baseball thought experiment

I’m trying to hate this blog a little less, lately, and part of the way I’m gonna do that is just write stuff that I think is interesting and worthwhile to me, to be a little more esoteric and a little more weird. For most people, this may be translated as “even more self-indulgent.”

In that spirit, I’d like to propose a baseball hypothetical to you.

Suppose you had a pitcher who you knew would, unerringly, give up one run every inning. It’s irrelevant how that run gets scored, although for the sake of argument let’s say it’s always an earned run. And let’s stipulate that while he gives up one run an inning, he always gives up a run in any inning in which he appears — if you bring him in for one out after another pitcher retires the first two batters, he will unerring give up a run before getting that out. If you keep him in the game after that, he will retire the next two batters but without fail give up a run to the third batter, keeping his average of a run an inning intact.

So: is this pitcher a terrible asset? The most valuable asset in baseball? Or where in-between?

By conventional metrics, he’s terrible. His ERA is 9, after all, a number so high as to get you drummed out of baseball entirely, not just the majors. You’d never, ever want to start him; the team with the highest runs scored per game in 2015, the Toronto Blue Jays, only scores 5.48 runs a game, meaning they’d expect to lose any particular game started by this pitcher by 3 or 4 runs. In fact, the best run-scoring team in the “modern” era, the 1930 Yankees, scored 1,067 runs in a 154 game season for an average of 6.92, again meaning that you’d expect to lose the average game this pitcher started by a couple runs, and even if you only intend to trot him out there for 6 or 7 innings, he’s leaving the game out of reach of all but the best offenses.

And yet! Think about his value as a reliever: if you ever take a lead that’s greater than the number of innings left in the game, you know that you will win that game. He’s the ultimate closer, because if you have a two run lead going into the 9th, or a three run lead going into the 8th, or a four run lead going into the 7th, the game is over, regardless of what the offense does. He’s the ultimate long reliever in a game that you blow open in the early innings, allowing you to rest your starters and let him trot out there and mop up the remainder of the game. And he’s a great firefighter; if you’re in a terrible jam, with the bases loaded and the game’s most feared hitter at the plate with nobody out, you can bring him in knowing you’ll only give up the one run, even if you decide that’s the only batter he’ll face all game.

Clearly, his value changes depending on your team’s offense. He’s much less valuable on a team with an anemic run scoring ability. But otherwise, what do you think: how valuable is he? What would be an appropriate salary? What kind of assets would you give up to get him in a trade?



  1. In Coors, he’s worth a ton. Just load up on hitters and then many games are over.

    I assume runs are scored mostly in clusters which adds to his value.

    He’s probably worth more than Kershaw.

    1. Very good point. If this guy could pitch every day, he’d have an impact on how you assembled the rest of the team, the lineup, everything — and a corresponding impact on the opposing team, though smaller because they only see him in <10 games/year.

      If your goal is to maximize the odds that at some point you'll be up by more than the # of innings remaining, rather than your odds of being up at the end of nine (though I realize both are oversimplifications), it would probably have impacts on strategy and counter-strategy that are hard to predict in advance. Pro sports strategy & payroll, especially in the Moneyball era, are getting to be a pretty complex system; something like this could be a kind of butterfly effect.

      While this player is just a thought experiment, I wonder whether every sport should deliberately introduce some twist to the rules every five years just to keep things interesting…

  2. This is interesting, and if the 1-run rule would hold without exception and he wouldn’t need the same amount of rest as other pitchers, I think he’d be much closer to the best asset in the league than just an average one. If his perfect consistency would lead to his team winning 10 extra games, then he’d probably be deserving of the MVP (Bryce Harper’s WAR is 10.3 currently). Even he’d only lead to an extra 6 wins, that still puts him as valuable as a guy like Buster Posey, who’s currently the best catcher in the sport.

    Also, there’s the issue of scarcity. It’s not easy to find a Zach Greinke to be in your bullpen, but it’s not too hard to find a player who’s 60% as good as him. But in this scenario, it doesn’t seem like another player like this exists, so his relative value would be much higher.

  3. I’d say he has some value, but it’s pretty limited.

    You can only use him in games where you lead by more runs than there are innings left or when you’re getting blown out and are just throwing in the towel.

    Obviously, in the first situation he’s pretty valuable. But probably not that much more valuable than any decent pitcher.

    For example the odd of the home team winning if they’re up by two with one inning left to play is about 94%. With our hypothetical pitcher, the odds are now 100%. Is that 6% edge, roughly one extra win out of twenty chances, worth carrying this guy on your roster? Especially since you absolutely do not want to use him any other situation?

    1. A large amount of 9th inning saves are by more than 1 run. He’d be worth keeping on the roster, especially given that he apparently needs no rest and can be equally effective whenever he is used.

    2. You can only use him in games where you lead by more runs than there are innings left or when you’re getting blown out and are just throwing in the towel.

      Not true, though. Up 5-2 in the 7th, and you might use him to ensure you get to the 8th with a 2-run lead when your best relievers could be called on. Not saying it’s ideal or even wise usually, but it would happen.

  4. Based upon this:

    I would say that guy is worth, oh, about $35 million dollars a year.

    About 73% of the games are decided by 3 runs or less. That’s 118 wins, or enough to waltz backwards into the playoffs any year, decided by < 3 runs. About 70% of the games are decided by 2 runs or more, if you want to look at it the other way around, that's still over 110 wins.

    Either way, even without a more robust analysis of the score distributions by inning, I have to assume that the utility calculus of that player is beyond just measurable, it's substantial.

    Any game when you're at home, and you're ahead by 2 runs or more in the bottom of the 8th, that's a guaranteed win. Any game when you're away and you're ahead by 3 runs or more in the top of the 8th, that's a guaranteed win.

  5. I love this. I think the first thing you do is get someone to crunch the numbers on exactly how often the situation occurs over the course of a season that a team is leading by more runs than there are innings left (L), and then calculate the proportion of those situations that end up being converted into actual wins (c). The number of wins (W) this player is worth is then W=L(1-c). E.g. if on average a team finds itself in a situation of leading by more runs than innings remaining 20 times in a season, and under normal circumstances it goes on to win 75% of those games, then this player would be worth 5 wins per season. I think that’s the beginning of how you figure out how much to pay him, whether he’s worth a roster spot, etc.

    But here’s another wrinkle. If you’re the team’s owner instead of the manager, your calculation might be different. The manager basically just cares about winning games, but the owner has to think about dollars and cents. Putting this guy in effectively ends ball games, so you could expect a very predictable drop-off in viewership and ad revenue every time he goes into a game. If your team goes up 10-0 in the first inning then the stadium will empty out, nobody will stick around to buy overpriced beer and hot dogs, and the TV and radio broadcasts might as well just end. So even if the numbers indicate this player could be worth a lot of wins to a team, you might have trouble convincing the owner to sign him.

  6. Nah, he’d be worth a ton. You could lock games down guaranteed, and it would happen at least ten times in a year, probably more. So he’d command a high salary.

  7. Many of your best posts were definitely in 2008-2011, when you weren’t refreshing Twitter every 30 seconds … more posts like this!

  8. Is he always available, impervious to overwork? If you got a 7 run lead in the third inning would you send him in for the guaranteed win if that means he’ll be unavailable for the next four or five games? Or are you saving him strictly for later inning situations since you’re already at high probability of winning that game? How good is the rest of your bullpen? If they’re really good how many late inning multiple run leads will they blow in the course of a season? How much has management budgeted for players? What is the nature of the ability that allows him to not allow any runs after one, and could it be used more profitably in some other aspect of baseball (or horse racing for that matter)?

    1. Assuming the other teams are aware of his ability, why shouldn’t they simply concede the game and leave the field once he enters? Would MLB allow this to go on?

  9. He’s got a superpower: certainty. Guaranteed to win under the right circumstances, and those circumstances are clear to anybody who understands arithmetic. However, the presence of this superpower subverts the whole point of the game, which is uncertainty. One reason people choose to watch baseball and not pro wrestling is because the outcome is not a foregone conclusion. So in the long run the existence of such a superpower would probably degrade interest in the game, but in the short term you’d be crazy not to keep this guy on the roster.

  10. First of all, you can find win probability calculators on the internet, such as The win probability of a team winning by two going into the ninth is 94.41%, nothing at all like the 75% proposed above. So if I team is in this position 20 times, that’s one win. For three runs and two innings, it is 94.94%. It is also easy enough to determine, from publicly available statistics on the internet, the number of times per year a particular team leads by these numbers at these times of the game. If someone wants to do the work, the total number of games this pitcher permits the team to win over the number they would actually win is easy enough to calculate. I’m guessing the total number would be less than five, probably around three, but I wouldn’t be surprised, which brings us to the second of all.

    Wins Against Replacement, the various implementations of which are known to any baseball fan who is into stats, tells us how many pitchers are worth three wins per season. There are three main websites that calculate WAR in three different implementations (actually four, since one has two different implementations); but all run roughly the same numbers of pitchers that reach given levels. Without going into details about how WAR is calculated by different sites, I’ll mention that according to Fangraphs year to date numbers as of today, there have been 29 pitchers worth at least three wins this season, and 11 worth at least five wins. So that puts in some perspective the value of this proposed pitcher.

    Third, the thing that makes this pitcher more valuable than his ERA is called sequencing, which is ruled largely by random variation. Of course if you could control your sequencing perfectly in baseball (only hit a home runs when the bases are loaded, not when the bases are empty, and hit the same number of home runs), your number of runs scored and number of wins would outpace your statistics. Numerous people have studied this, under rubrics such as “clutch”, and determined that there is no such sustainable skill in the sport. (I.e. a high clutch score in 2014 has no correlation to a high clutch score in 2015.) Would teams pay for performances that were predictably “clutch”? Of course – through the roof.

    Finally, and this is the real point: if you’re a stats nerd and a baseball fan, there are plenty of places to satisfy your curiosity – and much more interesting questions to ask.

    1. Actually, there is a miscalculation in these numbers stemming from the fact that the number of absolutely wins for the pitcher in question with the perfect sequencing is an a number of wins against average, not a number against replacement. The “average” of number of wins a player is worth in baseball stats is 1.33 (1000/750). (750 is the number of players on an active roster at a time; 1000 is the total number of wins above replacement allotted in a season, which is another of those things I just have to point you to Fangraphs for if you want the explanation.) So I’ve understated the value of this pitcher by 1.33 wins – probably he’s worth around 4.33 WAR, according to my best guess.

    2. It’s a thought experiment, not a question about real baseball.

      It would take a lot of work to figure out his WAR because he does more than come in in the 7th when you are up 4. He comes in with the bases loaded and no outs in earlier innings. Assuming he doesn’t need rest, I would guess his WAR would be sky high.

  11. A few years back, the Detroit Tigers could have used a guy like this in the bullpen, they would have probably made the World Series. In the playoffs, they had a few games that had brilliant starting pitching, then the bullpen crapped the bed.

  12. I was originally thinking of this in terms of only his ability to close out games, and I think the bulk of the commentary is correct that, from that angle, he’s only little better than the other assets you could use for that purpose. I think a very cursory way of understanding this is that the average pitcher saves the vast majority of games that this dude saves. He helps you clean up a pretty small tail, and is useless in a lot of other situations.

    But Adam’s comment really changed the way I look at it. There are a ton of situations throughout the season where this dude is better than the average pitcher, and cleans up a sizable tail of very bad outcomes. Bases loaded; no outs; heart of the order coming up–this guy is a godsend.

    Finally, when you get to the playoffs and the law of the large numbers takes a back seat to certainty, he’d also start to look valuable. I wouldn’t be surprised if the right answer is to try to keep him in your minor league system, and call him up every September.

  13. Without knowing much about baseball, the obvious advantage I see is that this pitcher levels the defense skills of the entire team. The team building consequences of that are huge.

    if this guy always pitches, no other player on the team needs to be able to catch a ball. That means every single bit of training is specialized to getting runs – preventing them is covered, even if it’s covered poorly.

    Beyond training, there’s also hiring. The only thing the scouts need to worry about is someone’s ability to get runs, not prevent them. That means they never need to think about tradeoffs – x number of points worth Y dollars. It also opens a big pool.

    Now my lack of knowledge is showing, but when you only look for one skill, you open a wide pool of talent that other teams don’t have access to. You can hire hard hitters with no sense for the game, people who can’t catch, people who get tunnel vision, etc. Other teams don’t have access to that pool (or at least can’t afford to outbid you in it).

    I’m out of my domain here, but here’s the simple question: if you had to build a team that usually scored more than one run per inning, and didn’t have to optimize for anything else, how hard would it be?

    1. Legendarily hard. I used an online Runs Created calculator and inserted Ted Williams’ lifetime stats: A team of nine Ted Williams would score 1.02 runs per inning.

      Suffice to say, nine Ted Williams are not available via trade or free agency.

      1. … more realistically (heh), a team of nine players identical to Mike Trout in his Rookie of the Year season (when he led the league in both OPS+ and steals) would score only 0.82 runs per inning.

  14. Practically worthless. There are two aspects of this pitcher:

    First, as a pitcher who gives up neither more nor fewer than one run per inning, there is an illusory tactical appeal—a manager could bring him in to ice out the game any time his team had a five-run lead in the top of the sixth. However, the value of that compared to the alternative is almost nil. In 2015, MLB teams already win 83% of games when they enter the sixth inning with any lead (my command of baseball-reference’s Play Index isn’t strong enough to find out the percent of 5+ leads held, but it’s got to be much higher). Nearly 75% of innings result in zero runs score; only 12% result in more than one run. The mythical pitcher described gives up a tremendous amount of zero-run upside to guard against a very small downside, and more importantly, a downside that even replacement-level pitching already controls quite well.

    There’s a tremendous overvaluation of closers in baseball, and it’s kind of interesting to wonder why. From a Bayesian perspective, it seems like the expected value of the fireman comes from a conflation of things teams do when they’re winning and things teams do in order to win. To analogize to football, the team whose quarterback is kneeling is almost always winning the game, but a team whose QB did nothing but kneel would never win. We’ve all seen Mariano Rivera come in with a 3-run lead and win the game and we think that he “saved” it, but the game was 95% won and lost before he even started stretching, and far lesser closers usually notch the same save.

    Then there’s the second aspect of this pitcher, that any inning in which he appears results in at most one run after he enters. That is considerably more interesting—but I’m pretty sure it’s cheating. The hypothetical pitcher is a 9.00 ERA pitcher, and I understand the terms of the hypothetical, but it’s worth remembering that inherited runs are not charged to the inheriting but to the bequeathing pitcher. To say that a pitcher like this could enter with no outs, the bases loaded, and 2001’s Barry Bonds at bat and still only give up a single run is to combine two distinct and remarkable concepts: the pitcher that gives up one run per inning with eerie consistency, on the one hand, and secondarily also the magical ability to erase inherited baserunners, presumably in a manner reminiscent of the opening scene of The Last Boy Scout. That is dramatically out of scale with the other parameters of the experiment, and it strikes me as an accidental misstatement of what you’re actually trying to get at.

    1. Let’s assume that the hypothetical pitcher doesn’t have magical abilities to erase inherited runners, but will always give up 1 ER upon entering a game with the bases empty.

      The hypothetical guy still has value, especially in the era of the 7- and 8- man bullpen. Lloyd McClendon (and I think any manager) would love to lock down the 2-run and 3-run leads in the top of the 9th, rather than run the risk of the blown save. Fernando Rodney had 7 appearances in which he gave up 2 or more runs in an inning, and while those 7 games wouldn’t have vaulted the Mariners to the playoffs, they would at least have put them above 0.500.

      There’s a lot to be said for a tool that provides you with a guaranteed win given defined parameters, even if the parameters appear infrequently!* Jeff Banister would have killed for this guy on the second-to-last day of the season.

      *let’s say only 10% of the time, although if you dig around in play-index looking for leads of 4 plus entering the 7th, 3+ entering the 8th, and 2+ entering the 9th I’ll bet it’s more than that. And then you blow the lead 10% of that time…1.6 wins on the year would be very meaningful to the Twins or Angels.

      1. You’ve got to read the posts by Kenny above—pro teams do not blow anything like 10% of those leads. This is the essential point that most people, even some baseball fans, don’t grasp: A league-average reliever will convert nearly all of those hold and save opportunities of Johnny Consistently Rotten (that is, 95% of his strategic advantage), while also doing things the imaginary pitcher can’t—like pitch a five-out save with a one-run advantage, or pitch two clean frames in extras to give the offense a chance to come back. Mariano Rivera can do those things, and so can guys like Brandon McCarthy—but the imaginary pitcher can’t.

        I still can’t find out how many games, exactly, an average team manager would be able to use such a pitcher strategically, but looking at the playoffs we discover a manger could have brought him in several times, but he only would have been of use once (for the Astros in Game 4 against KC):

        ALCS: Game 1 in the 8th (Royals up 3-0; KC eventually scored again to win 5-0), Game 2 in the 9th (Royals up 6-3, the eventual score), Game 3 in the 4th (Blue Jays up 9-2; they eventually won 11-8 after four KC runs in the 9th), Game 4, 8th (Royals up 9-2, they won), Game 5, 7th (Blue Jays up 5-0; Toronto won), and in Game 6 he could not have been brought in at any moment without blowing the lead.

        NLCS: Games 1 & 2 in the 8th (Mets up 4-1 each), Game 3, 8th (Mets up 5-2), Game 4 in the 6th (Mets up 6-1). Obviously the sweep means it wouldn’t have made a difference.

        KC-HOU: Game 1, 9th (Astros up 5-2), Game 2 he could not have been brought in without blowing the lead, Game 3, 8th (Astros up 4-1, Houston won), Game 5, 9th (KC up 7-2, KC won).

        In Game 4, neither team had a greater than a one run lead after the top of the 2nd until the 7th, when Houston scored three times to take 6-2. The ‘pen blew the lead by giving up 7, and the mythical pitcher would have preserved a 6-4 win.

        TEX-TOR, Game 1, 9th (Tex. up 2 and won), Game 2, 14th (Tex. up 2 and won), Game 3, 7th (Tor. up 5 and won), Game 4, 5th (Tor. up 6 and won), Game 5, 8th (Tor. up 3 and won).

        Cubs-Cards, Game 1, 9th (Cards up 4 and won), Game 2, 8th (Cubs up 3 and won), Game 3, 8th (Cubs up 3 and won), Game 4, 9th (Cubs up 2 and won).

        NY-LA, Game 1 in the 8th (Mets up 3 and won), Game 2 in the 8th (Dodgers up 3 and won), Game 3 in the 5th (Mets up 7 and won), Game 4 in the 9th (Dodgers up 2 and won), Game 5 he could not have been brought in at any point without blowing the lead.

        Wild cards, the Cubs and Astros could have brought him in in the 7th and 8th, yet somehow they managed to protect their 4-0 and 3-0 leads without Mr. Rotten’s assistance by blanking the Bucs and Yanks.

        Wow, so terrifically valuable, right? If only the Astros had devoted that one roster spot, they might be in the World Series for the first time since… ’05, I think? Except the other 9 teams would have found him useless. Teams played 258 1/2 innings (well, 317, since we’re pretending he could have pitched for either team), and this pitcher would have spelled guaranteed defeat in all but 53 of them. In three games, he could not have been used by either team without throwing the game away. The Yankees and Pirates never could have used him. The Cards could have used him for a single inning in which the actual Cards bullpen did not allow a run. The Royals actual bullpen blanked their opposition in all these theoretically useful innings but one, and the Dodgers blanked them in all. Toronto’s actual pen threw zeroes in all such situations except a game they were up 7-1 (ultimately winning 8-4). The Mets’ pen is pretty lousy, but even they put up zeroes in all but three frames that the hypothetical pitcher could have taken up (twice giving up one run and once giving up two in a laugher), and they won all results.

        Joe Posnanski has a good summary of basic scoring tendencies:

        It is far, far easier to get through an inning of a ballgame than is appreciated even by fans. Over time, teams have scored, on average, half a run per inning, maybe a little more or a little less.

        This guy is borderline useless.

  15. With the caveat that I’m not a huge baseball fan, I’m not surprised that the baseball metrics break down in this situation, since they were designed to quantify uncertainty (“given this insanely detailed history of Player A’s past performance, here’s an estimate of his future performance…”). Our hypothetical One-Run Johnny is perfectly predictable and reliable every time, so it’s sort of like Newtonian physics breaking down at relativistic scales – it’s a generally useful system being applied to a totally incompatible situation, and it produces accordingly bizarro results.

  16. Here’s the scenario that I love: conceivably, if this person existed, you could pitch them two innings per game, every game, for a total of 324 innings. If he walked the first five batters every time, that would be 324 innings without giving up a single hit.

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