Range, Equity, Maximize

Ok, so in tonight's Red Sox-Twins game Boston has a 6-5 lead in the top of the 9th, with Jonathan Papelbon in to close. Since Papelbon had apparently left his split-finger fastball in the bullpen, he throws about 48 straight fastballs to Nick Punto, and Punto fouls them all off. On pitch 49, Punto dumps a soft line drive into center field. Coco Crisp gets not such a good break on the ball, charges in, dives for the ball, and the ball lands in front of him and trickles by, giving Punto second base for free. The next guy bunts Punto over, giving the Twins a man on 3rd with one out.

Fortunately for the Red Sox, Papelbon strikes out the next chump and completes the inning for the save. Shortly after the game, it is reported by some guy on SoSH (actually, I'm 99.9% sure that it's the guy who writes this personal poker blog) that

Dave McCarty compared coco's dive to being pot committed in poker. He says once you get past a certain point you can't play it safe so you might as well take the chance.

I think we need to take up a collection to buy Covelli a copy of Professional No Limit Volume 1. Dude needs to make a commitment plan earlier in the hand.

After I saw a replay it seemed to my that Coco's last-ditch dive for the ball was pretty strong; it didn't look like he was of two minds split between diving and playing it safe. However, there's another side to his decision-making process to consider.

One of the first concepts introduced in Professional No-Limit Hold 'Em Volume 1 is the REM process, meaning Range-Equity-Maximize. The idea is that anytime you have a decision to make, you want to assess the Range of possible hands your opponent(s) have and the Equity (expected profit) resulting from your possible moves. You then want to choose the move (checking/betting/calling/raising/folding) that Maximizes your Equity in the hand.

Let's look at Coco's REM analysis. The Range of possibilities is as follows: if he pulls up short, the Twins have a man on first with no one out. If he dives and makes the catch, the Twins have no one on base and one out. If he dives and doesn't make the catch, the ball probably gets by him for a double, as it did tonight, and the Twins have a man on second with no outs.

The Equity of the various outcomes can only be calculated approximately, but on Fangraphs one can obtain figures for expected win percentage based on a given situation in the game. After poking around a few sample games, we can obtain win expectation values as follows (I think Fangraphs always assumes that each batter and pitcher is league-average): 71.7% (man on first, 0 outs), 90.5% (no one on, 1 out), 63.6% (man on second, 0 out).

So if Coco estimates that his probability of making the catch is x, then diving for the ball Maximizes the Red Sox win equity if

x(0.905) + (1-x)(0.636) > 0.717

or if

x > 0.301

Coco has to have a 30% chance of making that catch for diving to be correct. Now if we assume that there's some chance of the hit not getting past him for a double if he dives, then that tips the scales in favor of diving. Furthermore, Papelbon is an elite strikeout closer, so relatively speaking, having a man on second base with 0 out isn't quite as disastrous as it is for most pitchers. Of course, Papelbon increases the win expectancy in each situation, but I suspect the man-on-2nd, 0-out situation gets increased the most.

On the other hand some of the Red Sox win equity comes from the game going into extra innings, which imposes additional costs upon the Sox bullpen. If instead of using win expectancies, we decide we want to finish the game in the top of the 9th, then we want to consider the probabilities that the Twins do not score in the inning. According to Baseball Prospectus, they are 58.3% (man on first, 0 outs), 83.5% (no one on, 1 out), 37.5% (man on second, 0 out). Using this criterion, Coco's break-even requirement for catching the ball is 45%, which is a stiffer threshold. Again, this all assumes league-average batters and pitchers; with Papelbon on the mound I think the requirement would be lower.

All in all though, I think Coco's decision to dive for the ball is more sound than it seemed at the time.

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