I would like to see the derivation if you don't mind. I realize that we should be varying our bet sizes based partially on the blinds stacks, but didn't know there was a nice little equation for it.

also, the <= means less than or equal too, right?

Chuck, <= means less than or equal to.

arty:

CORRECTION!!! CORRECTION!!! CORRECTION!!!

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arty:

It turns out I made a mistake in the post above

.

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| THE CORRECT FORMULAS ARE:

| -------------------------------

| X = 5B if the BB moves in

| X = 5B + 1.5 if the SB moves in

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arty:

Here is the derivation:

The pot is unopened and contains the blinds which is 1.5 big blinds. All units from now on are in big blinds.

You bet B. The pot is now 1.5 + B

The Big Blind goes all-in for his stack X. The pot now is 1.5 + B + X

You have to call 1 + X - B

So the

pot odds are (1.5 + B + X)/(1 + X - B)

If you are 40% dog, your odds of losing are 60/40 = 3/2

So the maximum you can call is when pot odds equal the odds of losing, i.e.

(1.5 + B + X)/(1 + X - B) = 3/2

Manipulating the equation leads the following:

2(1.5 + B + X) = 3(1 + X - B)

3 + 2B + 2X = 3 + 3X -3B

2B + 2X = 3X -3B

5B + 2X = 3X

X = 5B for BB moving in

------

If the Small Blind moves in, then the pot is still (1.5B + B + X)

but you only have to call (X + 0.5 - B)

then the maximum to call at 40% dog is

(1.5B + B + X)/(X + 0.5 - B) = 3/2

2(1.5 + B + X) = 3(X + 0.5 - B)

3 + 2B + 2X = 3X + 1.5 -3B

1.5 + 2B + 2X = 3X -3B

1.5 + 5B + 2X = 3X

X = 5B + 1.5 for SB moving in

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Let us verify the first formula:

Say you have been raising 2.5 big blinds. The blinds are 100/200 and the BB has a stack of 2,500. You open-raise to 500 and the BB moves in for the remaining 2,500.

The pot is 100 + 200 + 500 + 2,500 = 3,300

You have to call 2,500 + 200 - 500 = 2,200

As you can see the pot odds are 3,300/2,200 = 3/2

so if you are greater than 40% to win, you must call.

Practical usage:

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First note that in the example above, when the stack of the Big Blind is exactly 5 times your bet, it doesn't matter whether you call or fold since the EV=0. In tournaments, you can use these close decisions to manipulate your image and choose the action that is best. If your stack is not very big, obviously folding is preferred.

If the blinds have about 10BBs left, and you usually bet 3BB, then it is better to move in to maximize FE.

Here is an example:

Blinds are 100/200, BB has 2,000, you usually raise 3BB=600

According to the formula, X=5*600=3,000. Since the BBs stack is

2,000 < 3,000, you have +EV to call if the BB moves in. Let's verify it.

You open raise to 600 and BB re-raises all-in for 2,000

The pot is 100+200+600+2,000 = 2,900

You have to call 2,000 + 200 - 600 = 1,600

The pot odds are 2,900/1,600.

The required winning percentage to call is:

% to win = 1,600/(2,900+1,600) = 1,600/4,500 = 36%

Since we expect to win 40% of the time, it is an automatic call.

This post got quite a bit off topic. Perhaps it needs its own thread.