[gumpzilla]

[ QUOTE ]

I'm pretty sure the notion of betting to give yourself odds is flawed - the basic idea is that while the EV of making a call goes up if you bet more in the first place, you have to also account for the fact that you've sunk more money into the pot already. I'm lazy and not going to work this all out right now, but take it into account from the beginning of the hand.

[/ QUOTE ]

I changed my mind; here it is.

Let's consider an artificial example to make it very clear. Say there is $10 in the pot, you have $30 behind, and I cover you. I can choose among betting $5 and calling a shove, or betting $10 and calling a shove, and let's say I win 1/3 (we'll ignore splits.) Further simplifying, let's say that you shove 25% of the time, and fold 75% of the time.

What are the EVs of these two lines?

Line 1 : bet $5, call shove.
Line 2 : bet $10, call shove.

75% of the time, I win $10 in either case. The remaining 25% of the time, you shove. When you shove, I can consider the money that I've put into the pot lost, and then assess what I gain from a call.

If I bet $5, then you shoving $30 means there is $45 in the pot and it costs me $25 to call. 1/3 * 45 - 2/3 * 25 = - 5/3. If I bet $10, then there is $50 in the pot after your shove, and it costs me $20 to call. 1/3 * 50 - 2/3 * 20 = 10/3.

So, the EVs of the lines:

line 1: EV = .75*$10 + .25*(-5 - 5/3)
line 2: EV = .75*$10 + .25*(-10 + 10/3)

Sure enough, they are equal. This is because in both cases, I am getting all of my money in 25% of the the time with the same equity in the pot.

There are a couple of things to note. If I do not give myself odds, then I can do better by folding than calling the shove. So, all other things being equal, if you're truly bluffing, not giving yourself odds and folding to the shove will be a better play than giving yourself odds and calling. However, all other things will not be equal. The main variable that changes is your response to my bet. Obviously in the case where I bet $.25, there is very little chance that you fold. Bet sizing will play a big part in determining the likelihood of your responses, and so the analysis above is oversimplified.

The main thing, though, is that betting to give yourself odds doesn't really work. Betting pretty large is good, though, because if you're not 3betting light, then you're going to want to be betting legit hands most of the time in this scenario, so you leak less information that way.