Willfull suspension of disbelief as it pertains to river play

[fizzleboink]

Ok I'm going to try and solve this problem using some math.

Let X be the probability that Hero is ahead of UTG
Let Y be the probability that Hero is behind UTG

Let A be the probability that Button overcalls 1 BB
Let B be the probability that Button calls 2 BB

Let C be the probability that UTG calls your raise when he is behind
Let D be the probability that UTG 3-bets your raise when he is ahead

Constraint:
X + Y = 1

Equation 1: EV of calling UTG's bet = (1 + A)X - Y
Equation 2: EV of raising UTG's bet = (2B + 2C)X - (3D)Y

This is assuming you always call a 3-bet from UTG. UTG never 3-bets a worse hand and never folds a better one.

I'm going to tackle this problem from the angle where I don't know how often Hero is ahead or behind here, but I have some idea of how the calling station will act, as well as UTG.

So I will let X and Y be the unknowns.

Basically I want to find the tipping point where both raising and calling have the same EV. If they have the same EV, then I can combine equations 1 and 2 together:

(1 + A)X - Y = (2B + 2C)X - (3D)Y

Using the constraint equation (X + Y = 1) I will put Y in terms of X.

(1 + A)X - (1 - X) = (2B + 2C)X - (3D)(1 - X)

Solving for X yields:

X = (1 - 3D) / (2 + A - 2B - 2C - 3D)

If you want X in terms of percent, just multiply it by 100%.

I think this equation can be used re-used in this kind of situation all the time, and it is a situation that does happen somewhat often. I'd like to maybe try and generalize it some more later on with a variable number of possible overcallers, when I have a bit of free time.

Now just an example, here are some numbers that might be reasonable:

I will let:
A = 0.8 (= 80%)
B = 0.1 (= 10%)
C = 1 (= 100%)
D = 1 (= 100%)

If you feel that these are unreasonable assignments of the variables, you can change them and plug them into the equation above and see how the results change.

Anyways, it gives us a result that X ~ 83%. So Hero needs to be ahead of UTG greater than 83% of the time in order for raising to be superior to going for the overcall.

The next step would be to put a weighted range of hands on UTG, and to see how that range compares to X. Ofcourse you can argue the values that I chose for A, B, C, and D are wrong. You can also get into long-winded arguments about how often Hero is ahead. I think the best way to do it is to play around with the numbers and see how the result changes. Assume a bad case for Hero and if the answer is still clear cut, then fiddling with the numbers a bit is irrelevant.

I slapped together a quick spreadsheet so that you can play around with the numbers and see the result a lot quicker:

http://rapidshare.de/files/10003408/...Raise.xls.html