Wednesday, May 25, 2005

Possible addition to the chapter on Position

This section may be added to the end of the chapter on Position in the second printing.

The Value of Position: An example against a simple opponent

The value of any position is tough to determine. It is as difficult to determine as the question “How much should I expect to win with AA?” Simulations against reliable opponents would be a good way to estimate these answers. Without simulations, it is educational to see the value of position by using a sample hand against a simple opponent.

Assume your opponent is loose and passive. He normally will not bet unless he has a strong hand. This means when he bets, he usually has you beat. Since he is loose, he is willing to call with many hands that others would raise or fold. When you have a better hand, you will win more money from him than against normal players because he will call.

In this sample hand, you have T9s. Your opponent has been typically passive throughout the hand. This could mean you have the better hand. You are not sure against this opponent since he would play in the same passive manner with a pair of T’s with a better kicker.

The board on the River is: T-8-5-3-Q rainbow

Though your opponent is passive, you can still count on him to bet if he catches something good on the River. In this hand, you can count on him to bet if he caught a straight or a top pair of Q’s. We will assume he will not check-raise.

If your opponent did not improve on the River, then it does not matter in this hand whether you act first or last.

If your opponent caught a straight or a top pair of Q’s on the River….

….and you act first, then you still have to bet because he will call with worse hands as well. He is more likely to have a worse hand than a better hand.
….and you act last, then you have the opportunity to see what he does first. If he bets, you can fold and save a bet that you would have lost if you had acted first.

Let’s put some hypothetical percentages to the type of hand your opponent has.

25%: A missed draw
50%: A worse hand than yours, but one that will call a bet
12.5%: A better hand than yours by a little bit
12.5%: A better hand than yours by a lot

When he has a better hand than yours by a lot, he will be happy to bet or raise. In the sample hand, his hand could be J9 for the straight. If he has a better hand than yours by a little, he will just check or call. In the sample hand, his hand could be JT.

Here is a table that shows how much you will win or lose on the River when you act first or last.

Your opponent’s hand Prob. You Act First You Act Last
A missed draw 25% 0 0
A worse hand than yours 50% 1 1
Better hand by a little 12.5% -1 -1
Better hand by a lot 12.5% -1 0

If your opponent bets on the River, you can safely fold. You are not worried about this passive player bluffing on the River.
You should always bet when he checks, because the probability he has a worse hand than yours is greater than the probability that he has a better hand.
We are assuming that if your opponent has a better hand by a lot, it is because the River card improved his hand to either a straight, two pair or a top pair of Q’s.

With these assumptions, we can quantify the value of your position in this hand.

EV when you act first: (25% x 0) + (50% x 1) + (12.5% x -1) + (12.5% x -1) = +0.25
EV when you act last: (25% x 0) + (50% x 1) + (12.5% x-1) + (12.5% x 0) = +0.375

The value of acting last in this hand is worth 0.125 big bets. (0.375 – 0.25)

Situations like this will repeat themselves all the time. Gaining an extra 0.125 big bets here and there really adds up over time.

This is an example with a simple opponent and a simple outlay of cards. It should be clear that the decision making is easier when you act last versus when you act first. You have much more information to make a decision when you act last. The advantage in acting last is present in all hands. Only the strategy of check-raising can reduce the positional advantage to a small degree.

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