## The total variation of win probability, or: THE MAGNIFICENT TWINS-TIGERS GAME

Fangraphs has a neat feature:  they’ll make a chart showing how the probability of a home team win varies over the course of a game.  In a “normal” game this probability starts at .5 and slowly makes its way towards 0 or 1 as one team takes a lead and then holds on.

Today’s Twins-Tigers playoff is not a normal game.

The total variation in win probability over the course of a game is a good way of quantifying how much back-and-forth there’s been between the two teams.  You might take it as a loose measure of “excitingness.”

In this game, the Twins have gone from an 80% chance of winning to 20% to 73% to 20%, again up to 83% and then back down to 50%.  That’s a total variation of at least 2.62, all since the 6th inning!

I wonder what the all-time record for total variation in a single game is?  It would have to be a game with multiple extra innings in which runs scored, I’d think.

And we go to the bottom of the 11th, still tied 5-5.  Minnesota with a 64% win probability per FanGraphs.  Joe Mauer coming up third this inning.  Now that my own team is done playing for the year, I am allowed to say:  go Twins.

Update:  In the comments, Michael Lugo proves by science that the Tigers-Twins playoff (total variation:  7.69) was more exciting than game 7 of the 1991 World Series, but less exciting than this.

Update 2: A similar computation carried out in 2005 by Dennis Boznango at The Hardball Times. An even more similar discussion at FanGraphs.

## 9 thoughts on “The total variation of win probability, or: THE MAGNIFICENT TWINS-TIGERS GAME”

1. Michael Lugo says:

On your extra-inning suggestion: July 7, 1993 Phillies-Dodgers (7-6 in 20 innings) has total variation of 1035%. (Add up the absolute values of numbers in the wWPA column.) And May 9, 1984 Brewers-White Sox (7-6 in 25, with both teams scoring three in the 22nd) has total variation 1466%. (You wrote about this game before.)

It doesn’t seem quite fair to count extra-inning games, though. (Or at least one should normalize to total variation per inning?)

I’d nominate Game 4 of the 1993 World Series (476%) as most exciting game ever, but I’m kind of biased by the fact that I was there.

2. Jeff says:

If we’re talking about exciting World Series games, look no further than game 6 and Game 7 of the 1991 World Series. I’m one of those guys who generally finds low-scoring games exciting though

3. Michael Lugo says:

Also, this paper of Vecer, Ichiba, and Laudanovic (from the web site of a class by Aldous) gives your total variation distance as a measure of excitement in games, and work out some expected total variations for soccer games (where apparently scoring can be modeled well by a Poisson process.)

The paper is from the Journal of Quantitative Analysis in Sports, which is going to waste too much of my time in the near future.

4. JSE says:

Michael: did you add up those absolute values yourself or do you have a script that does it? What’s the TV for game 7 of the 1991 WS? I too would count that among the most exciting games I’ve ever watched, but it has a different profile — “relatively high TV given that WE stays within a small band around 50% most of the game” might be an interesting criterion to look at.

5. Michael Lugo says:

Added it up myself. (Someone should write a script, but that someone is not me.) 91 game 6 is 426%; 91 game 7 is 412%. This is actually kind of surprising to me, that those come so close to the high-scoring game I mentioned.

Rangers 30, Orioles 3 (which you probably don’t want to be reminded of) is 216%. I was kind of thinking this would be the least exciting game ever, but I forgot that the Orioles actually led 3-0 at one point in that game. Presumably the least exciting game ever by this measure is a game in which the visiting team gets out to a large lead in the top of the first and there’s no scoring afterwards.

6. JSE says:

Well, the theoretical minimum is 50%. I wonder what’s the closest to this ever attained? Somebody who knows how to do such things should write a script to pull this information from the Retrosheet box scores.

7. Anonymous says:

I have anti-Twins feelings (although stronger anti-Yankees feelings), mostly because of a slightly fractious Athletics-Twins play off series. On the other hand, the greatest offender on that occasion was A J Pierzynski, who doesn’t even play in Minnesota anymore.

As for the “perfect 50% game”, I remember the Red Sox scoring 10 runs without an out in the first inning, and going on to some huge victory, except that may have been at Fenway.

8. Michael Lugo says:

anonymous 7: Marlins – Red Sox, June 27, 2003 was at Fenway. The Red Sox scored 14 in the bottom of the first. Total variation: 70%. (The Marlins scored in the top of the first in that one.)

Thinking that a big first inning is a key, and Googling a few things, a better candidate is a Phillies-Reds game from earlier this year — the Phillies scored 10 in the bottom of the first to get a 10-0 lead. Total variation is 56%. This Reds-Mets game from 1988 also is at 56%, also with 10 runs in the bottom of the first.

But in 2005 the Dodgers scored 10 in the top of the first in Cincinnati — and by the method of calculation I claimed above, this gets 50%. (I suspect, though, that it may be possible to get below 50% by this method due to rounding error.) At this point I claim that we’re not going to find a candidate for the game that absolutely minimizes this quantity, and that win probability isn’t defined well enough to make that possible. Furthermore, 50% exactly (without rounding error) might not be possible; in such a game every plate appearance would have to increase the winning team’s win probability, so it’s not clear to me how an inning would ever end! It should be possible to get arbitrarily close to 50%, however, by scoring large numbers of runs in the first inning without an out.

As for the game that started this: yesterday’s Twins-Tigers game is on Baseball Reference now and scores at 769%.

9. I don’t think you can really compare this game to the 1991 Game 7 in this manner. I would think in the playoffs, you’d also have to take into account the variance of pWINWORLDSERIES. The Twins-Tigers games was ridiculous exciting in and of itself, but I think, all else being equal, a game in which pWINWORLDSERIES can go from 0-1 should have an advantage over a game in which pWINWORLDSERIES can essentially go from 0-.125 (or so).