Billy Hamilton, On Second, With No Outs

When baseball returns next month, it will be a little weird. Not just because of the empty stands, though of course that will be weird too. Not because of the NL DH — despite the bellyaching about the sanctity of the game, baseball with a DH feels more or less the same as baseball without one. No, I’m talking about the new extra innings rule, which will place a runner on second base to start each half inning beyond the ninth.

A runner on second by itself isn’t weird, but having it happen every inning without a leadoff double will definitely take some getting used to. It’s not all dark clouds, however, because weird baseball rules create weird baseball situations. Effectively Wild listener Brett Mobly wrote in about a particularly interesting angle, and via the magic of Ben-to-Ben communication (read: Meg emailed me about it), here we are.

Mobly wondered about a hypothetical that Jeremy Frank posed on Twitter. What if Billy Hamilton comes to the plate with two outs and no one aboard in the bottom of the ninth? Given that the runner who starts on second base is, by rule, the player on the batting team due up last in the batting order, a Hamilton out comes with a huge carrot: the best baserunner in baseball starting the next inning in scoring position. How should that change his behavior at the plate?

We can start by eliminating the extreme scenario of Hamilton purposefully making an out. Even if he were intent on starting the next inning on second base, he could do better by taking a regular at-bat and then simply running until he’s thrown out. Single? Steal second, then try to steal third, then try to steal home. The end result will either be an out — the same result as purposefully making an out — or a game-winning run. Heck, he might luck into a home run, unlikely as that sounds.

Alright, so the manager isn’t throwing up the intentional strikeout sign. That doesn’t have to stop our fun. We can still work out how valuable it is to have Billy Hamilton on as an extra-innings runner, as well as how aggressive he should be on the basepaths.

First things first: how valuable is that runner on second base? We can go to 2019 run expectancy data to find out. From the position of a runner on second base and no one out, teams scored 1.18 runs until the end of the inning (excluding all innings past the eighth). With the bases empty, that number is only 0.54. Hey, the runner is worth 0.64 runs, easy!

We can do better, though. Run expectancy isn’t exactly what we want — we want the odds of scoring one run, two runs, and so on. Luckily, that’s easy too. All we have to do is look at every time a runner reached second base with no one out and note how many runs scored in the rest of the half-inning. Then, rather than taking the sum of the runs scored, we can turn it into a frequency of each possible outcome, like so:

Next, let’s ask another question: how valuable is Billy Hamilton as the runner on second base? Over the course of his career, pretty valuable, as it turns out. He’s been the runner on second in this situation — man on second, no outs — 131 times in his career and scored 75.6% of the time. That compares to 60.8% of the time for the league as a whole. It’s a small sample, to be sure, but unsurprisingly having the fastest man in baseball in that spot is helpful.

From here, it’s time to work backwards. Let’s assume that we’re going into extra innings and starting with Hamilton on second. The opponents get to bat first. We know, from above, that we’ll enter our half inning tied 39.2% of the time, down a run 31.7% of the time, and so on.

Next, we need to figure out our team’s run distribution. Rather than use Hamilton’s exact distribution of runs, I’m going to use a little bit of a cheat. I’m going to start with the league average run distribution and simply add one-run innings to get up to his 75.6% success rate. Why? Because I can’t see Hamilton’s speed turning a three-run inning into a four-run inning very often; presumably the vast majority of times he turns a dry inning into a successful one, he’s moving the team from zero to one on the scoreboard.

From there, we can simply work out the odds of winning given our deficit at the start of our half inning. Take this example, when we enter the inning down by a run:

If we assume that ending the inning tied gives our team a 50% win probability, we can simply sum up the likelihoods of each outcome to get our win probability entering the bottom of the 10th with a one-run deficit: 52.4%. That’s right: trailing by one makes us favored in this new weird world of baseball, at least with the best runner in the game on second base (we’d be 45% to win with a random runner on second).

By adding up all the possible scenarios and multiplying by their likelihood, we get to an answer: merely by changing the identity of the runner on second from random baseball player to Hamilton, our team’s odds of winning rise from 50% to 55.2%. That’s a shockingly big win probability swing, but Billy Hamilton is a shockingly fast player, and he’s in a situation where those skills are most necessary: in a close game when he gets on base automatically.

With those extra-innings odds in mind, we can simply work backwards. We know that with Hamilton batting in the ninth, his team’s chances of winning are at least 55.2%. If they were lower, he would simply start stealing bases until he got caught or won the game. Even after that, though, we’ll need some adjustments.

The normal win expectancy for a team with a runner on first base and two outs, batting in the bottom of the ninth in a tie game, is 56.5%. So should Hamilton just cool it after he reaches first base? After all, 56.5% is greater than 55.2%.

Of course not! To explain why not, we’ll need to do a little more working backwards. Let’s say Hamilton reaches second base. From there, not quite any hit will score him, but it’s close to that. For his career, he’s scored from second on singles 74.2% of the time. Let’s punch that up to 85% of the time with the better jump and devil-may-care baserunning that go hand in hand with the two-out game state.

Next, let’s give the next batter an exactly average batting line (for a non-pitcher, what with the universal DH and all). That leaves the odds looking something like this:

From there, we need to iterate again. Our team’s odds of winning are clear after an out (50%) or after Hamilton scores (100%), but the other two need further work. We can just use generic win expectancy from the first and third situation, because the identity of the runner on third barely matters with two outs. That’s 64.4%. For first and second, we need to do another quick calculation using the same odds of Hamilton scoring on a single. If there’s another one-base advancement, then we can just use the odds of scoring with the bases loaded. I’ll save you the nitty-gritty, but our team wins 63% of the time from a first-and-second, two outs, Hamilton on second setup.

Blending all of that together, our win expectancy is 61.9% when Hamilton stands on second base with two outs. This sets up one of the most lopsided steal situations imaginable. With Hamilton on first, we’re 56.5% to win. Make it to second, and our odds increase to 61.9%, a 5.4 percentage point increase. If he gets thrown out, our odds only fall by 1.3 percentage points. That means he only needs to be successful stealing second 19.4% of the time to break even.

To put it lightly, Hamilton is more than 19.4% likely to succeed in a base theft. For his career, he’s been successful 79.9% of the time when attempting to steal second. Of course, he doesn’t run literally every time, and we need him to do so in this situation. Let’s knock the odds down to 70% to account for the times where he runs despite a poor jump. That still means that if Hamilton reaches first base and then automatically attempts to steal, our team has a 59.9% chance of winning the game.

We’re now close — so close!! — to solving a question which no one asked but I still wondered — how likely is our team to win in this position? Hamilton at the plate, tie game in the bottom of the ninth — forget the goofy scenarios and automatic outs and whatnot, how likely is our team to win?

To solve this one, we need an extra layer: Hamilton’s career production at the plate. We’ll simply reduce it into which base he ends up on after his plate appearance, as well as the team’s odds of winning from that point:

Overall, that makes our team’s odds of winning a whopping 57%. A regular batter in that spot checks in at a puny 53.9%. Hamilton improves our odds of winning by 3% — and he’d improve our odds of winning by one percentage point even if he made an out every time he came to the plate!

That’s a wild thing to say out loud. Give a team the option between a league average player getting a plate appearance or Hamilton automatically being out, and they should take Hamilton. His speed on the bases is simply that important.

We can still do better! Starting any situation with “Billy Hamilton at the plate” is a big disadvantage. Let’s hold him in reserve as a pinch runner or defensive replacement. Our generic league average hitter will bat. If he reaches base, we’ll sub Hamilton in to go hog wild on the basepaths. If he makes an out, it’s double switch time — Hamilton is coming in to the game as a defensive replacement and to start the tenth inning standing on second base.

In this situation, our Fightin’ Hamiltons win 58.3% of the time. Why? The average batter hit a home run on 3.74% of plate appearances in 2019, as compared to 0.69% for Hamilton. That’s an extra three percentage points worth of home runs. Coincidentally, Hamilton gets on base three percentage points less often than the league average hitter — in other words, he has an OBP 30 points lower. Turn three percentage points of outs (55.2% win probability afterwards) into homers (100% win probability), and that’s .03 * (1-.552) or 1.3 percentage points of bonus win probability.

The bottom line is, this rule was made for Hamilton. It very specifically makes the things he’s best at more important, while minimizing his greatest weaknesses. If you were predisposed to dislike this rule, that might be a good reason for you to really dislike it. The rules of the game have been twisted to make this formerly marginal player a superstar. Boo! You knew a runner on second was ruining baseball!

Of course, you’d be wrong. This new rule doesn’t transform Hamilton into a superstar. Putting him on base is the most value you can get out of this rule, and it’s simply not that much. Consider: 8.6% of games required extra innings in 2019. That works out to roughly five extra inning games per team in a 60-game season. Hamilton is worth 5% win probability as the free baserunner in those games. Toss in an extra 1% win probability for the chance he gets to pinch run in the ninth, and that’s 6% per extra inning game. That’s a whopping 0.3 wins, in expectation, over the course of the season.

Now, 0.3 wins isn’t the same as zero wins. It’s worth as much, roughly, as turning two random plate appearances into home runs. But that’s the top end of an estimate of Hamilton’s extra value. It’s likely lower in reality, because he can only sub in as a defensive replacement and pinch runner if he isn’t already in the game. Not only that, but we’ve only calculated the value for Hamilton playing on the home team. Sub in as a visitor, and the opposition knows the target it needs to hit before starting the inning. If Hamilton manufactures a run with speed, the home team can play for a single run and try the whole thing again next inning without the best baserunner in baseball starting on base.

In the end, is Billy Hamilton a really cool corner application of the new extra innings rules? Unequivocally yes. But his impact is small, and the impact of this rule in general is small. It’s not distorting the balance of the game. If anything, it’s adding a little fun. Do you want to see the Giants employ this strategy? I can’t wait! And all it costs, in terms of changing the outcome of games, is a piddling 0.3 wins per 60 games. I’d pay that price to see Hamilton on first, taking off regardless of the odds against him because of a strange situation that incentivizes making an out, in a heartbeat.