# RE+: Factoring Player & Team Hitting Ability Into Run Expectancy and the True Value of a Stolen Base

There are 24 different “states” in baseball. The three bases can be filled in eight different ways, and there can be 0, 1, or 2 outs at any given moment. Each of these 24 base-out states has an expected run value associated with them. Each value represents the average number of runs that the team is expected to score by the end of the inning. These values change each season depending on the run environment, but they generally don’t vary much.

2019 Average Run Expectancy by State
STATE 0 outs 1 out 2 outs
000 0.53 0.29 0.11
100 0.94 0.56 0.24
010 1.17 0.72 0.33
001 1.43 1.00 0.38
110 1.55 1.00 0.46
101 1.80 1.23 0.54
011 2.04 1.42 0.60
111 2.32 1.63 0.77

Consider the following situation: Lorenzo Cain is on first base with two outs. Now consider two possible hitters, one being Christian Yelich and the other being Ryan Braun. According to the 2019 averages, the run expectancy in this base-out state was 0.24, regardless of the hitter. While both players had impressive seasons, Yelich is unquestionably the superior player at this point in time.

2019 Player Comparison
Player wOBA ISO
Ryan Braun .354 .220
Christian Yelich .442 .342

As a result of their differences, the run expectancy should be higher when Yelich is at the plate. Consequently, the benefit Milwaukee gets from Cain attempting to steal second base should be adjusted as well. Why is this the case? Given Braun’s inferior power and hitting ability, there is more to gain from Cain putting himself in scoring position, but more importantly, there is less to lose if he were to get caught. On the other hand, Yelich is much more likely to drive the ball. With Yelich at the plate, the increase in run expectancy from a stolen base is slightly smaller than if Braun were hitting. However, the decrease in run expectancy from being caught is significantly greater. This is why we need RE+.

### Background

Like most sports, there is one way to win a baseball game: score more runs than the other team. It’s as simple as that. That being said, the way each team goes about scoring their runs has fluctuated drastically over the last 40 years. In the 1980s, teams valued stolen bases significantly more than they do now. On the contrary, home runs were not hit nearly as frequently as they soon would be. In the 1990s, the home run totals soared as the league entered the infamous “steroid era.” Even with the increase in home runs, stolen base totals remained fairly constant. In the twenty years thereafter, home run totals continued to increase while stolen base totals have decreased significantly in recent years.

Average HR and SB Each Season*
1980s 3,462 3,248
1990s 4,227 3,235
2000s 5,214 2,795
2010s 5,193 2,724
*Not including strike-shortened 1981 and 1994 seasons.

This change can be attributed to the evolution of advanced analytics. Teams have been operating under the assumption that a runner needs to steal approximately two bases to make up for each time he is caught. Thus not only are fewer players attempting to steal bases, but those who excel at it are stealing less frequently. This can be seen by the declining number of players with 30 stolen bases each season.

Average Number of Players with 30 Stolen Bases Each Season*
1980s 25
1990s 21
2000s 14
2010s 13
*Not including strike-shortened 1981 and 1994 seasons.

### Methodology

RE+ uses the current base-out state, the hitter’s wOBA (weighted according to his number of plate appearances), and the team’s collective wOBA (excluding the batter) as a means of determining the number of runs that are expected to score before the inning is over. In order to ensure that players with a minimal number of plate appearances were not skewing the data, I first analyzed the players with at least 350 plate appearances. According to Russell Carleton’s studies on reliability and Cronbach’s Alpha, a given statistic can be considered “reliable” if the corresponding Alpha crosses the .49 threshold. When a statistic is reliable, it does not mean it is an exact representation of the given player’s abilities, but it is a fair representation of them. wOBA crosses the .49 threshold at approximately 350 plate appearances, so I fit a beta distribution by maximum likelihood for each season from 2010 through 2019. Subsequently, I used the empirical Bayes shrinkage estimator to weigh each player’s wOBA according to the number of plate appearances they accumulated. By using this method, I prevented players with fewer plate appearances from skewing the data, while still properly crediting those with more plate appearances. These weighted wOBA values (which I refer to as wOBA+) are the values that will be used in the RE+ formula.

I then calculated each team’s collective wOBA minus the current batter (which I refer to as twOBA-). Given the number of plate appearances teams accumulate each season, I did not need to adjust these values. Prior to my research, batter-specific run expectancy had been explored, but the rest of the team was not taken into account. Regardless of the hitter, why should the Yankees, who scored 943 runs in 2019, have the same run expectancy in a given base-out state as the Tigers, who scored 582 runs?

Lastly, I used the 24 base-out states to determine how the batter’s wOBA+ and twOBA- impact the run expectancy and the run value provided or lost by a given stolen base attempt. The run value that a play creates is determined by subtracting the run value associated with the beginning base-out state from that of the ending state, plus any runs that scored on the play.

Run Value of a Play = RE After The Play – RE Before The Play + Runs Scored

For example, a home run with no outs and a runner on third would not be worth 2 runs, as the run expectancy, according to 2019 averages, was already 1.43. Therefore, the run value of the home run would be:

0.53 – 1.43 + 2 = 1.1

The base state is certainly important in determining each statistic’s impact on the run expectancy, but the number of outs plays an even larger role. When there are no outs, it’s evident that the twOBA- plays a larger role because there are other hitters that will come to the plate. However, twOBA- plays a smaller role when there is one out. Why? Well, there are a few reasons. First, fewer hitters are expected to come to the plate when there is already one out. Additionally, a batter may hit into an inning-ending double play. As a result, wOBA+ has a greater impact on the run expectancy than twOBA-. Finally, as one would expect, wOBA+ has a significantly larger impact than twOBA- when there are two outs because the fate of the inning is in the batter’s hands.

While I focused my research on deriving the true value of each stolen base attempt, I should note that I did not use the baserunner’s characteristics in this model. While this is something I hope to do down the road, there are a lot of variables to consider when assessing a given runner’s base-stealing abilities.

### Player Analysis

When I put this system together, I focused on making it understandable to those who both are and aren’t well-versed in statistics. If the previous paragraphs confused you, this is how RE+ works:

• Weighted on-base average (wOBA) is a function of the hitter’s total unintentional walks, hit by pitches, singles, doubles, triples, and home runs.
• Consider two different players with the same on-base percentage and batting average. Hitter A primarily reaches base by singles and walks, while Hitter B more frequently hits for extra bases. Hitter A’s wOBA will be relatively lower because singles and walks carry less weight than extra-base hits.
• Additionally, the weight, or run value, assigned to each outcome (single, double, etc.) is specific to each season.
• Therefore, a relatively lower wOBA is generally a result of some combination of getting out more frequently or having fewer valuable outcomes (like those previously mentioned).

Let’s assume Trea Turner is on first base. Next, consider which events would and wouldn’t score him. If the batter were to walk or be hit by a pitch, Turner would move up just one base. If the batter hits a single, Turner would move to second or third base, but it is unlikely that he would score (in 2019, it happened in just 0.6% of the instances in which a single was hit with a man on first). Clearly, the probability of Turner scoring on a double, triple, or home run is significantly higher. Therefore, given this knowledge, it wouldn’t make sense to risk being caught stealing if the batter frequently walks or drives the ball.

Below is a preview of a tool that can be used to calculate several different values that are impacted by the base-out state and the hitter. First, input a batter. Next, choose the base state and the number of outs. After doing this, select whether you’d like to consider the ninth inning or not. If there are two outs and it is NOT the ninth inning, the next hitter will be taken into account. Why? If the runner is caught stealing, we must account for the current hitter now leading off the following inning. If he is not caught, the batter who is on deck is more likely to leadoff. In the ninth inning, this doesn’t matter, so the following hitter will not be taken into account. If there are two outs and it is not the ninth inning, select the next batter.

Finally, choose whether you would like to see the impact of stealing a base (SB) or being caught stealing (CS). After filling out these inputs, you will see three different results. The first value is the RE+. This value does not consider any stolen base attempts, as it is simply the RE+ in the situation that was constructed. The next value is the change in RE+ from either stealing the base successfully or being caught. Note that this value will only appear when the base state is 100, 010, or 101. The final value indicates the probability of success that produces no expected change in RE+. If a runner’s probability of success is believed to be greater than this value, attempting to steal the base would have a positive expected change in RE+.

If you would like to find these values using your own wOBA+ and twOBA- values, use the “wOBA+ Specific” tab. All four sliders are set to .320, as this was the league wOBA in 2019. Click here for access to this tool.

In 2019, Yelich posted a wOBA of .442 and his twOBA- was .312. On the other hand, Braun’s wOBA was .354 and his twOBA- was .321. Yelich had an MVP-caliber season, but Braun still managed to put up respectable numbers and was a viable threat at the plate. These differences can affect the value of a stolen base drastically, especially when there are two outs. To remain constant, let’s assume it is the ninth inning. When there was a runner on first base with two outs, a stolen base increased the RE+ by 0.1 when Braun was at the plate. When Yelich was batting, the benefit was slightly lower, at .09. The difference in value added from stealing the base was minimal, but the value lost from being caught was substantial. When Braun was batting, it cost the team .25 runs if a runner was caught stealing second base in the aforementioned base-out state. When Yelich was batting, the team lost .35 runs. In other words, when Braun hit in these situations, the Brewers should have only attempted to steal second base if they felt the runner’s probability of doing so successfully was greater than 72.5%. When Yelich was batting, this value increased to 79.6%.

Both of these “necessary” success rates are fairly high because both players were relatively productive at the plate. However, when a less intimidating player was hitting, like Orlando Arcia, the values change tremendously. When Arcia was up in this situation, a stolen base still created just .1 runs, but being caught only cost the team .15 runs, which is .20 fewer runs than it would have cost the team if Yelich were at the plate. With Arcia hitting, a baserunner needed just a 60.7% probability of success to produce a positive expected change in RE+.

These values differ from those suggested by the 2019 average run expectancy values. The average values suggest that a stolen base in this state increased the run expectancy by .09, while being caught decreased the run expectancy by .24. While the value gained from stealing second base was fairly similar, the value lost from being caught was notably different because it doesn’t account for the hitter. The 2019 averages suggest that, regardless of the hitter, a runner should only attempt to steal second base in this situation if they believe they have a 72.7% probability of success. On the contrary, RE+ suggests that some hitters require a much higher probability (Yelich) while others require a much smaller probability (Arcia).

How did the Brewers do with just a runner on first base and the aforementioned players at the plate? With Yelich batting, Lorenzo Cain was 5-for-8 when attempting to steal second base. Even so, according to RE+, the Brewers lost one run in those attempts. With Arcia at the plate, five different players went a combined 5-for-6. The Brewers’ RE+ went up 0.19 runs from these attempts. Lastly, with Braun at the plate, Yelich was 7-for-7 and Cain was successful once. The Brewers’ RE+ only increased by 1 run.

To dive even deeper, Cain was caught stealing second twice with nobody out and once with one out when Yelich was at the plate with no one else on base. Those three instances combined for a loss of 1.88 runs. If he was successful all three times, the benefit would have been just 0.62 runs combined. Even with Milwaukee’s high success rate (in this base state) of 76%, they lost 0.21 runs as a team.

### Team Analysis

In the context of stealing bases, RE+ tells us a lot about both the runner’s and team’s decision-making skills. Below is a preview of a tool that can be used to determine how many runs were lost or gained from attempting to steal bases in a given season (2010-19)*. Simply enter the teamID and the year to calculate these values. Click here for access to this tool.

*These values do not account for the on-deck batter because we cannot be certain as to whether the manager planned on using a pinch hitter. Therefore, it’s possible the data would be skewed, so I removed the on-deck batter from the formula.

In 2019, the Mariners had the most stolen bases in baseball (89) and were caught 41 times when the only baserunner was on first. Even though they stole more bases than every other club, their RE+ went up just 1.28 runs, which was still the seventh highest value in baseball. On the other hand, the Indians stole 13 fewer bases but had a higher success rate. Even though they were successful more frequently than Seattle, the Indians saw a .14 decrease in RE+.

Once again, the 2019 average values for run expectancy tell a different story. According to those values, Seattle’s total run expectancy decreased by .99 runs in these situations. Cleveland is still considered to have lost runs, but these values suggest their total run expectancy decreased by 1.24 runs.

Why was Seattle’s change in RE+ so much better than the average run expectancy values suggest? The average required success rate the Mariners needed (for these attempts) in order to increase their RE+ was 70.5%, which was the fifth-lowest in baseball. Additionally, 11 of their attempts took place when the minimum required success rate was below 65%, which was the most in baseball. On top of that, they didn’t attempt to steal if they needed a probability of success greater than 75%. On the contrary, the Royals, White Sox, and Pirates attempted to steal bases in those situations more than 10 times each. None of these teams had a positive RE+, losing at least 2.33 runs each. Complete data tables with this information can be found here. Note that the percent columns provide the number of stolen base attempts at a given “minimum success rate.”

### Application

The RE+ model, unlike basic base-out state run expectancy models, rightfully accounts for the hitter and the rest of the team’s offensive prowess. RE+ can be used for both evaluation and decision-making purposes. From an evaluation standpoint, the model can be used to assess the decisions made by both players and managers. More importantly, RE+ can be used to ensure that teams are not frequently running at inopportune times moving forward. When RE+ suggests it may be a good time to steal a base late in a game, manager’s may also use that knowledge to their advantage by considering a pinch-runner or a pinch-hitter.

### Future Work

There are several other factors that I would like to incorporate into RE+ moving forward.

• Accounting for a specific runner or type of runner.
• Rather than using twOBA-, I would like to focus on the hitter(s) that immediately succeed the batter in all calculations. I could potentially use OBP to determine the probability of them coming to the plate.
• A Bayesian formula that determines exactly when to run by incorporating the current and upcoming batters, stolen base success rates, the catcher, the pitcher, and the frequency in which the pitcher throws each of his pitches.
• Finding the value that specific catchers provide.
• Using RE+ to determine how “clutch” a player or team may be.
• Including xwOBA in calculations.

### Technical Appendix

Below is an example of how RE+ values, and the values for minimum success probabilities in regard to stealing bases, are calculated. For the sake of the example, let’s assume Mike Trout is batting and there is a runner on first base with one out.

Thus far, we have the base-out state (100 1), Mike Trout’s wOBA+ (.413), and his twOBA- (.304). There are base-state specific coefficients, so the mentioned coefficient will change depending on the state. This first value we will calculate is the beginning RE+.

RE+ = (twOBA-)*(twOBA- Coefficient) + (wOBA+)*(1Out:wOBA+ Coefficient) + (twOBA-)*(1Out:twOBA-Coefficient)

RE+ = (.304)*(0.930253) + (.413)*(1.726721) + (.304)*(-1.096573) = 0.6625745

Next, in order to determine the Change in RE+ from a SB, we will run the same formula, but as if he had stolen the base (010 1), and subtract the beginning RE+.

RE+ (SB) = (.304)*(1.922088) + (.413)*(1.92697) + (.304)*(-1.824641) = 0.8254625

Change in RE+ (SB) = 0.8254625 – 0.6625745 = 0.162888

When there are two outs, the RE+ from being caught stealing depends on who the next inning’s leadoff hitter would have been. This is because the current hitter would now lead off the next inning. There is only one out in this scenario, so we conduct the same formula, but as if he had been caught stealing (000 2). We then subtract the beginning RE+ value from the new RE+ value.

RE+ (CS) = (.304)*(0.875425) + (.413)*(0.586377) + (.304)*(-1.161231) = 0.1552887

Change in RE+ (CS) = 0.1552887 – 0.6625745 = -0.5072858

Finally, to calculate the value for the minimum success probability (in regard to stealing bases) that results in a positive expected change in RE+, we divide the Change in RE+ from being caught by the difference between the Change in RE+ from being caught and the Change in RE+ from stealing the base successfully.

Minimum Probability of Successful SB = (Change in RE+ (CS)) / ((Change in RE+ (CS)) – (Change in RE+ (SB)))

Minimum Probability of Successful SB = (-0.5072858) / (-0.5072858 – 0.162888) = 0.7569466

Therefore, when there was a man on first with one out and Mike Trout at the plate in 2019, no one should have attempted to steal second base if they believed their probability of doing so successfully was less than 75.7%.

All numbers are updated through the 2019 Season. Data sourced from FanGraphs, Baseball Reference, and Retrosheet.

Inline Feedbacks
Jimmember
1 year ago

Excellent.

feddy
1 year ago

good stuff

Green Mountain Boy
1 year ago

Good article! You don’t have to be a rocket scientist to know that runs will be easier to score against Joe Blow than Gerrit Cole, and likewise you don’t have to be a rocket scientist to know that Mookie Betts will perform better against a particular pitcher than Pedro Severino. So a manager can’t make a decision based on “averages” for that game situation, but also has to take into account the participants in the battle and how much they swing the deviation from “average”. Frankly, I’m surprised someone hadn’t come up with this type of analysis sooner. And beyond that, I’d love to see how “hot” and “cold” streaks affect things.

For what it’s worth, in 2013 I conducted a study. I started with the assumptions that every base attained is worth 1/4 run, no matter how it’s attained, and every batter has 4 bases available to them + whatever bases are available due to runners on base. For example, with no one on, a walk or a single both attain 1 base out of 4, or 25%. A steal while on first attains another 1 of the initial 4, or +25% for a total of 50% (0.5 runs). A caught stealing takes away a full AB, or 4 potential bases. So… a single and steal of second = 2/4 or .50 runs. A single and unsuccessful steal = 1/8; 1/4 for the single and 0/4 for the caught stealing. Sum those situational stats and divide by PA to get bases attained per PA. Then you have something to work with!

Anyway, when all was said and done, it turned out that every base successfully attained was worth almost exactly 1/4 run, league-wide. The best teams (for 2013 anyway) turned bases into runs at 30-32%, the worst at 20-22%. Which was a cool result. I would have liked to follow up to see if teams reverted to the mean of 25% in subsequent seasons, but never had the time, as I was unemployed for much of 2013 and had nothing BUT time.

I’d love to see more research on all of these areas. Great job, Matt!

Green Mountain Boy
1 year ago

First, I don’t claim to be any sort of statistician. Took one stats course in college to fulfill a requirement. Did well, I’ve been interested since, but I’m a 2-year-old in a sea of professors, I readily agree.

Second, I chose a single and unsuccessful stolen base as 1/8 as follows. A single with nobody on is one of four potential bases attained. The CS takes away four potential bases from the subsequent batter, because of the actions of the first batter/runner, so it goes to him. Similarly, a nobody on single or walk attains 1 of 4 bases for the batter, but the subsequent batter who hits into a double play gets 0 of 7 (0 of his own 4 bases plus 0 of the runner’s remaining 3 bases, PLUS 0 of the 4 bases the second out took away from the next guy, so batter 1 singles for 1/4, batter 2 GIDP for 0/7 + 0/4 for 0/11.

Yes, percentage-wise 0/4 = 0/infinity, but that’s why I used the summation in the numerator divided by the total PA in the denominator.

I know it’s not sophisticated. I never meant it to be. It was actually a first step toward answering “Should you bunt down a run with your second hitter in the last inning of a one run game when the first batter reaches first base?”

I actually had more to it which took into consideration the number of outs, the inning, and the score, but that result never bore consistent fruit like the run value of 1 base seemed to, although I still think I was on the right track.

Ultimately, I was looking for a “unified theory of gravity”, where one number could tell you about both pitchers and hitters using the same scale. I do think I was on the right trail, but again, not being a statistician I made a lot of rudimentary assumptions and readily admit I may have walked right by the golden nugget.

If you think there’s a kernel of potential useful information here, I’m open to sharing ideas and exploring options. Me: Theory? Yes! Math? Yes! Baseball junkie? Yes! Statistical acumen? No, unless “going by feel” counts!

Daniel Eckmember
1 year ago

This is really great work!

I have some questions about the underpinnings of the methodology.

1) I understand that you use all 2010-2019 data to obtain a reliable estimate of the two parameters in the Beta distribution. However, I am concerned that including this much data may result in weird fits because of the evolution of baseball over this span. For example, 2013-2015 are notorious pitcher dominant seasons and these seasons influenced the sprint to the three true outcomes type of baseball we see today. Perhaps wOBA is robust to such changing conditions, but perhaps not. Do you think that your analysis would change much if you used say the last three years of data instead of the last ten?

2) The empirical Bayes approach is pretty slick. However, I am not sure if the entire population of players is the right prior for every player in the dataset. As a specific example, players with low numbers of PAs will be treated as average players under this approach. This seems to exhibit a bias in their favor, I would guess that more often than not a player with a low number of PAs is either worse than average or is developing (is worse than average when the data was collected). Do you think that it would be worth while to consider a more granular prior distribution for separate “talent” levels within the overall player pool? I understand that this is difficult.

Daniel Eckmember
1 year ago