Discounting Outs

Poker Concepts, Strategy

September 23, 2007

In last week’s article on odds, I went through a complicated example of deciding if you should call with a draw. The point of this example was to show, in a big-picture sort of way, how odds are used to make poker decisions. Now I want to explore one small aspect of that more closely.

Specifically I want to talk about discounting outs. As a reminder, here’s the situation:

You’re playing 20/40 limit holdem and are dealt Ah7h on the button. Preflop, action is folded to you, and you raise on a semi-steal. A tight player in the small blind re-raises, the big blind folds, and you call. The pot is now $140 and action is heads-up. The flop comes Qh4d5h. The small blind checks, you bet on the semi-bluff, and the small blind raises. You call (note that raising here is not a bad choice either). Pot is now $220. Turn is 2c. Villain bets $40. Do you call?

Now, there’s a large number cards that improve hero’s hand here. Any heart gives hero a flush. Any 3 gives hero a straight. Any A or 7 is a pair. That’s a total of 18 cards. In some sense, all of these are outs – they’re definitely all outs to improve. But we’re not interested in how often our hand improves – what matters is how often it wins. The first thing to notice is since this is the last card to come, if hero makes the nuts then hero will win the hand. Of the 18 outs, only 8 are the nuts – all of the hearts except 4h. The 3’s lose to anyone holding a 6, and tie with anyone holding an A. An A is beat by anyone holding a better Ax or any of the possible sets or two pair. A 7 is beat by all of those plus any Qx or pocket pair above 7.

Now, as you can see, whether or not our non-nut outs are good depends heavily on what villain holds. Of course we don’t know for sure what that is, but we do have quite a bit of evidence. Villain has adopted a very aggressive line on ever single street and we’ve always been forced to close the betting by calling. This would seem to be out of character as a blind defense tactic for a tight player holding air or a marginal hand, so the preponderance of evidence is that villain holds a hand that figured to be best on every street. That probably means AA, KK, QQ or AQ. 44 and 55 are possibilities too, but seem unlikely for a tight opponent. We can’t rule out a strong preflop hand that missed or now has a weak draw (such as AK or JJ) either, but that seems less likely. A combo draw such as KhQh is possible as well.

So how do our various non-nut outs fare against villain’s likely holdings? First, it should be pretty obvious that a 7 is a bad card for us. It doesn’t beat any made hand villain could be holding. So in this case, 7’s really aren’t outs. If one hits we might call if villain bets, but it would be solely as a bluff catcher. 3’s are substantially better – we’re almost certain villain could not get to this point with a 6, so they can’t really lose. There’s a reasonable chance they’ll tie with another hand that has an A. Aces will be good only if villain holds KK or some unusual draw. The 4h will be good unless villain holds a set or 2 pair including a pair of fours (and the two pair would be a huge surprise). So it’s quite likely good.

Once you’ve made some sort of determination as to how good your non-nut outs are, the challenge is to translate this into a numerical form that will allow you to do odds reasoning. Luckily, this isn’t hard at all. Simply move outs from the “win” side of the odds to the “lose” side in proportion to how likely it is you think you’ll be beat if you make that hand. Continuing the example above, there are ace outs, but over half of villain’s holdings will be 2 pair or better if another A comes. So we count the aces as one winner and two losers. Similar logic is applied to the other non-nut outs. Note that for outs which may split, you only move half as much over to the losing side – this is a bit of a hack, but it works in practice. So if you’ve got 4 outs that win half the time and split half the time, that’s roughly the equivalent of 3 outs.

A similar method of discounting outs can be used when you’re drawing to the nuts, but your opponent has draw back to beat you. For example, this can easily occur in stud when you’re drawing to a straight or flush and believe your opponent has two pair or trips. The arithmetic is just like above – you adjust the odds by moving elements from the win side to the loss side in proportion to the odds that villain hits his re-draw. So if you’re drawing on 6th street to a flush with odds of 31:5 against, and villain’s drawing back at you with 2 pair at odds of 32:4 against, you know your odds to win are more like 65:9 against. Note that the arithmetic behind this is discussed in more detail here.

There’s one caveat about discounting outs: the method is best suited to limit or small spread-limit games and big bet games only when you are about to permenantly close the action. In big bet games with money behind those outs where you improve but don’t make the nuts can cost you not only the existing pot, but a large amount of that money behind as well. As such they should sometimes not be treated as outs at all, and when they are considered outs they shouldn’t be given even the discounted value calculated with the method in this article. I’ll explain more in a future article.

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