Many basic baseball strategy decisions involve trade-offs between scoring or preventing one run, and scoring or preventing multiple runs. An intentional walk to a good singles hitter with one out and a runner on second, for example, will decrease the chance that the offense scores one run by reducing the likelihood of a base hit, and by creating a double-play opportunity. But it will increase the chance of a big inning by putting another runner on base, when he’d most likely have made an out if he’d been pitched to. Similarly, the sacrifice bunt and the stolen base are ploys to score one additional run that reduce the chance of a multi-run inning by potentially giving up an out.

There are some situations in which the decision to play for one run or multiple runs is straightforward. With the score tied in the bottom of the ninth inning, for example, one run will make all the difference, and teams are correct to employ strategies like the stolen base, sacrifice bunt and intentional walk very liberally. Most of the time, though, the situation is much more ambiguous. To take a couple of examples:

- The home team trails by one run in the bottom of the fifth, and has a runner on first base. Is this a good time to send the runner or drop down a bunt at the risk of sacrificing an out? Or is it still early enough in the game that the homers should seek to play for a big inning?
- The home team leads by two runs in the bottom of the seventh, with runners on second and third and one out, and a dangerous hitter at the plate. Should the visitors attempt to contain the damage by walking the hitter and hoping to complete a double play, or is putting an additional runner on and loading the bases compounding the problem?

In order to evaluate these questions, we need to consider the probability of a team winning the game given a certain scoring margin and in a certain inning. For example, assuming two sides of average strength, a home team that leads by one run heading into the bottom of the sixth inning will win the game about three-quarters of the time. This estimate, as well as those that will be used throughout the balance of this article, are derived from a model that I created here, which establishes win probabilities based on a historical distribution of MLB run scoring by inning (e.g. a team will score exactly three runs in an inning about 3.5% of the time). We will assume for purposes of this article that both the home and the road teams have average run-scoring and run-prevention abilities.

Let’s take a look, for example, at the probability that the home team wins the game given a particular scoring margin following the bottom of the seventh inning. The model estimates the home team’s winning percentage as follows:

Home Team Win Probabilities following 7th inning Score Home Win % +5 runs 98.2% +4 runs 96.3% +3 runs 92.6% +2 runs 86.0% +1 runs 74.1% Tied 50.0% -1 runs 25.9% -2 runs 14.0% -3 runs 7.3% -4 runs 3.7% -5 runs 1.8%

In other words, a team trailing by two runs after seven innings will come back to win the game about 14% of the time. Here, derived from the table above, is the marginal benefit to the home team’s win probability from scoring exactly one additional run during the bottom of the seventh given a particular scoring margin:

Marginal Benefit of One Additional Run for Home Team, after Top of 7th inning Score Home Win % Marginal Increase +5 runs 98.2% -- +4 runs 96.3% +1.9% +3 runs 92.6% +3.6% +2 runs 86.0% +6.6% +1 runs 74.1% +11.9% Tied 50.0% +24.1% -1 runs 25.9% +24.1% -2 runs 14.0% +11.9% -3 runs 7.3% +6.6% -4 runs 3.7% +3.6% -5 runs 1.8% +1.9%

As we would anticipate at this stage of the game, there is a profound inflection point around tied scores and one-run margins. A team that trails by one run and ties the score, for example, will go on to win the game about an additional 24% of the time. On the other hand, a team that trails by five runs and scores one run to cut the deficit to four will only win about an extra 2% of the time by making their job somewhat easier in the eighth and ninth innings. All of this should be fairly intuitive.

Note also that the distribution of the marginal increases in win probabilities is symmetrical: the go-ahead run is exactly as valuable as the tying run. This should be intuitive as well, as we assume that the teams are of equal strength, and that the home team will win when it enters the eighth inning with a 3-2 lead exactly as often as the visiting team will win when it enters the eighth inning with a 3-2 lead in its favor.

Along the same lines, we can evaluate the marginal benefit of scoring *two* additional runs:

Marginal Benefit of One Additional Run for Home Team, after Top of 7th inning Marginal Increase in Win Probability Score Home Win % +1 run +2 runs +5 runs 98.2% -- -- +4 runs 96.3% +1.9% -- +3 runs 92.6% +3.6% +5.5% +2 runs 86.0% +6.6% +10.3% +1 runs 74.1% +11.9% +18.6% Tied 50.0% +24.1% +36.0% -1 runs 25.9% +24.1% +48.2% -2 runs 14.0% +11.9% +36.0% -3 runs 7.3% +6.6% +18.6% -4 runs 3.7% +3.6% +10.3% -5 runs 1.8% +1.9% +5.5%

So, if the score is tied in the bottom of the seventh, scoring one additional run will increase the home team’s win probability by 24.1%, while scoring two additional runs will increase the home team’s win probability by 36.0%. Another way to look at this: when the home team hits a two-run home run with the score tied in the bottom of the seventh, the first run that crosses the plate is worth a 24.1% increase in winning percentage, while the second run is worth an additional 11.9% increase in winning percentage, producing the 36.0% total. In other words, in terms of its impact on win probability, the first run is worth about twice as much as the second. Taking a tied game and turning it into a one-run lead makes a huge difference; scoring an additional run beyond that is helpful, but not nearly as much so.

Of course, if the home team does manage to hit a two-run homer in the bottom of the seventh, it is going to be plenty happy about that, and not particularly concerned with these sorts of nuances. Most of the time, however, it is not quite that simple. If, say, **Johnny Damon** is at first base with two outs and the score tied in the bottom of the seventh, and **Manny Ramirez** is at the plate, the Red Sox face a decision about whether to have Damon steal, increasing the probability of a one-run gain (since a base hit will usually now score him) but reducing the probability of a two-run gain (since he might get thrown out, denying Ramirez the opportunity to hit a home run). Our results suggest that the Red Sox should be inclined to play for one run in this situation, as we would expect.

We can summarize the impact of these marginal effects on win probabilities by creating a ratio, which I will call the One-Run Value Yield, or ORVY. ORVY is a measure of the relative values of scoring exactly one and two additional runs in a given scenario. As we have seen, with the score tied in the bottom of the seventh:

- The first additional run is worth a 24.1% increase in win probability;
- The second additional run is worth a further 11.9% increase in win probability;
- The ORVY in this situation is the first number divided by the second number, or 24.1% divided by 11.9%, or 2.02. That is, as I have suggested above, the first additional run is worth about twice as much as the second.

The higher the ORVY, the more inclined a team should be to play to score or prevent one run, at the risk of sacrificing multiple runs.

The table below provides the ORVY for the home team given various scoring margins in the bottom half of each of the nine innings.

I am not sure how much the color-coding will help, so let me cover some of the more interesting implications of this table:

- The later we go into the game, the more attractive one-run strategies become. Again, this should be considered a highly intuitive result. But it would be easy to understate just how profound the differences are. In the bottom of the first inning of a tied game, the ORVY is just 1.09, which implies that the first run that scores is only about 9% more valuable than the second run that scores. Once we move to the bottom of the eighth inning, however, the ORVY is 3.64 in a tied game, meaning that the first run is around 260% more valuable than the second. Managers are surely behaving in error if they give up too much to sacrifice bunt early in the game, when a second run is nearly as valuable as the first. At the same time, statheads surely underestimate how viable one-run strategies can be when the game is late, and the score is close. Generally speaking, it seems that the inflection point for one-run strategies in today’s baseball environment comes in about the bottom of the sixth inning.
- The ORVY when the home team trails by one run is always exactly 1.00. That is, the go-ahead run is just as valuable as the tying run. Contrary to popular wisdom, a team should not be particularly eager to employ a one-run strategy when it trails by exactly one run, if this involves a significant reduction in the probability of scoring two runs and taking the lead.
- Employing a one-run strategy when a team trails by
*more*than one run is fundamentally awful baseball strategy. In some abstract sense, it may seem that it’s easier to score one run twice (teams score one run in about 15.7% of innings) than two runs once (teams score two or more runs in about 13.1% of innings). However, those times that a team has a good opportunity to score one run–say the leadoff hitter gets on–it will also usually have a good opportunity to score multiple runs. Sacrificing this runner to second when trailing by two runs in the bottom of the seventh, hoping to score one run now and pick up another one later, is not the sort of tactic that championship baseball teams employ. - On the other hand, a team frequently would be correct to play for an additional run once it already has a lead in the game. Coming back from a two-run deficit is a lot more difficult late in the game than coming back from a one-run deficit. If the home team leads by one heading into the bottom of the eighth, for example, scoring one more run and taking a two-run lead provides it with a huge advantage, more than halving the probability of losing from 16.8% to 7.8% (translating to an ORVY of 2.17). It should be willing to sacrifice some probability of a multiple-run inning in order to procure this. Strategies like sacrifice bunts and stolen bases are much more efficient when a team leads in the game than when it trails, even if it trails by just one run. Conversely, a team trailing in the game should work like mad to prevent
*any*additional runs from scoring (by employing the intentional walk, playing the infield at double-play depth, and so forth), even if this carries some risk of producing a big inning that would put the game entirely out of reach.

As a final aside, I should address the question of reliever usage. It is important to note that ORVY is meant to describe relative, within-game tradeoffs, rather than between-game tradeoffs. If it were the seventh game of the World Series, for example, a team would be correct to put in its closer when trailing by five runs, since as dire as its situation might be, trailing by five runs is relatively much worse than trailing by six runs. After all, it doesn’t matter if the closer tires himself out, since there aren’t any more games to play.

Ordinarily, however, the cost of using the closer is not measured in terms of trading off one run for multiple runs, but rather in trading off an increased win probability in one game for an increased win probability in future games. That is, it would not be wise to use the closer when trailing by five runs since at best it improves the chances of winning to very, very remote from very, very, very remote, and it would be useful to have him rested for more competitive games in subsequent days. On the other hand, the model that I’ve developed suggests that it is just as important to use the closer when the game is tied then when a team is ahead by one run, since the impact on win probability of allowing an additional run to score is the same.

#### Thank you for reading

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