raising out players with 2nd best holding
 

In ToP, Sklansky speaks of how when the pot is big, you should raise to drive out opponents even if your holding is an underdog.

The example he gives is that with 4 players total you are a 30% underdog and that if you raise you can narrow it down to 2 players and have a 45% chance of winning the hand.

I never thought I would say this, but I wish his book had more math to back this up. I am trying to understand the relationships between pot size, your odds of winning, and your profitibility based on which action you choose (whether to raise to drive out opponents or just call).

I guess I'll try to work through some examples so you guys can comment.

Let's say it's fourth street in .25/.50. The pot is $4. 3 opponents. Your hand has a 30% chance of winning with 3 opponents, a 35% chance with 2 opponents and 45% with 1 opponent. Each opponent on fifth street is good for calling one bet if you make the winning hand, and you will fold if you don't.

The opponent to your right is betting fourth street, you need to decide to call or raise

Situation 1:
You call, other 2 villains call
30% of the time you hit and win $7
70% of the time you lose .50
an average of $1.75 profit each time you call here

Situation 2:
You raise, 1 opponent cold calls, 1 folds, original bettor calls
35% of the time you win $7
65% of the time you lose $1
an average profit of $1.80

Situation 3:
You raise and only the original bettor calls
45% of the time you win $5.50
55% of the time you lose $1
an average profit of about $1.93

Situation 4:
You raise, other 2 fold, original bettor re-raises, you call
45% of the time you win $6
55% of the time you lose $1.50
average profit about $1.58

Now lets say the pot starts at $2 instead

Situation 1:
You call, other 2 villains call
30% of the time you hit and win $5
70% of the time you lose .50
an average of $1.15 profit each time you call here

Situation 2:
You raise, 1 opponent cold calls, 1 folds, original bettor calls
35% of the time you win $5
65% of the time you lose $1
an average profit of $1.15

Situation 3:
You raise and only the original bettor calls
45% of the time you win $3.50
55% of the time you lose $1
an average profit of about $1.03

Situation 4:
You raise, original bettor re-raises, you call
45% of the time you win $4
55% of the time you lose $1.50
average profit about $0.97

hmm... so it appears that if you have a "big pot" then it can be more profitable to raise IF you are not likely to get re-raised. If it is a small pot it may be more profitable to be unagressive and let fish call down. Of course there are a lot more variables.

There has to be some sort of mathematical formula to figure out whether you want people staying in the hand or not.

I'll have to keep thinking about this one...