World Series Win Probability Added

Among the quirkier new-age baseball statistics is Win Probability Added. This statistic uses the assumption that all players are equally skilled, along with some knowledge about the overall run environment in which a given game is being played, to generate each team's expected chances of winning a game based on the relative score, the inning, the number of outs, and which bases are occupied. So, for instance, at the beginning of the game each team's Win Expectancy is 50%, but if the away team puts up 15 runs in the top of the first their Win Expectancy shoots to about 99.9%. Any given play, therefore, will alter one or all of the score, the inning, the outs, or the bases, and therefore the Win Expectancy, and you can give the play a Win Probability Added score based on the change in Win Expectancy. Attributing that change can be tricky in cases of, say, stolen bases or errors or whatever, but as a general rule you attribute WPA to pitchers and hitters. And the idea is that WPA tells the story of a game, though it's well-known that it's not a great way to evaluate players overall. But the story of a baseball season isn't just isolated games as part of an anthology. They are, rather, episodes, which add up to tell an overarching narrative, at the end of which someone wins the World Series. From a purely competitive standpoint, that's the goal, not just to win games but to win it all in very early November. Adding to your team's odds of winning a game is nice, but you really want to boost its chances of winning the World Series.


Can we, therefore, create an analogous statistic to account for the way that individual game stories add up to become season stories? Well, maybe we could in an overarching way, but there's at least one conceptual challenge, and then whichever way we resolve that there's a practical challenge. The concept of WPA is, as I mentioned, the neutral assumptions. Every player is just as good as every other player. So, when the 116-win Mariners take a TARDIS trip from 2001 back to 1962 to play the 40-win Mets, the Win Expectancy at the start of the game is 50% for each side. And walking Barry Bonds to pitch to Pedro Feliz doesn't raise a team's odds of winning, even though it's probably always the right call, as National League pitchers declared en masse in 2004. These neutral assumptions conflict with the dominant approach to generating playoff odds. Both Fangraphs and Baseball Prospectus use their statistical projections of each team's talent level to inform those probabilities. Sometimes the team with the better record has the worse odds of making the playoffs, if there's reason to think they're not really a better team; conversely, a race that looks close in the standings might not be very close in these odds, if we think the leading team is clearly superior going forward.

That approach almost certainly increases the accuracy of these predictions, but it is at odds with WPA's neutral assumptions. After all, you don't want to penalize good teams for having been expected to be good. So perhaps we would want to use instead the results of what is now Fangraphs' "Coin Flip Mode," where you assume every game to be a coin-flip. That's fair enough, I suppose. It still requires calculating team playoff expectancies, and now with the second Wild Card the separate expectancies of winning the division versus winning a Wild Card spot. And that's hard. Perhaps not in the grand scheme of things; after all, Fangraphs does run their Coin Flip Mode projections after every game of the season. It might be tricky to calculate this in real time, but retrospectively it's only harder than I can do, not harder than anyone could do if they really wanted to. One final complication does occur to me, though, namely that a team's odds of making the postseason any given morning depend on a lot more than what happened in that team's own game the night before. If a team's main division rival wins their game, everyone on that team will have much lower WSWPA figures from their game than if their rival had lost, which seems to contradict the idea that we want players' statistics to reflect their own performances.

There is one context, however, that is utterly devoid of these difficulties, and that's the post-season. Under the neutral assumption that every game is a coin flip, which is not a bad assumption in October, we can state with quite little computation the precise World Series Win Expectancy for teams at every stage of the post-season. There are two steps to this: first, it's simple enough to know the stakes of each round of the post-season. Anyone who loses goes home, i.e. goes to 0% WSWE. Anyone who wins the World Series, obviously, wins it all, i.e. 100% WSWE. Winners of League Championship Series get to the World Series, where they have 50% WSWE. Winners of the Division Series get to the LCS, where they accordingly have 25% WSWE. Division winners and winners of Wild Card games, therefore, start with 12.5% WSWE, and therefore each Wild Card team has 6.25% WSWE on the day the season ends.

The second step concerns what happens within a series. There are only fifteen possible states a 5-game series can find itself in, and only twenty-four states for a 7-game series. If we assume each game to be a 50/50 proposition, it's trivially easy to derive odds of winning in each of these states. What you can then do is combine the first two steps. If X is the WSWE conditional on winning a given postseason series, and Y is the odds of winning the series conditional on how the series stands, the conditional WSWE at that point in that series is just XY. Simple! Then, for the final bonus stage, we can have WSWPA within each game. Here we just treat the interval from 0% to 100% covering the ordinary WPA scale as running from the WSWE if a team loses a given game to the WSWE if they win it. Tonight's game between the Cardinals and the Red Sox, with the Cards leading the Series 2-1, will either end with an 88% chance of Cardinal victory or with a 50% one. So an event within this game that has 10% WPA, for instance, would have 3.8% WSWPA. You can do that for every game throughout the post-season, and in principle it should be pretty easy to compute WSWPA numbers for any post-season game, as it's going or in retrospect. Since the post-season is, like relief pitching, intrinsically a narrative event, and since we care a lot less about projecting future results than about seeing who came through in the biggest spots, this might be the best way to evaluate post-season performances. I'd love to see someone with better computational powers than I do something with it.

Oh, and just for reference, here are the Series Win Expectancies for each possible state of both a 7-game and a 5-game series. These are presented from the point of view of the team leading the series, since the trailing team just has the complimentary Win Expectancy. Also omitted are tied states, because obviously that's just 50/50, and states where the series is over, because, well, you know.

7-Game Series
1-0: 65.6%
2-0: 81.3%
2-1: 68.8%
3-0: 93.8%
3-1: 87.5%

5-Game Series
1-0: 68.8%
2-0: 87.5%
2-1: 75.0%

It's amazing how little I need to write to convey all the information! There are 39 combined possible states, and I only need to write out 9 of them for the other 30 to fill themselves in.