[ QUOTE ]
four rounds of cards being dealt, and four rounds of betting,

[/ QUOTE ]

Poker players usually say four streets of cards being dealt with three raises each street.

Re: simple game theory question
 

[ QUOTE ]
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?

[/ QUOTE ]

the first player bets.

The second player's action vector looks like this.

|______________call_______________|________fold___ ___|

When should the second player raise?

|_____raise___|_____call___________|___fold_______ __|br|

This pure bluff raise looks spectacular when it works.
But the math says never bluff raise from the fold area.

|____raise____|____call_________|br|________fold__ _____|

The bluff raise from the tail end of the call area does
better.

If that's what you meant by pure bluff, never pure bluff.

Re: simple game theory question
 

[ QUOTE ]

This pure bluff raise looks spectacular when it works.
But the math says never bluff raise from the fold area.

|____raise____|____call_________|br|________fold__ _____|

The bluff raise from the tail end of the call area does
better.

If that's what you meant by pure bluff, never pure bluff.


[/ QUOTE ]


jogsxyz, Snowball is asking a gametheory question. All of your advice categorically runs contra to gametheory. You don't bluff a hand that gets +EV from calling on the river.

In gametheory, on the river, u only bluff raise from the "fold area" - or more precisely, hands that do not show a call value greater than zero - no where else. The math says that...and its completely contra to your advice.

----_Dirty&Litigious_----

Re: simple game theory question
 

Hi nick,

Will you be focusing on poker-related research while earning your degree? Working with the poker research group in Alberta?

If you are...i got a problem or two i could use some input on [img]/images/graemlins/smile.gif[/img]

----_Dirty&Litigious_----

Re: simple game theory question
 

[ QUOTE ]
[ QUOTE ]

This pure bluff raise looks spectacular when it works.
But the math says never bluff raise from the fold area.

|____raise____|____call_________|br|________fold__ _____|

The bluff raise from the tail end of the call area does
better.

If that's what you meant by pure bluff, never pure bluff.


[/ QUOTE ]


jogsxyz, Snowball is asking a gametheory question. All of your advice categorically runs contra to gametheory. You don't bluff a hand that gets +EV from calling on the river.

In gametheory, on the river, u only bluff raise from the "fold area" - or more precisely, hands that do not show a call value greater than zero - no where else. The math says that...and its completely contra to your advice.

----_Dirty&Litigious_----

[/ QUOTE ]

Just do the math. Bluffing from the call region is higher EV than calling from the fold region.

Re: simple game theory question
 

The main author Martin Zinkevich (who came up with the new method) has since moved to work for Yahoo so unfortunately, he's not here anymore. The others, however, are all still here (although the Godfather and creator of the group, Darse Billings, is about to move to work for Full Tilt Poker). I attend their weekly Poker Group meetings.

Regarding the second question, on the next paragraph, he mentions exactly what you are talking about (what he meant by four betting rounds was that on each street, there is also betting, as some research previously has omitted one round of betting due to computational resource constraints):

[ QUOTE ]
Early poker abstractions [2, 4] involved limiting
the possible sequences of bets, e.g., only allowing three bets per round, or replacing all first-round
decisions with a fixed policy. More recently, abstractions involving full four round games with the
full four bets per round have proven to be a significant improvement [7, 6]. We also will keep the
full game’s betting structure and focus abstraction on the dealt cards.

[/ QUOTE ]

Re: simple game theory question
 

@ nickaboutrisk:

i just briefly looked over the paper but what is really new about and superior about that solution?

is it correct that you trade memory that a LP Solution needs with CPU power, but you won't reach a really optimal state because it is just an approximation (of the optimum of the approximated game)?

there might be the advantage that you calculate a sequential nash equilibrium that you won't always get if you use the Koller algorithm but i don't know (just flew through the pages of your paper) but there is a paper somewhere how to correct that...

anyways it doesn't perform really better than the Koller algorithm so therefore its not really a progress?

Re: simple game theory question
 

[ QUOTE ]
the discussion about the maximal strategy is at least as theoretical and unpractical as talking about the optimal strategy


[/ QUOTE ]

Not really. The OP's question is about a controlled environment. Only one street and static relative equities.
These solutions are known and solvable.

With less than 10bbs in NL the jam or fold charts are solutions to the headup game. The key is to reduce the complexity of the problem.

Re: simple game theory question
 

yeah i got that allready

i am just saying that if people mention that equilibriums have no practical sence in "real poker" and mention the concept of the maximal strategy they are also wrong because in todays poker its more about "keep the edge, don't let the villain adjust" and not about max strategies meaning exploiting his as hard as you can

i know that there are solutions for lots of pokerlike games and tourneysituations

and i know that its the key to abstract the game in lossless way to compute the nash equilibrium and i know that all that wasn't OPs question which has allready been answered and relies heavily on the definition of "pure bluff".

anyway i really think that all academic work so far has not focused enough about those abstractions because todays game theory solving is mostly done by computer scientists and not by economists and in computer sience its more about the algorithm than the acutal solving of the game so there has to be put in some serious research about correct abstraction since those Linear Programming Stuff is kinda shooting birds with cannons without abstractions..

all those guys trying to solve the game have different motivations than "solving the game"

basicly since koller's paper about the sequential form of games in 1995 there hasn't been a major breakthrough even though computers have evolved massivly.

Re: simple game theory question
 

[ QUOTE ]
yeah i got that allready

i am just saying that if people mention that equilibriums have no practical sence in "real poker" and mention the concept of the maximal strategy they are also wrong because in todays poker its more about "keep the edge, don't let the villain adjust" and not about max strategies meaning exploiting his as hard as you can

[/ QUOTE ]

Any true student of the game should learn nash equilibriums for one street games. But that doesn't mean he should follow them religiously. When opponents deviate even only slightly from optimal, often but not always, students would do better by using best exploitive strategy.

[ QUOTE ]

anyway i really think that all academic work so far has not focused enough about those abstractions because todays game theory solving is mostly done by computer scientists and not by economists and in computer sience its more about the algorithm than the acutal solving of the game so there has to be put in some serious research about correct abstraction since those Linear Programming Stuff is kinda shooting birds with cannons without abstractions..

all those guys trying to solve the game have different motivations than "solving the game"

basicly since koller's paper about the sequential form of games in 1995 there hasn't been a major breakthrough even though computers have evolved massivly.

[/ QUOTE ]

Those are just the wrong authors. RGP has threads by poker players. Ferguson, both Chris and his brother Tom, Weideman, Chen, Ankenman all contribute about the micro aspects of the game.
Still waiting for a paper on the two street, dynamic game. Don't think anyone has solved it yet.

Re: simple game theory question
 

first: i of course know you should adjust to an opponent to exploit him if he plays weak. this has been said OVER AND OVER by various posters including me. i am just saying that its highly inpracitcal to exploit him maximaly because max strategies usually are very extreme (the example where preflop 3bet with every hand made earlier) and the opponent will adjust so its more about "keeping the edge without the opponent adjusting" as i said before. so exploit him, but not as hard as you can...

and no these are not the wrong autors...

the papers of ferguson, weidman chen und ankenman are very very basic models

there are forms of pokers that are already solved (2 streets and more). you are really behind, the group the guy has postet a paper from are far further, there is also gilpin and sandholm who have solved a form of poker called "rhode island poker" which is a simple hold'em

basicly if you want to solve big games you need that koller algorithm, and you'd have to solve it with linear programming

ferguson and the others haven't really developed something new, these [0,1] models are known since 1944 and they are even solvable by hand.

sience is really WAY AHEAD of this.

there are multiple solutions of pretty big dynamic games

Re: simple game theory question
 

http://www.cs.cmu.edu/~sandholm/RIHo...roceedings.pdf

This paper only gives an overview for solving RI holdem.
There's no micro details.

Ferguson explains the painful step-by-step process.

-----------------

On this exploiting thing on the final bet the call/fold
decision is completely extreme exploitive. If the
bettor overbluffs even slightly, you should call
100% of the time(assumes you actually hold a hand
that can beat a bluff). If the bettor underbluffs,
you should fold 100%.
It's only best to use optimal strategy(mixing calling
and folding) against an unknown opponent or a very
strong opponent capable of mixing his strategy.

Re: simple game theory question
 

[ QUOTE ]
http://www.cs.cmu.edu/~sandholm/RIHo...roceedings.pdf

This paper only gives an overview for solving RI holdem.
There's no micro details.

Ferguson explains the painstaking step-by-step process.

-----------------

On this exploiting thing on the final bet the call/fold
decision is completely extreme exploitive. If the
bettor overbluffs even slightly, you should call
100% of the time(assumes you actually hold a hand
that can beat a bluff). If the bettor underbluffs,
you should fold 100%.
It's only best to use optimal strategy(mixing calling
and folding) against an unknown opponent or a very
strong opponent capable of mixing his strategy.

[/ QUOTE ]

Re: simple game theory question
 

there is other exploiting than just calling a bluff on the river which happen every 50 hands...

there is also often exploitable spots in preflop play or flop play (fold to often) that makes it the maximal strategy to raise 100% of your hands. you dont want to do that because your opponent will adjust very fast...

@ gilpin and sandholm papers: go to gilpins homepage, there are 3 papers with technical reports and proofs and everything, you can even play the optimal solution.

Re: simple game theory question
 

Haven't read all the responses, but I don't see how there could be a theoretical cap. If there were, than the last bet made would be exploitable.