## 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.