The Disappointment Zone

Musings from a Cleveland sports fan

Can we construct a model for optimized decision-makinging? Should we?

Posted by disappointmentzone on 8 August 2006

We have available to us the expected run value for every possible situation in baseball—base loaded with no outs (2.40476 runs), first and third with two outs (.47882 runs), runner on first with one out (.93357 runs), etc. We also have available detailed statistics on every batter in baseball and what he does in different situations—two outs, runners in scoring position, against left handed pitcher, etc. It is with this data (and a dose of common sense) that managers are able to figure out that with the bases loaded and two outs in the bottom of the ninth in a one run game with a right-handed pitcher on the mound they are better off pinch-hitting for the .057/.057/.105 batter with the guy who’s hitting .356/.398/.513 against right-handed pitching. Which is why we often see managers pinch-hit in such situations. If proving that the expected run value of pinch-hitting in that situation is worth more than not pinch-hitting is relatively easy to do—and it is; there is practically no math involved—and if we have all of these data sets available, might we be able to calculate, for each situation in a baseball game, the optimal decision a manager should make?

I think we can, but these data sets alone are insufficient for such an aim. Before moving on to what one would need to even begin to assemble a reasonable model of optimized decision-making for every situation in baseball—for each at bat in a game, that is—it is worth articulating why expected run tables and detailed batting statistics are insufficient, for in such an explanation a clearer image of what would be needed—and why—emerges.

Here is an example. It is the bottom of the third inning in a 2-2 game. Mike Mussina is pitching for the Yankees. The Indians have loaded the bases and there is one out (expected run value: 1.65062). Jhonny Peralta comes to the plate. It’s an early September game and if the Indians win the team will clinch the AL Central title. Heading into this at bat Peralta is hitting .156/.175/.306 against right-handed pitching (RHP) this season and has grounded into 59 double plays. On the bench is Shin-Soo Choo, who is batting .267/.308/.416 against RHP and has only grounded into eight double plays all season—a testament to his speed more than anything. Choo is a marginally better batter than Peralta, in other words, and accordingly would have a better chance at driving in the runners on base. What should manager Eric Wedge do?

Well, that’s what we are trying to figure out—what is the optimum decision Wedge could make in this situation? Since we don’t know the answer to this, perhaps the better question at this point is what would Wedge probably do? Well, Wedge would probably not pinch-hit for Peralta—even though not pinch-hitting for Peralta is likely costing his team runs in that situation. So the question becomes: why not pinch-hit for Peralta? The answer is to pinch hit for Peralta in the third inning with Choo would mean that Joe Inglett would have to enter the game to play short stop, since Choo is unable to replace Peralta defensively and no one else on the team is, either (Luna is hurt). Inglett is an above-replacement level player, but just slightly in the field and just barely at the plate. Inglett represents a significant drop-off from Peralta. Choo, meanwhile, can only play in the outfield, where he can hold his own. But there is a reason he wasn’t given the start in this game—Michaels, Sizemore, and Blake all bat 100 points higher against RHP than Choo. To leave Choo in the game would mean to remove one of these men. The only way Choo doesn’t stay in the game is if Inglett replaces Choo in the lineup as soon as the inning is over. So for the rest of the game Inglett is playing short stop and taking Peralta’s at bats and Wedge is short one pinch-hitter for later in the game since Choo cannot pinch-hit twice.

Now we are getting closer to what sort of data sets would be necessary to begin an optimized decision-making model. In the example above, in addition to an expected runs table and split batting statistics and defensive statistics, Wedge would need an expected runs table for Peralta against Inglett, ideally two expected runs tables—one for defense and one for offense. If Inglett will likely get three more at bats in the game, what level of offensive drop-off can Wedge expect to see from Peralta to Inglett? To know this Wedge would also need to know who is in the Yankeee’ bullpen and which pitchers he could expect to see enter the game. If Wedge knew, for example, that Inglett was only going to see RHP for the rest of the game, then figuring out the drop-off from Peralta to Inglett might be as simple as comparing their splits against RHP. But if Inglett might face a LHP—and Inglett is even worse against LHP than he is against RHP; bad enough that he’s well below replacement level—then Wedge needs to know how pinch-hitting Choo impacts his bench with respect to pinch-hitting for Inglett. If there is no one who can pinch-hit for Inglett, perhaps because there is no defensive replacement for him or because the only decent batter against LHP is Choo, then the drop-off between hitters might be even larger. For example, had Wedge not pinch-hit for Peralta in the third inning he could have batted Choo in Peralta’s spot in the eighth—with the bases full, again, and one out—and then replaced Choo in the field with Inglett, who would, one hopes, only play defense for one inning and then not bat in the game, because Choo would get a hit in that situation, giving the Indians the lead, and Fuasto Carmona would earn the save in the ninth (how far can I take this hypothetical flight of fancy?!?). So now the difference between batters is the difference between Choo against LHP—against which he thrives—and Inglett against LHP, not just the difference between Paralta and Inglett against LHP.

So I’m counting at least an additional four expected runs tables (for the sake of simplicity all batting and defensive statistics are being converted to expected runs). And that’s just for one situation—whether or not to pinch-hit for Peralta in the third. Each moment in the game the same decision faces Wedge—whether or not to pinch-hit for the batter. In a lot of situations that decision will already be optimized—or so one would hope. The best option—all things considered—to lead off the game will actually lead off the game. Optimizing decision-making becomes more complex the more complex the situation becomes—multiple runners on base; early in the game; number of outs; etc. The Peralta example was about as extreme an example as one could come up with (possible the only way to make it more extreme would be if it were the first inning), but complex situations abound in normal baseball games. Oftentimes teams are in situations when, for one reason or another, pinch-hitting might be, at first glance, a reasonable option to explore, if not implement. And for each of these situations additional expected runs tables will be required, most of them unique to the situation. So that’s hundreds, if not thousands, of additional statistics necessary to construct a model.

So this is what one would need before constructing something even near a serviceable model of optimized decision-making–and I can think of other data sets that would be necessary as well. And then there is the other half of baseball—pitching—which is a whole other can of worms. And I’m too tired to consider what that might entail. Needless to say, managing a baseball game requires a lot of little decisions to be made. Managers make most of these decisions either by consciously not acting (e.g., not pinch-hitting for Hafner in the first inning) or by indifference—not even considering the range of possible decisions that could be made in every moment. Managers generally make a couple big decisions at the start of the game—who to start where and in what order to bat them—and that’s it until late in the game. For now about all that one can reasonably assess are the decisions managers make at the start of the game, which is what my writing on platooning lineups is about, and what decisions managers should or do make in late-game situations when the range of feasibly intelligent options is narrow. All the stuff in between falls in the realm where statisticians don’t tread—probably because the results would fit snugly with what managers already do. So Wedge doesn’t pinch-hit for Peralta in the third inning because it’s the intuitive decision to make. Fully justifying that intuition, statistically, would require a huge undertaking. Sometimes clichés just work better.

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