The final plate appearance of last night's game ended with Jonathan Diaz grounding into a bases loaded double play. In the recap, I semi-joked that Diaz should have just stood there with the bat on his shoulder, taken the K, and let Reyes determine the team's fate. Let's look at some numbers to see if that is actually a reasonable course of action.
Warning: Number heavy (though uncomplicated) post ahead.
Thanks to Tom Tango's run expectancy matrices, we know the expected value of each base out state. For example, the Jays were expected to score 1.631 runs with the bases loaded and 1 out. For ease not having to create my own Markov model, we're going to assume Jonathan Diaz (and everyone else on the team) is an average hitter and that the numbers in the chart apply to them exactly.
The average pitcher-hitter walk rate in recent years is about 3.2%. Let's say that Jonathan Diaz strikes a little more fear into the heart of a pitcher and that he's twice as likely to be walked even when actively giving himself up.
As the table shows, the Jays' run expectancy would have dropped by a tick over 0.7 runs in the above scenario. Note that the RE value for a walk is 2.631 instead of 1.631 because it includes the run that would score on the walk.
In the second scenario, in which Diaz swings away as he did last night, we have to project the chances of a whole slew of outcomes: K, BB, GB out, GIDP, FB out, LB out, GB single, FB single, LD single, FB double, LD double, and sac flies.
Regressing his K% 25% of the way to his weighted (don't bother asking how I weighted it) career Major and Minor League average gives a K rate of 25%. Doing the same for his walk rate gives us 10%, and leaves 65% of his plate appearances ending with a ball in play. Regressing Diaz's GB, LD, and FB rates 50% of the way to the MLB averages since 2008 gives rates of about 49%, 15%, and 36% respectively, which results in probabilities of 31.5%, 9.8%, and 23.6% when multiplied by the .65 proportion of PAs with balls in play.
Using data from FanGraphs and Beyond the Box Score and some intuition, we arrive at the table below that shows the projected probability of each outcome as a subset of the batted ball type. Below that is a table showing the run expectancy values for each possible outcome. As before, the RE values for events that score a runner (or two) have the value of the run(s) already baked into them.
After determining the probability of each outcome in the first table occurring overall, rather than as a subset of each hit type (the table is omitted for your sake), we multiply each probability by the RE values in the second table, and arrive at the following table which shows the expected run value for each hit type by outcome.
Summing up all those values gives us 1.160 runs. 1.631 - 1.160 = -0.471 runs by allowing Diaz to hit. Subtracting the -0.701 in the free strikeout scenario from the -0.471 above returns a net gain (well, a net reduction of loss) of .230 runs by allowing Diaz to swing away. Even accounting for the fact that Diaz is a poor hitter and Reyes is a strong hitter would not be nearly enough to flip the decision.
As much as it stings to lose an a bases loaded double play, not forcing Diaz to pretend to be an NL pitcher was, unsurprisingly, the right decision by a pretty wide margin.