• Bronze
Joined: 15.06.2009
Below is the answer to the GTO Quiz in the now closed thread with that name. Difficult questions tend to stir up feelings when posed and not being understood. I was hoping for some interesting discussion, but little of that happened before hell broke lose.

I'll post the original question in the next post (you can look it up in the closed thread in the meanwhile).

***

The answer is that one should fold 33.377748167888074616922051965356...% of the time.

Bizarre huh? Getting pot odds of 15001:1...

To see this, one must consider the problem from two perspectives, ask oneself a couple of questions, and realize the answers to those questions. One must also make two more realizations.

The first question to ask is: How much is the opponent really bluffing here?

The answer to this question is straight from the standard "simplest scenario". This can probably be found in one of our articles. Since he is, initially, playing GTO, and since he is presenting us with pot odds of 15001:1, the odds against him bluffing is 15001:1. (This isn't very often.)

Now a realization. Since the opponent is (still) playing GTO, it doesn't matter how much one calls or folds with the bluff catcher from this initial perspective. One can call 100%, 50%, 0%. The result is the same.

I'll prove this explicitly. It is an easy EV calculation. Take a sample of a very large multiple of 15002 situations. Suppose one calls 100%. One loses \$1 15001 times (times the multiple) and win \$15001 1 time (times the multiple). All on average of course. Ergo EV = 0. That EV = 0 for folding 100% is immediate.

Calling X% is just a tiny little bit algebra. This general case EV for calling X is X*EV for calling 100% + (100 - X)*EV for calling 0% = 0 + 0 = 0. QED.

***

Realization: If I call 100%, my opponent will sooner or later realize this. He can - and will - exploit me. For always folding, we have the same. (He'll notice that whenever we call, we have more than a bluff catcher). Since this game is going to be played for a long time, we must find the optimal answer that cannot be exploited.

Question: What is a suitable percentage to call in order to be unexploitable?

Answer: Once we call exactly so much that his bluffs break even, then he can find no way of exploiting us.

Proof: If his bluffs profit, he can bluff at every opportunity and exploit us. If they lose, he can exploit us by never bluffing (we always pay off his value hands with an extra dollar).

Note too, since it doesn't matter for us whether we call 0% or 100%, whenever his bluffs profit, his value part loses and vice versa.

Almost done. Then how much exactly?

Answer: When he bluffs, he invests \$5001 to win \$10 000. He must succeed (standard formula) 5001/(5001 + 10 000)*100% = 33,377748167888074616922051965356...% of the time. Thus we fold that percentage of the time. QED

***

Conclusion: When playing a very good balanced player, you must not look yourself blind at the pot odds you get for a call when holding a bluff catcher when faced with a river raise.

In fact, the pot odds should have zero impact in a totally polarized situation on part of your opponent. It is the opponents investment in a possible bluff that counts.
• 17 replies
• Bronze
Joined: 15.06.2009
The OP in the now closed thread.

***

Some will find this easy (and often be deadly wrong), others will find it quite hard.

Your task: You are going to give an input parameter to a computer program that is going to play a quadrillion (10 to the power of 15 = a million billion) hands against another program.

Note: The quadrillion hands may be replaced with a possibly larger number for it to be statistically significant, but this is of of no particular importance for the problem at hand.

The situation you get to control is the following: On the river, there is \$5000 in the pot. You bet with a pot-sized bet, presumably for value. There is now \$10 000 in the pot.

Your opponent now moves all in on you. To your amusement, he raises with \$1. (You cover him )

There is now \$15 001 in the pot, and it will cost you \$1 to call. You have a hand that can only beat a bluff and it beats all conceivable bluffs.

The question is this: Should you call? Should you FOLD You are allowed to sometimes call and sometimes fold. If you chose this option, then you must present a percentage. Round off to one decimal place.

Parenthesis: Normal hands play out as well. These can be totally ignored because you only get to set precisely the parameter I have already defined. (In normal play, the competing programs are exactly equal.)

There is one more premise. Your opponent (the player actually giving a corresponding input parameter to the opposing computer program) is a GTO expert, but knows a little bit about exploitation as well. He may change his parameter as often as he wishes, but you may not.

If after a quadrillion played hands you are ahead, you get the dough (and if roughly break even (this takes care of a possible round off error in a possible percentage given), \$100 000 000 will be paid to you). A loss costs you nothing.

Now, call, fold, or a mixture, and if so - give the percentages.

• Bronze
Joined: 22.01.2015
He must succeed (standard formula) 501/(501 + 10 000)*100% = 33,377748167888074616922051965356...% of the time.

I believe you meant to write 5,001/(5,001+10,000)*100% =33.34%

If we call just enough to defeat his potential rebluff equity, won't that mean even stevens, hence we do not win the prize?
• Bronze
Joined: 15.06.2009
Thanks for the correction 501-5001 in ten places allover. Was awfully tired when I wrote that - or Microsoft decided to make 5001 to 501 when I pressed save.

Yep, even stevens. That's what you can achieve vs GTO - no more. But even would have been good enough for the prize.
• Bronze
Joined: 15.06.2009

Okay, villain must succeed 33.37% of the time for the bluff to be BE, so far clear.

Why does that imply that we must fold 33.37% of the time?

If we don't, then we deviate from GTO. Villain will notice and will adjust. According to the rules of this game, we can't readjust, and either villain will profitably always bluff or never bluff depending on how we deviate.
• Bronze
Joined: 01.06.2014
doesn't this answer ignore the fact that if you incentivise your opponent to stop bluffing then you will have many hand repetitions where your opponent just folds river? Or did you just not clarify this point for confusion - ie. we ignore those hands - even though they will give us higher overall profit but not maximizing profit in the hands where villain goes all in?
• Bronze
Joined: 15.06.2009
I don't understand what you mean, but whatever you mean you are wrong because you talk about improving your profit. This is impossible. You can break even and that's it. You break even if you use the percentages I gave.
• Bronze
Joined: 01.06.2014
Originally posted by YohanN7
I don't understand what you mean, but whatever you mean you are wrong
gg

obviously no point in me pressing the issue.
• Bronze
Joined: 15.06.2009
Please don't quote half sentences of mine. You took away the point and changed its meaning.

What I am saying is that deviation from the given percentages makes you exploitable. By calling with the given frequency you are unexploitable, just like the opponent. It is a Nash equilibrium.
• Bronze
Joined: 01.06.2014
Originally posted by YohanN7
Please don't quote half sentences of mine. You took away the point and changed its meaning.

What I am saying is that deviation from the given percentages makes you exploitable. By calling with the given frequency you are unexploitable, just like the opponent. It is a Nash equilibrium.
look i can't retype my original question/comment any clearer than it already is right there
• Bronze
Joined: 28.01.2012
Well that's 2 mins of my life I'll never get back. Hopefully I can play better in those spots when villain jams and I get 15k:1 pot odds tho.
• Bronze
Joined: 15.06.2009
@NothingIsForSho

I did try to decipher your OP. You are trying to exploit someone playing GTO. There is a lack of understanding here. GTO cannot be exploited.
• Bronze
Joined: 30.04.2009
just to say, I found it an interesting theoretical puzzle, even tho' I didn't post any answers

• Bronze
Joined: 15.06.2009
Thank you sherriffatman, at least one poster does not hate the thread.

It is not only a logical puzzle, it has something to say about NL Hold'em in real life. When you get raised on the river, you should not primarily consider the pot odds, but rather the opponents investment in a possible bluff. Looking only at pot odds makes you into a calling station.
• Bronze
Joined: 15.06.2009
Originally posted by metza
Well that's 2 mins of my life I'll never get back. Hopefully I can play better in those spots when villain jams and I get 15k:1 pot odds tho.
Well, you never were any of the brighter figures around here, and abstraction is something not everybody is intellectually equipped to deal with. It takes talent (but not much really) or maturity or even both.
• Bronze
Joined: 01.06.2014
I understand the answer better after thinking about it more and the point is that it makes no difference either of what the breakdown of the villain's nut/bluff ratio is or how he can choose to construct his all in/check back range.

These aren't obvious if you haven't thought about it before, and saying "it's the right answer because it's GTO" (ie. because it's the right answer) isn't going to help you if you haven't already figured out the details yourself.
• Bronze
Joined: 15.06.2009
Game Theory Optimal is a strategy that cannot, by definition, be exploited. The strategy exists for all poker games. This was proved by John Nash about fifty sixty years ago. If two or more players are playing GTO, then it is impossible for any of the players to change his strategy as to make a profit.

The opponent in this case is playing GTO, so there is nothing we could do that would change the outcome (I prove this in the OP for the case of calling 0 or 100% and anything in between). The only thing we can achieve by, say calling more or less than we should, is that we would become exploitable because now our opponent can change his strategy to get an edge. Also, it was part of the problem that we cannot change strategy, but our opponent can.

One should also not fall into the trap thinking that the opponent will not bluff for a while if we caught him bluffing. The odds for him bluffing are always 15001:1 against - as long as he sticks to GTO. He can adjust and bluff zero or plenty, we cannot readjust as part of the rules.
• Bronze
Joined: 22.01.2015
Anyway, I still maintain that this scenario cannot be programmed and ran today.