came to mind vividly.

It was the 8th inning, the Nats were down a run, and there was a runner on first with nobody out. The manager decided it was time for a bunt, and called for Livan Hernandez, an expert bunter. Trouble was, he was the next night’s starting pitcher, and was back in the clubhouse watching the game on TV, and not fully dressed (starting pitchers play only every 4 or 5 games). After a lengthy delay, he straggles to the plate, tugging at his trousers. He bunts at the first pitch – strike. Disoriented, he fishes his batting gloves out of his pocket and puts them on. He bunts again, another strike. He notices that his shoes are not tied, and takes time out to tie them. Finally, he seems ready, and lays down a perfect bunt, advancing the runner to second. Job done, he returns to the clubhouse, and resumes watching the game on TV.

We usually think of “expected value” in terms of money – a weighted average of possible revenue (or cost) outcomes, where the weights are the probabilities that those outcomes will occur. To financial analysts, it’s “present value” or “discounted present value,” which adds a discounting factor for money that does not arrive (or leave) until later.

To baseball quants (“sabermetricians”), the parallel concept is “expected runs” or “expected run value.” The most useful application is in assessing game situations – e.g. “runner on second, 2 out.” On average, a team will score 0.348 runs in the rest of the inning, when facing that situation. 0.348 is the expected run value of “runner on second, 2 out.” This information comes from the 1984 book by John Thorn and Pete Palmer, “The Hidden Game of Baseball.” The authors report that Palmer simulated all baseball games between 1900 and 1977 to arrive at the expected run value of all 24 runners/outs situations (0, 1, 2 outs, and 0, 1, 2, and 3 runners on all combinations of bases).

One noteworthy result concerns the sacrifice bunt, in which the batter taps the ball a short distance, making an out himself, but placing the ball in a location that makes it possible for a runner on base to safely advance to the next base. It turns out that the expected run value of “nobody out, runner on first,” (the pre-bunt situation) is 0.783, while the expected run value of “one out, runner on second” (the situation after a sacrifice bunt) is 0.699. On average, teams score more runs in the former situation than in the latter. So, a sacrifice bunt to advance the runner, in terms of global aggregates, costs runs.

Runs scored is what baseball people pay attention to, so the concept of “expected run value” makes sense to them. Of course, “probability of winning” is ultimately what matters, and it turns out the story is the same – the sac bunt reduces the probability of winning.

So far, we have been talking of averages – what has happened over all the games played from 1900 to 1977. The characteristics of individual players make a difference. If it’s the pitcher coming up to hit with a runner on first, the expected run value that he is facing will be less than if it’s the leadoff hitter coming up to hit, so the long-term averages do not apply (pitchers are typically poor hitters.) In such a case, it makes sense to bunt. The long-term averages are baselines against which to make strategic decisions, based on the specific capabilities of individual players.

And the Nats? Despite the perfect bunt, they failed to score the runner from second. And Livan Hernandez returned to the field the next day as scheduled, and pitched his 50,000th pitch during the course of the game.