Saturday, August 26, 2006

The 75% Rule

[This is a re-edit of part of the previous post, chopped into its own post for easier linking from my "Archive Highlights." I've also turned comments on for this post: Do I really have something here?]

There's a particular sort of player that Gil has long found vexing, that I ran into this weekend in Chicago. I was nailed a couple of times by this on what was otherwise a very good table. The problem player is one who plays very passively with good hands, only to raise on the river. I'll posit a hand to demonstrate:

You're under the gun with AK. You raise. You get two cold-callers, and the big blind calls as well. The flop is K88 (we'll assume throughout that no flush is possible). You bet, and one of the cold-callers and the big blind call. The turn is a Q. You bet, the cold-caller calls, and the big blind folds. The river is a 4.

Ordinarily, I'd read the cold-caller for a smaller King—say KJ—or for a pair, probably 99 or TT. A Queen is possible; that he took off the turn and got "married" to his hand by the Queen. Nothing the Villian has done has led me to believe he's got an Eight. So I'd bet.

Often enough, the Villian raises, I call for the size of the pot, and I'm shown an Eight. Sometimes the Villian doesn't even raise, and I lose a slightly smaller pot.

Gil has long questioned whether one should bet the river in this situation. In fact, I've seen him not bet the river in a number of situations where I thought it was clear that he had the best hand, probably because he was remembering hands like this one.

Has the Villian, intentionally or otherwise, found a way to counter my aggressiveness? I can't believe his play is optimal—except against me. Against me, he wins the maximum with his Eight.

So what is my option? Checking the river seems pretty feeble, too. I'd have to call if the Villian bet. I lose slightly less this way if I'm beat, but I also win less if I'm ahead. And nothing that's happened so far would lead me to believe that it's more likely that I'm beat, than that I'm ahead. Just so that we can assign numbers to this, let's say it's 60:40 that I'm ahead.

If I bet the river, I win zero bets if I'm ahead and he folds, or I win one bet if I'm ahead and he calls. He probably folds a good percentage of the time, say 50%: if he has 99 or TT, or a Queen, he might choose to give up here. The only hand that beats me that could possibly fold is KQ.

If I bet the river, I lose one bet if I'm behind and he calls, or I lose two bets if I'm behind and he raises. If he has that Eight, I think he raises about two-thirds of the time: say 65%. (I think I'd raise 100% of the time here in his situation if I didn't raise the turn, but since I'd probably raise the turn 100% of the time, it seems moot.)

Taking all of the numbers I've assigned to this, I get {0.6[(0.5×0)+(0.5×1)]}+{0.4[(0.35×(−1))+(0.65×(−2))]}, which works out to an EV of −0.65, or a 2/3-bet loss from betting out. (Internet Explorer puts the line-break for that problem in a funny spot, for me. If you're following along with the math, hopefully that doesn't add to your confusion.) (It's also been pointed out to me that the math is difficult to follow along with in any case. All I'm doing is taking the percentage chance of something occurring, as I've estimated it, and multiplying it by the number of bets I'd win (or lose) in that case. If I keep doing this, I get my theoretical EV for the entire play. I've deleted some of the intermediate steps before arriving at my answer, which makes the problem look shorter and (hopefully) less intimidating.)

My suspicion is that checking is also −EV, but less so. If I check and I'm ahead, I win zero bets. I think we can discount the possibility of a bluff from a player who's played this passively thus far, but if the Villian does bet, I'm going to call. Maybe the Villian bets a hand worse than mine about 20% of the time when I check. So if I check, and I'm ahead, I win 0.2 bets on average.

If I check and I'm behind (almost certainly to an Eight), I think the Villian will bet at least 80% of the time. Since I'm going to call, I lose 0.8 bets on average. (This assumption might mar the general applicability of the rule I develop below: Although the Villian probably raises in the hand I've posited, in other cases that would be less likely. In that case the final answer would be lower than 75%.)

Taking the (fewer) numbers I've assigned to this, I get (0.6×0.2) + (0.4×(−0.8)) = −0.20, or an EV of a 1/5-bet loss from checking.

What does all of this mean? If I correctly divine the odds at 60:40 that I'm best, checking is the superior play. This is counterintuitive, but it seems to be borne out by my thumbnail math. The question becomes this, then: How sure must I be that I'm best, before a bet becomes +EV? Let's assume that the other numbers are the same. The algebraic problem is this:

0 =
[ x((0.5×0)+(0.5×1)) + (1−x)((0.35×−1)+(0.65×−2)) ]
− [ x(0.2) + (1−x)(−0.8) ]

You can work the algebra yourself if you like but the value of x turns out to be 0.739+. Since this is thumbnail math anyway, we'll say that the breakeven point is 75%. If you're more than 75% sure you're ahead, bet. If you're not, check.

I'm probably going to use this rule of thumb I've developed, but the problem includes a number of other variables I simply assume that I've estimated correctly. However, the Villian might see the showdown more often, might bluff less often, might raise more often with the Eight. All of these would change the answer I get, that you must be 75% sure you're best. However, this is all thumbnail math anyway; under fire you probably won't say, "Lord Geznikor's magic number is 75%. I'm only 72.5% sure I'm best. Therefore, I check." If you really can estimate the chances you're best that closely, then you'll have better numbers for all of the other things I've estimated, and your answer will change.

This is a problem Gil and I argued over in the car, on the way back from Chicago. It's a long drive. My feeling then was that you had to bet the river, and Gil (somewhat ambivalently) favored a check. As much as I hate finding out that Gil is right about something, I can change my mind when it can be shown logically that I'm wrong. So, even if nobody else followed this topic all the way through, Gil did, and since he turns out to have been right, he'll never let me forget it.


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