## Tuesday, December 06, 2005

### Defining Value

I have been thinking lately about the question of how to define “value” in baseball. I have an article on my website entitled “Ability v. Value” which discusses the differences between what sabermetricians mean when we talk about ability or value.

In the article, I define two classes of value, “Literal Value” and the regular “Value”. Literal value, as I define it, involves only methods that track actual changes in run and win expectancy, like Value-Added Batting Runs or Win Probability Added. Value includes methods which use composite season statistics, but give credit for things like hitting with runners in scoring position or a pitcher who pitches in a lot of high leverage situation.

I also broke down ability into “Ability” and “Literal Ability”. Ability is defined as “theoretical value”, i.e. the value that a player would be expected to accumulate, on average, if he played in a given set of circumstances. Usually this would be our expectation for a player in a neutral park, but it could be “ability to help the team win games in Coors Field” or “in 1915” or “batting fifth in a lineup with A, B, C, and D hitting ahead of him and E, F, G, and H hitting behind him”. There are all sorts of different ways you could define ability, but the mathematical result you get will be specific for the context you choose.

Literal ability goes even further, and attempts to distill the player’s skill in a given area of the game (such as power, or speed, or drawing walks), or his “overall ability”. This is very tricky, because nothing happens in a vacuum, everything happens in some sort of context, and so divorcing a metric from context is pretty much impossible. Therefore literal ability is more of a theoretical concept and not a measurable quantity (although methods like Speed Score are an attempt to measure literal ability in speed, but of course are acknowledged by their creators as approximations).

Anyway, to generalize, value is backwards-looking, and ability is forwards-looking (or at least what might have happened in a different context given the same production in a given timeframe).

The recent signing of BJ Ryan to a large contract by the Blue Jays has put the issue of when to time the value measurement into my head. Literal value methods like Win Probability Added value on a real-time basis. If at a given moment the probability of winning is 60%, and after the next play it increases to 62%, then the player responsible for that play is said to have added .02 wins. So a closer, who pitches as the highest leverage time, will come out with a higher WPA then a starter who had the same performance in the same number of innings.

But if we are ascertaining value after the fact, why do we have to do it in real time? Suppose that Scott Shields is called in to pitch on the road in the bottom of the seventh inning with a one-run lead. According to Tango Tiger’s WE chart (linked below), the win probability is .647. He retires the side and at the end of the inning, the probability is .732, so he is +.085. He starts the eighth inning with a probability of .704, retires the side, and leaves with a probability of .842, so he is +.138 for the inning and +.223 for the game. In the bottom of the ninth, it is still a one-run game and Francisco Rodriguez is summoned with a win probability of .806. He finishes it off and of course the win probability is then 1, so he is +.194 wins. So Shields, for two innings of scoreless work, only gets .029 more wins then Rodriguez did in one inning. Is this fair? Sure, if you define value real-time. Rodriguez pitched in a more critical situation and his performance did more to increase the real-time win probability.

But since we are looking backwards, why can’t we step back and, now, omniscient about what happened in the game, ascertain what value the events actually had? Each out in the game had a win value of 1/27, and since neither allowed any runs or anything else, we don’t have to consider that. So Shields should have added 6/27 wins and Rodriguez 3/27. Viewed from the post-game perspective, Shields performance is much more valuable then Rodriguez’. Now you could also argue that if you took this perspective far enough, any event that didn’t lead to a run in the end(like a hit that does not score) has no value. And that’s a possible outcome.

Now the point is not that real-time value determinations are incorrect or invalid. They are simply a different way of defining literal value. But I would contend that they are not the only way to define literal value. It is one of the easiest to explain and define, and it certainly makes sense. I’m not arguing against it, just arguing that it is not an undeniable choice for what I have called “literal value”. Of course, you can define “value” or “literal value” reasonably, in such a way as to make it an obvious choice.

Tango Tiger's Win Expectancy Chart

1. FYI: I cannot find your article on Ability v. Value using the Blog search.

I have had the same idea/question about assessing wins after the fact. Why not assign "win value" based on how each run was scored? Thus a double scoring the winning run in a 1-0 contest would get half of a win. But a similar double in a 2-0 contest would get only a quarter of a win

Unfortunately, I've been offense oriented so that's as far as I ever got with that thought, I have no idea how to incorporate defense and pitching to this concept.

2. That approach would be fairly similar to the "Victory-Important RBI" invented by Bill James in the 1980s as a response to the Game Winning RBI. The problem I have with such an approach is that by crediting only the payoff event, the one that drives home the run, you ignore the work done to set up the run(i.e. the guy who walked and then was driven home on the double).

Also, events in innings that don't lead to a run still help to score runs. Getting on base allows another batter to come to the plate. Suppose your cleanup hitter bats with 2 outs in the bottom of the 9th and drives home the winning run. Had a batter who reached base in the 1st inning not done so, your cleanup hitter would have never gotten to the plate.

The Ability v. Value article is not on this blog but is on my other website. I can see why the way I put it could have been confusing. Here is a link:

http://gosu02.tripod.com/id11.html

I reserve the right to reject any comment for any reason.