[ QUOTE ]
Suppose you are playing NL, and there are two rounds of betting. There is already $1 in the pot. You each have $x. There is no drawing. You can assume that you are playing high card with at least 5 cards, and your opponent has the second highest card face up.
[/ QUOTE ]
In the optimal strategy, you bet some amount y some of the time, including every time you have the nuts and some bluffs. When you are called, you either push for the remaining x-y on the second round or give up.
You should bluff so that your opponent is indifferent to calling or folding.
On the first round, the pot odds are (y+1):y, so you should bet and give up y times for every y+1 times you plan to push on the second round.
On the second round, the pot odds are (x+y+1)
x-y), so you should bluff x-y times for every x+y+1 times you push with the nuts.
You get to bet (1+y/(y+1))(1+(x-y)/(x+y+1)) times as frequently as you have the nuts. This is proportionate to your equity. To maximize your equity, set y=(sqrt(1+2x)-1)/2. Qualitatively, you should bet so that your opponent gets the same pot odds on each round.
Here is a concrete example drawn from
this thread: Suppose the pot is $25, and you have $40 left to bet. If you push, you can only bluff 40/65 = .615 as often as you value bet. If you bet $20 at a time, you can bluff .784 times as often as you value bet. The optimum is to make your bets on the first round $13.11, which allows you to bluff .807 times as often as you value bet. (You bet about half of the pot on each round.) Of course, the model is not perfectly accurate, but it shows that it can be good to use the extra round of betting.