Game Theory: Down Angle

Let’s talk managerial strategy. Imagine runners on 2nd & 3rd with 1 out and the defense is “playing in.” As a baserunner at 3rd, how are you instructed to react on a groundball? Most likely, you’re taught to “see it through the infield.” But should you? Let’s evaluate the decision using some numbers.

Before we begin, we need to understand a concept called “run expectancy” (RE). Given any base/out situation, the RE is the average number of runs scored from that moment until the end of that inning. This can be found empirically (off historical data) or through simulations (using Markov chains). Tom Tango founded this idea using MLB data more than a decade ago. We’ve followed his lead using the empirical approach with NCAA data since the adoption of BBCOR bat restrictions.

Back to our situation: 2nd & 3rd with 1 out. The NCAA RE in this situation is 1.562 runs, but that’s not so important. What we want to evaluate is the RE after this play is complete. Depending on the baserunning decision our final RE may be different. To keep the complexity of this evaluation to a minimum, we need to establish a few assumptions.

Assumption 1. Only one error may be made on a play, and any error made does not advance baserunners any further than the base they already exist. (Think throwing error caught by 1B that pulls him off the bag.)

Assumption 2. The 3rd base coach has clairvoyance in deciding the fate of the runner at 2nd (and 2nd only). In other words, on a single, he’s always perfect in his decision on whether or not to send the runner home when rounding 3rd. If he’ll be out, he holds him at 3rd. If he’ll score, he waves him home.

Assumption 3. The hitter will never end up past further than 1st base. Clearly there are several scenarios in which this could occur, however to eliminate complexity, we’ll stick with this as our final assumption.

Now we can move to our decision. As described earlier, we have two options. First, we have the traditional “see it through the infield” choice. Second, we have our “down angle” philosophy. Each decision has it’s related outcomes.

Option 1. Runners freeze until the ball is through the infield, then advance. Given a routine groundball, this option has four basic possible outcomes A – D.

A – Hitter grounds out. Runners remain at 2nd & 3rd. 0 runs scored. 2 outs.

B – Hitter grounds to a fielder who makes an error. Runners remain at 2nd & 3rd, and the hitter is safe at 1st. 0 runs scored. 1 out.

C – Hitter singles through the infield. Because they froze to see the ball through, runners move up one base each. Hitter is safe at 1st. Baserunners now at 1st & 3rd. 1 run scored. 1 out.

D – Hitter singles through the infield. Although they froze, both runners ended up scoring. Hitter is safe at 1st. 2 runs scored. 1 out.

Option 2. Runners move on down angle contact. That is, they advance upon a downward launch angle regardless of the spray angle. Given a routine groundball, this option has five potential outcomes E – J.

E – Hitter grounds into a FC as a fielder throws home to tag the runner out. Runners now at 1st & 3rd. 0 runs scored. 2 outs.

F – Hitter grounds to a fielder who makes an error. Runners now at 1st & 3rd. 1 run scored. 1 out.

G – Hitter singles through the infield. Although they were moving on contact, only one runner scored. Hitter is safe at 1st. Baserunners now at 1st & 3rd. 1 run scored. 1 out.

H – Hitter singles through the infield. Because they were moving on contact, both runners were able to score. Hitter is safe at 1st. 2 runs scored. 1 out.

J – Hitter grounds to a fielder who is unable to make the play at home, but can make the play at 1st. Runner from 2nd advances to 3rd. 1 run scored. 2 outs.

Assigning Probabilities:

Now that we’ve identified the two options and their corresponding potential outcomes, it’s time to apply some probabilities of occurrence for each specific outcome. Then, we can complete the decisional analysis. Our justification:

1. The average NCAA fielding percentages on all plays is 0.965 (thus, the error in decision #1 has a 3.5% chance of occurring). However, the play at home is unique and rare. Therefore, we’ll assume that an error here occurs twice as often, so a 7.0% chance of occurrence. (Runner has a lead, is off on contact, and it’s a tag play at home. Many more variables than a standard 6-3 putout.)

2. Given the ball is hit on the ground with the infield in, the ball will get through for a single 40% (estimated) of the time.

3. Of all singles with 1 out, a runner on 2nd will score on 47.6% of them when a runner reacts normally (doesn’t freeze to see the ball through the infield). When freezing, the runner on 2nd will score approximately 25% of the time.

4. 10% of the groundballs will result in a fielder being unable to make a play at home, but will be able to get the out at 1st (think diving stop, but not enough time to throw home).

Analyzing the traditional “see it through the infield” decision:

Decision Outcome Bases Occupied Runs Scored Outs Remaining RE of End State Runs + RE Probability of Occurance e-Value
1 A _23 0 2 0.684 0.684 0.665 0.45
1 B 123 0 1 1.792 1.792 0.035 0.06
1 C 1_3 1 1 1.357 2.357 0.300 0.71
1 D 1__ 2 1 0.630 2.630 0.100 0.26

And now the “down angle” method:

Decision Outcome Bases Occupied Runs Scored Outs Remaining RE of End State Runs + RE Probability of Occurance e-Value
2 E 1_3 0 2 0.611 0.611 0.430 0.26
2 F 1_3 1 1 1.357 2.357 0.070 0.16
2 G 1_3 1 1 1.357 2.357 0.210 0.49
2 H 1__ 2 1 0.630 2.630 0.190 0.50
2 J __3 1 2 0.449 1.449 0.100 0.14

With the runs, outs, RE’s, and probabilities identified, we can compute an expected value, simply (Runs + RE) * probability. Without getting too technical, the final e-value is our new RE the moment the ball leaves the hitter’s bat on a negative launch angle. As we find a value for each decision, we want to continually choose the higher (assuming risk neutrality) to generate more runs over time. Thus, long term, the down angle philosophy will increase RE by 0.08 runs. Next week, we’ll provide a sensitivity analysis to determine how dependent our findings are to the various assigned probabilities.

Remember, you can’t evaluate a decision based on it’s outcome. Separate the two. In summary, go down angle… and go home.

RBI Hitting Approach: Should it Change?

Having had a lengthy discussion with the three other coaches on our collegiate staff regarding a hitter’s approach with RISP, I thought I’d make an attempt to analyze how professional hitters might alter their approach in RBI situations. Specifically, we’re focusing on runners at 3rd or 2nd/3rd with less than 2 outs. We have eliminated all situations in which a runner resides on 1st since a coach/manager may elect to employ a strategic play (e.g. a hit & run). We have also eliminated all bunt attempts from this study. We truly want to evaluate the approach a hitter may take without external factors such as managerial strategy.

With a proven, established approach already in place in our college program, a new perspective provided the suggestion to be more aggressive and “eat the RBI early.” That is, a hitter in an RBI situation ought to look for something he can hit early in the count. This philosophy to “look for something you can hit versus drive” may allow one to avoid more strikeouts, but will it lead to success?

First, let’s establish the “standard” approach. Seems to me, the simplest situation is 0 on, 0 out. A hitter’s goal is to simply get on base. With 500,011 MLB PA’s from 2002-2012, this is clearly enough data to establish our standard. (Other situations such as 0 on, 1 out could be considered, however certain nuances of the game could affect the P/PA. For example, a 1-pitch AB by the leadoff hitter may force the second hitter of the inning to take a pitch to avoid the quick inning.) Nevertheless, we find 3.8175 P/PA with 0 on and 0 outs.  In case you’re curious, for hitters with at least 20 PA’s, I’ve plotted the leadoff hitting approach below.


Now we’ll compare to our RBI situations. See the table below for the base states and the corresponding P/PA for each of the population as a whole.

Outs Bases Occupied P/PA
< 2 3rd 3.8175
< 2 2nd & 3rd 3.8255

Such small differences in P/PA appear to provide no statistical difference from the P/PA in our standard situation. However, to finally answer the systemic question, we need to create a hypothesis of the overlapping data and test the statistical significance. To do so, we could do old school calculations using Student’s t-test, or we can use modern day software to simply tell us that these two distributions are actually statistically the same (that is, with 95% confidence, they are derived from identical means).  However, let’s take a look at how individual hitters (with at least 30 PA’s) alter their approach from leadoff to RBI situations.ppa_diff

The plot above shows how some individual hitters will, in fact, alter their approach in RBI situations.  Keep in mind that pitchers are taught to pitch less to contact with RISP.  That is, they will attempt to execute pitches with more selectivity.  Typically, they are taught one (or more) of three philosophies in potential RBI situations:

  1. work the “lower U” of the strike zone
  2. attempt to achieve pitch height AND width (i.e. hit the corners of the zone)
  3. go to offspeed, typically down, and especially if it’s a swing and miss pitch (resulting in more pitches out of the zone)

A quick look at PITCHf/x data tells us that a pitcher will throw strikes in the zone approximately 52% of the time to a leadoff hitter and just 45% of the time in our noted RBI situations.  Thus, a causal increase will undoubtedly occur in the hitter’s P/PA in these situations.  Next week, we’ll take a look at the RBI success rate as a function of the altered approach. Stay tuned…