# Rates of Bases/Outs States in an Inning

### Introduction – Review of Runs Expectancy

About a year ago, I posted on how one can summarize a Runs Expectancy Matrix. For each of the possible inning situations defined by runners on base and number of outs, the Runs Expectancy matrix gives the expected runs in the remainder of the inning. The matrix using 2019 data is displayed below. Looking at the entry in the Outs = 1 row and the 103 column, we can see that there will be, on average, 1.23 runs scored in the remainder of the inning when there are runners on 1st and 3rd with one out.

This matrix can be used to measure the Runs Value of any play using the formula:

Runs Value = Runs Value (after play) – Runs Value (before play) + Runs Scored on Play

For example, suppose there are runners on 1st and 2nd with one out and the batter hits a double, scoring both runners.

• Before the play we have runners on 1st and 2nd with one out — by table, Runs Value = 1.00.
• After the play, we have runner only on 2nd with one out — by table, Runs Value = 0.72
• Two runs scored on the play.

So the Runs Value of this particular play (double with runners on 1st and 2nd with one out) would be

Runs Value = 0.72 – 1.00 + 2 = 1.72

Using this recipe, we can find the Runs Value for all plays in a particular season.

### Runs Potential and Likelihood of States

The runs expectancy matrix is helpful for understand the potential for scoring runs in any bases/outs situation. But this matrix says nothing about the likelihood of reaching any of these bases/outs states during a game. For example, if it is rare to have, say, runners on 2nd and 3rd with two outs, then a team will not have the opportunity to score runs in this situation. We know one needs “ducks on the pond” (runners in scoring position) for a single to result in any runs scored.

In this post I take a historical look at the rates of being in different bases/outs situations. We will see some obvious trends in this rates that will indicate some issues with modern MLB baseball.

### Measuring Excitement of a Bases/Outs State

A simple question: What bases/outs situations are exciting in baseball? I think most of us would agree that a particular situation is exciting if the play outcome can result in a wide range of outcomes. For example, suppose your team has the bases loaded with two outs. You might observe a single, scoring two runs, or instead observe a strikeout, ending the inning. The wide variation of outcomes (one very positive and one very negative) makes that particular situation exciting. In contrast, bases empty with two outs would not be exciting since the range of possible outcomes is limited — the only way a run can score from a bases-empty situation is a home run.

We can measure the value of any play by its Runs Value. Suppose, for each of the 24 possible bases/outs situations, we collect the Runs Values for all plays that are bat events. We define the excitement of a particular bases/outs situation as the standard deviation of the Runs Values:

Excitement (State) = Standard Deviation(Runs Value)

Using data from the 2021 season, I plot the standard deviation of the Runs Values for all states below. As you see, the highest standard deviations correspond to the bases loaded states — in fact, bases loaded with two outs (coded “123 2”) has the highest standard deviation. In contrast, the bases empty states with 0, 1, and 2 outs (“000 0”, “000 1”, “000 2”) have the smallest standard deviations. This seems to be a reasonable way of ordering states by excitement level.

### Historical Look at State Rates

There has been a dramatic change in the frequencies of different bases/outs states in recent seasons of baseball. From Retrosheet data, I collected the frequencies of all 24 possible bases/outs states for each of the seasons 2000 to 2021 and converted the frequencies to rates. For each of the 24 states, I plot the rate of that state against the season and add a smoothing curve. I first display the rates for the eight different bases states for no outs, then display the rates for one out, and then display the rates for two outs. (I’m suppressing the rate values on the vertical axis since we are focusing on the changes in these rates over season.)

There are some clear takeaways from inspecting these graphs:

• The rates of particular exciting (high standard deviation) runners on base situations such as “020”, “003”, “120”, “103” have significantly dropped in the period from 2000 to 2021. Generally these are the situations with runners in scoring position. These decreasing patterns don’t change by the number of outs.
• In contrast, we see a rise in the rates of particular “boring” (low standard deviation) runner on base situations such as “000” (bases empty) in this period of baseball. Also we see an increase in the rate of particular one-runner situations such as “100” (runner on 1st) with 1 and 2 outs.
• The general takeaway is that the exciting bases/outs situations are becoming less common in current baseball and the boring situations such as bases empty and runner on first with two outs are more prevalent.