Memorial Day was memorable for Padres fans for the wrong reason, like too many days recently: the Padres lost another frustrating game, this time on a walk-off hit by Albert Pujols against Craig Kimbrel.

There was some lively debate after the game regarding two separate decisions made by Bud Black: having Quackenbush start the 9th inning and intentionally walking Mike Trout with two outs to load the bases for Albert Pujols.

While I believe Kimbrel should have pitched the whole inning, it’s a much more subtle mathematical and game theory argument than the decision to walk Trout intentionally, which is very easily proven incorrect.

If Trout walks, the Padres get an additional opportunity to retire Pujols. If Pujols walks, the Padres lose. Trout gets less hits per plate appearance than Pujols does hits plus walks.

The math

Against a perfectly average pitcher – the average of every pitcher to face Mike Trout – Mike Trout gets out 60.5% of the time and gets a hit 26.0% of the time; these are basically his weighted career averages. 12.5% of the time, he walks to bring up Pujols anyway. The other ~1% of the time, he reaches via HBP or sac bunts. The out total above includes reached base on error, which also ends the game, so I’ll round down to 60% for out to get rid of both that and the sac bunt oddity in the data. (The calculation isn’t close, so a rounding error here doesn’t influence the decision making anyway.)

Bad Albert Pujols – ie. the one over the past few seasons – gets a hit 23.6% of the time and otherwise doesn’t make an out (via walk or HBP) another 8.5-9.5% of the time. Combined, the odds of Pujols winning the game (which is really just the odds of him reaching base or an error being committed) is roughly 33-34%.

That’s considerably higher than the odds of Trout getting a base hit, since he only gets a hit in ~26% of plate appearances. If Trout walks, though, it brings up Pujols, so the final Trout odds are .26 + . (.14*.33) =~ .307 =~ 30.7%.

In other words, intentionally walking Trout to face Pujols with the bases loaded lowered the Padres odds of winning by roughly 3%, assuming average pitchers.

Kimbrel is no average pitcher, but the effect actually leans further in favor of pitching to Trout: Kimbrel issues more walks than the typical pitcher while lowering the hit odds of both players somewhat proportionately.

Other considerations – the ability to force at third, the mental side of “not letting the best player beat you, and so forth – are either mathematically impractical or mathematically unprovable.

Conclusion

Ultimately, the decision didn’t lose the game for the Padres: Pujols didn’t walk.

But that shouldn’t stop people from realizing that the decision Buddy made was still the wrong decision, which had an expected value of at least -3% of the game.

This is a case where bad process ended up with a bad result. Poetic justice.