Math... of poker, Chapter 2

[siegfriedandroy]

Okay, I just started reading Chen's book, and am curious about the first example in the second chapter on exploitive play.

Chen provides an example where Player A, first to act, has the nuts 20% of the time, and a losing hand the other 80%. He bets every time he has the nuts, and bluffs x percent of the other times. Player B has a mediocre hand. There are four big bets in the pot going to the river. What should x be to make it profitable for Player B to call?

.2(-1) + x(5) is the given equation, so if A bluffs more than 4% of the time, B is correct to call. I am confused b/c generally, when making riv decisions, I will think of it the way Sklansky has explained (i.e. Im getting 5 to 1- is Player A bluffing more than 17% of the time). But here a call is correct if A bluffs even 5% of the time. This is a substantial difference, and Im curious about why this is. I suspect it has something to do with the given assumption that he has either the nuts (20%) or air (80%), which Sklansky obviously does not assume in his general example.

So how do we reconcile the 4% with the 17%? Is there any practical value in sometimes using assumptions similar to Chen, that opponent either has the nuts a small % of the time, or air every other time? If so, what situations would such assumptions be justifiable?

[JaredL]

If the opponent has you beat with probability .2 and always bets in that situation and bets with probability p when you have him beat the probability that you are ahead given a bet is given by:
p*.8/(.2+p*.8)

So your calling EV is
-1(.2/(p*.8+.2)) + 5*p*.8/(p*.8+.2)

Solving the above equals zero for p gives us p = .05.

edit: I've used Bayes' rule to come up with the 5%. They did the same except their x is .8 times my p so it's the overall percentage of times they bluff.

Using pot odds they are offering you 5-1 so for you to breakeven they have to have you beat 5 times as often as you beat them. This means that if they are betting 20% of the time with a good hand they must bet 4% of the time overall with a bad hand. This corresponds to betting 5% of the time in the situation where they have a bad hand.

[JaredL]

What is the Sklansky example where the 17% came from?

[jogsxyz]

I misplaced my copy, but I'll try to answer.

20% nuts
4% betting misses.
76% not betting.

When facing a bet, player B is seeing 20 real hands to 4 bluffs. That's the same 5 to 1 you read from Sklansky.

[siegfriedandroy]

Okay, thanks. I just got confused b/c i would normally think, "I can call if I'm good once in 6 times", and basically equate that with, "i can call if he's bluffing at least once in six times (or 17%ish)". but then chen's equation says we can call if he's bluffing more than just 4%??

[JaredL]

[ QUOTE ]
Okay, thanks. I just got confused b/c i would normally think, "I can call if I'm good once in 6 times", and basically equate that with, "i can call if he's bluffing at least once in six times (or 17%ish)". but then chen's equation says we can call if he's bluffing more than just 4%??

[/ QUOTE ]

The way jog set it up is probably easiest to explain.

Your error is that you should be thinking in terms of conditional probabilities not total probabilities. You need to be good 1 time in 6. This means that given that he bet you need to be good 1 time in 6. He bets with a good hand 20% of the time so if you are going to be good 1 time in 6 then he has to bet with a bad hand 4% of the time.

[RobNottsUk]

You have bad hands far more frequently, than great hands.

So a surprisingly low sounding bluffing frequency, will make an oppenent's calls profitable. They have some kind of hand, but you'll have air too often rather than the hand you're representing.

[pokervintage]

[ QUOTE ]
Chen provides an example where Player A, first to act, has the nuts 20% of the time, and a losing hand the other 80%. He bets every time he has the nuts, and bluffs x percent of the other times. Player B has a mediocre hand. There are four big bets in the pot going to the river. What should x be to make it profitable for Player B to call?


[/ QUOTE ]

I am not sure what x should be, but I do believe that it does not need to be precise just close. I will, though, offer this bit of unsolicited but correct advice. If you run into an opponent that understands the above and bluffs close to the correct amount of times try your best not to play against him. Find another opponent or game. You will be wasting your time with him. That is what is important about Chen's arguement, when thinking about playing poker, nothing more.

okervintage