Answering My No Limit Question Plus Two More

[David Sklansky]

Again the situation is that you are headup on the river, first to act. I'll be more precise this time and specify that you have a made hand and know that your opponent is on a draw and will never call if he misses and always beat you if he doesn't. Also his draw is less than 50% to get there (a fine point that has relevance to a problem I will be posing). In limit there is no good reason to bet. (Unless he is a maniacal raise bluffer). Your last bet cannot make you money and cannot be a better strategy than always checking and calling. But there can be a good reason in No Limit.

The situation can occur when a check on your part elicits both a large bet when he makes his hand and an equally large bet some of the time when he misses. In other words he either checks behind you or bets big approximately according to game theory. If this same player in the face of a bet from you, will turn into a meeker player who will very rarely or never raise bluff (whether he calls or raises when he makes his hand is irrelevant since you lose the same amount) then a significantly smaller bet coming from you can yield a higer EV. Even though you can never win that bet.

To show this mathematically I pose two questions. Assuming that your opponent will never raise bluff and always bet the size of the pot (in a perfect game theory ratio) if he does bet after you check, what is the maximum size your "stop a bluff" bet can be where you are still better off than checking? (Obviously the smaller your bet the better. But in real life your opponent is more apt to see through tiny bets and won't be dissuaded from bluffing.) Answer should be in the form of x% of the pot.

Question Two is the same as Question One except that this time both you and your opponent have an unlimited amount in front of you, so he will be betting mega times the size of the pot when he bets. This time the maximum bet from you to stop his game theory play can approach being as high as what? (Again x% of the pot.)

(Normally these questions are answered too quickly by the Game Theory geeks who hang out on the Poker Theory Forum and thus you guys don't have a chance to think about them. I'll doubt they'll come her though since they are not interested in a bunch of "Why I actually play as well as, and am as nice as Gus Hanson" posts. On the other hand we now have a lot of bigshots here who not only win tournaments but have oodles of graduate degrees so one of them should quickly get the answer. To them I ask that they hold off posting until some of the masses reply.)

Oops. I just realized that I forgot to specify in my two questions the PROBABILITY that the drawing hand will hit. Well I don't have time to come up with a good number. So do the best you can with what you have.

[sdplayerb]

Isn't another important element what the board is and what possible hands the river card causes?

If it is a scare high card, that is one thing. But an innocuous low card, I would likely play it differently.

Also what kind of player I am against changes it as well. Is it somebody that is likely to bluff? If so, when a low card hits, like a 2...i am more apt to check hoping to induce a call.

There appears to me to be too many factors that need to take into account to possibly answer your question without explaining each situation..or else would need you define the scenario in much better detail.

[David Sklansky]

To avoid muddying the waters for the purpose of this math problem, let's specify that we are playing a game where the last card is face down.

The math works for holdem as well except that you can't know exactly what he is drawing to.

[West]

I *think* the answer to question 1 is just short of 50% of the pot. Betting half the pot would break even with checking, if opponent is bluffing a pot size bet according to game theory.

[eyekast]

when your done can you give a detailed answer as to how to figure this out because i don't even know where to start on this [img]/images/graemlins/tongue.gif[/img]

chris

[West]

I *think* the answer to question 2 is:

You could bet a percentage of the pot up to the percentage = to x/y, where y is the number of cards that make your opponent his draw, and x = y-1. So if your opponent would make his draw with 18 out of 44 cards (less than 50% chance), then I believe that according to game theory, he should be bluffing y-1, or 17 times out of the 26 times he doesn't hit his draw. In that case, you could bet up to 17/18% of the pot (94.444%), and be better off than checking. Or so I think now.

[1p0kerb0y]

As a member of the masses, I believe David has granted me the opportunity to go first. Maybe one day I'll be a bigshot though.

In question number one, David stated that your opponent will make an "equally large bet some of the time that he misses". Well, this percentage is absolutely relevant to the question at hand.

One theory would be that the more often your opponent is apt to bet a bluff on the end, the higher percentage of the pot you should bet. The theory behind this is that the more often he is willing to bluff, the looser and crazier he is, thus the bigger bet you will need to prohibit a bluff-raise, or even a future bluff on the end.

I.E. if he bluffs 40% of the time he misses his hand, you should bet 40% of the pot.

If he bluffs 20% of the time he misses his hand, you should bet 20% of the pot.

I may be way off here, but then again I am not a bigshot so it doesn't matter!

[kmvenne]

I agree on both problems, outside the fact that your X/Y% would have to be tweaked slightly down. Just like we don't bet exactly 50% in example one, we don't bet 2/3rds the pot paying off the three outer twice to save two pots worth of bluffs, we bet slightly less so our play is successful by the slimmest of margins. I guess the answer is X/Y%-c where c is the lowest denomination of chip we have at our disposal (That is why it's lower case [img]/images/graemlins/smile.gif[/img] ).

[burningyen]

Question 1:

What happens when you check?

a = # of cards that make your opponent's draw
b = # of cards on which your opponent will bluff based on game theory
c = # of cards left before the last street/river

Then:

(a+b)/c = % of the time that your opponent will bet the river
(c-a-b)/c = % of the time he will check behind
a/(a+b) = % of the time you will lose when you call a river bet
b/(a+b) = % of the time you will win when you call a river bet

EV of calling a river bet = [-pot * a/(a+b)] + [2*pot * b/(a+b)] = pot * (2b-a)/(a+b)

EV when your opponent checks behind = pot

So:

EV when you check = {[(a+b)/c] * [pot * (2b-a)/(a+b)]} + {[(c-a-b)/c] * pot} = pot * [(c-2a+b)/c]

So what's the smallest stop-a-bluff bet you can make to be better off than checking?

(c-a)/c = % of the time your opponent will fold
a/c = % of the time he will raise (and force you to fold) or call
x*pot = the amount you should bet

EV of betting x*pot = {pot * [(c-a)/c]} + {-x*pot * a/c} = pot * [c-a-(x*a)]/c

EV of betting x*pot >= EV of checking, so:

pot * [c-a-(x*a)]/c >= pot * [(c-2a+b)/c]

Which means:

x >= (a-b)/a

[burningyen]

Question 2:

I'll use the same variables as in my answer to Question 1. Presumably by a "mega" river bet you mean that the size of the pot approaches zero relative to the size of the opponent's river bet.

m = amount of the opponent's possible river bet

EV of calling a river bet = [-m * a/(a+b)] + [(m+pot) * b/(a+b)] = [(-m*a)+(m*b)+(pot*b)]/(a+b)

EV when your opponent checks behind = pot

So:

EV when you check planning to call = {[(a+b)/c] * [(-ma+mb+potb)/(a+b)]} + {[(c-a-b)/c] * pot} = [-m * (a+b)/c] + [pot * (c-a)/c]

So it would appear that checking with the expectation of calling an m-sized bet would be very bad as m approaches infinity. You would be better off checking and folding, giving you EV = 0.

So the trick is to find out what x will give you better EV than 0.

pot * [c-a-(x*a)]/c >= 0

Which means:

x =< (c-a)/a