Advancing an Extra Base on a Hit or an Error

The advantages of stealing based on the break-even probabilities also applies to taking an extra base on a hit or an error.

A base coach could use the break-even probability to determine whether to send the runner home from second on a single or send a runner home from first on a double.  He might also use it when considering whether to hold the runner from first at second or call him to third when the ball is hit to over his head to right field.

A base runner could use the break-even probabilities to determine whether to go from first to third on a single to left field or take an extra base on an error in the infield.

A batter could use these break-even probabilities to determine whether it is worthwhile to attempt to take an extra base on a hit.  For example, when it is worthwhile to attempt to turn a double into a triple or a single into a double.

Advantages of Stealing Based on Expected Runs

In my last post, I showed how to use the probability of winning to evaluate the break-even probability for a base stealing attempt to be worthwhile. That is, if the probability of successfully stealing the base is greater than the break-even probability, then the player should attempt to steal the base because in this situation the expected probability of winning the game would be higher than if he did not attempt to steal the base. If not, then the player should not attempt to steal the base because the expected probability of winning the game would be lower.

The problem with using the probability of winning to find the break-even probability is that the probability of winning table is very large. It provides a unique probability of winning for every possible state of the game.  Therefore, there would be a unique break-even probability for each possible state of the game.

A simpler approach to evaluating the break-even probability is to use the expected runs table that I developed in an earlier post. This could be easily memorized for every possible state in an inning. In this case, if the estimated probability of successfully stealing the base is greater than the break-even probability, the player should attempt to steal the base because this would increase the expected runs that would be scored in the inning.

Below is the expected runs table for any situation in an innning with the break-even probability for attempting to steal a base.

Outs	IB	2B	3B	Expected Runs	Break-even Probability
0	0	0	0	0.4885
1	0	0	0	0.2884
2	0	0	0	0.1156
0	1	0	0	0.8729	0.76
1	1	0	0	0.5043	0.73
2	1	0	0	0.2011	0.72
0	0	1	0	1.0611	0.77
1	0	1	0	0.6461	0.68
2	0	1	0	0.2793	0.69
0	1	1	0	1.4657	0.66
1	1	1	0	0.9244	0.62
2	1	1	0	0.4046	0.67
0	0	0	1	1.2919	0.84
1	0	0	1	0.8993	0.67
2	0	0	1	0.4069	0.36
0	1	0	1	1.7092	0.81
1	1	0	1	1.1436	0.81
2	1	0	1	0.5224	0.41
0	0	1	1	1.8950	0.71
1	0	1	1	1.3165	0.61
2	0	1	1	0.6006	0.43
0	1	1	1	2.3333	0.64
1	1	1	1	1.8205	0.71
2	1	1	1	0.7774	0.49

Notice that the break-even probabilities, in this case, are quite high for most the situations. 

However, stealing home from third base with two outs has a relatively lower break-even probability than the other situations. For example, with two outs and a runner on third base, the break-even probability is only 36%. This shows the potential of being ready to attempt to score from third base on a wild pitch or a passed ball.

Although the break-even probabilities are higher, with two outs and runners on first and third base (41%), runners on second and third base (43%) and bases loaded (49%), it seems that the probability of successfully making it home on a passed ball or wild pitch does not need to be even 50-50 to make it worthwhile to attempt it based on the expected runs calculation.

When to Attempt to Steal a Base

We can use the Markov Chain method to determine the potential benefit of attempting to steal a base.

Using the probablity of winning is one way to do this. We need to consider the probability of winning the game before the steal attempt. Then consider the probability of winning after a successful stolen base and the probability of winning if the player is caught stealing.

It is then a straightforward calculation to determine what the break-even probability needs to be to make it worthwhile to attempt to steal the base.

If the coach thinks the probability of the player successfully stealing the base is higher than the break-even probability, he should signal the player to attempt to steal the base. If the coach thinks the probability of the player successfully stealing the base is less than the break-even probability, then the coach should tell the player not to attempt to steal the base.

For example, consider the following situation. The score is tied in the bottom of the seventh inning and the home team has a runner on first base with two out. The probability of the home team winning the game at this point is calculated to be 56%.

If the player steals second base successfully, the home team would have a runner on second with two out. The probability of winning the game, in this case, is 60%.

If the player is caught stealing, the home team would be out of the inning and the game would go into extra innings. The probability of winning the game at this point would be 50% with two evenly matched teams.

The break-even probability would be 59%.

If the coach thought the player could steal the base successfully with more than a 59% chance, he should go ahead and signal for the player to attempt to steal the base.

Using the probability of winning values for all the possible situations in the game, we can calculate the break-even probability of attempting to steal a base.

Although I am using the Markov Chain results for two evenly matched teams, this approach allows for consideration of the speed of the particular baserunner and the arm of the actual opposing catcher.

Expected Runs found using Markov Chains

As mentioned in my last post, Markov Chains were used in The Book: Playing the Percentages in Baseball to calculate the probability of winning a MLB game for any situation that could occur.

Markov Chains can also be used to calculate the expected runs in the remainder of the inning for any situation.

I wrote a computer program to calculate the expected runs in an ISC fastpitch game using Markov Chains.  I will compare the results that I obtained to the results found for the MLB in the The Book.

In the table below, for the 24 possible situations that can occur in an inning, I show the expected runs that a team will score in the inning from that point on for MLB and for the International Softball Congress World Tournament.

    OUTS 1B   2B   3B  MLB    ISC

    0    0    0    0   0.555  0.489

    1    0    0    0   0.297  0.288

    2    0    0    0   0.116  0.116

    0    1    0    0   0.950  0.873

    1    1    0    0   0.567  0.504

    2    1    0    0   0.244  0.201

    0    0    1    0   1.192  1.061

    1    0    1    0   0.723  0.646

    2    0    1    0   0.343  0.279

    0    1    1    0   1.585  1.466

    1    1    1    0   0.982  0.942

    2    1    1    0   0.459  0.405

    0    0    0    1   1.445  1.292

    1    0    0    1   0.999  0.899

    2    0    0    1   0.387  0.407

    0    1    0    1   1.865  1.709

    1    1    0    1   1.249  1.144

    2    1    0    1   0.542  0.522

    0    0    1    1   2.075  1.895

    1    0    1    1   1.451  1.317

    2    0    1    1   0.624  0.601

    0    1    1    1   2.437  2.333

    1    1    1    1   1.671  1.821

    2    1    1    1   0.798  0.777

The numbers are quite similar although the ISC values are slightly smaller in all cases.

The expected runs scored in a nine inning MLB game is estimated to be 5.0.

The expected runs scored in a seven inning ISC game is estimated to be 3.4.

One way that we can use these results is to evaluate the value of a sacrifice bunt in terms of expected runs.

The expected runs with a man of first and none out is 0.950 for the MLB and 0.873 for the ISC.  If a team can lay down a successful sacrifice bunt, they would have a runner on second base with one out.  In this case, the expected runs drops to 0.723 for the MLB and 0.646 for the ISC.

There are no situations in which a successful sacrifice bunt will increase the expected runs in an inning.

Using the Probability of Winning to Evaluate Sacrifice Bunting

As I mentioned in my last post, I developed a computer program to estimate the probability of winning a game for every combination of score, inning, outs, and men on base.  This was an implementation of the Markov Chain methodology as described in The Book: Playing the Percentages in Baseball .

On pages 35 to 43 of The Book, there are tables of the probability of winning the game for all the possible situations based on Major League Baseball statistics.

For example, the probability of winning the game with a man on first base and none out in the bottom of the ninth inning when one run down is 0.353.

Let's assume that the home team lays down a successful sacrifice bunt in this situation.  Then, there would be a runner on second base and one out.  The calculated probablility of the home team winning the game at this point (in The Book) is now 0.296.

Thus, a successful bunt in this situation actuallly reduces the probability of winning the Major League Baseball game according to these calculations.

I collected data from the International Softball Congress World Tournament in 2011.  Then I used my computer program to do the same calculations.

I found the probability of winning the game in the bottom of the seventh inning when the home team is down by a run and they have a man on first base with none out.  It is calculated to be 0.344.

If the home team lays down a successful bunt, they will have a runner on second base with one out.  In this situation, the probability of winning the game is calculated to be 0.278.

So similar to the values found in The Book, a successful bunt, in this situation based on ISC data, actually reduces the probability of winning the game.

There are only a few situations in which a sacrifice bunt increases the probability of winning the game.  One is when the score is tied in the bottom of the ninth inning in the MLB or in the bottom of the seventh in the ISC with a runner on second and none out.

In this case, the probability of winning the game before the sacrifice bunt, in a MLB game, is 0.817 and the probability after a successful bunt that moves the runner to third with one out is 0.835.

The equivalent values for an ISC game are 0.803 before the bunt and 0.825 after a successful bunt.

Markov Chains and Monte Carlo Simulation

I have been dabbling with fastpitch statistics for a number of years.  I have been using fairly simple formulae such as runs created and linear weights as described in The Bill James Handbook 2013 and by Pete Palmer in The Hidden Game of Baseball .

Last winter, I started analyzing fastpitch statistics in a serious way.  I purchased two books that got me started.

The first book is Baseball Hacks: Tips & Tools for Analyzing and Winning with Statistics .  This book has many formulae and algorithms, as well as computer code, and explains how to use these formulae and algorithms to calculate sophisticated statistics.

The second book is The Book: Playing the Percentages in Baseball  which does not contain as many formulae but has many results using sophisticated computer programs and statistical analysis.

Two of the results in the first chapter of The Book were:

a. the average number of runs earned until the end of the inning for every combination of outs and runners on base, and

b. the probability of winning the game for every inning, score, outs and runners on base from the top of the first inning, no score, no outs, nobody on base to the bottom of the ninth inning with the score tied, two outs and the bases loaded.

I spent last winter writing computer programs to obtain similar results for fastpitch. The mathematical technique that I used is called Markov Chains.  The purpose of these calculations is to evaluate offensive strategies like bunting and stealing.

The results are interesting but counter-intuitive.  For example, there were very few situations in the results in which a sacrifice bunt improved a team's probability of winning the game.  In other words, even if the sacrifice bunt is successful in a particular situation, the probability of winning the game is reduced afterwards.

This is simply not sellable to fastpitch coaches and managers.  However, I think there is value in this result.  Especially, since I didn't consider the difficulty of laying down a successful sacrifice bunt.

The limitation to the Markov Chain approach is that we must use averages.  It considers two average teams playing each other and each team is made up of all average players.

Also, I was able to find the break-even point for attempting to steal a base, or attempting to take an extra base on a hit or an error, or advancing a base on a passed ball.  That is, if the estimated probability of successfully advancing to the next base is higher than the break-even point, it is advantageous to take the chance and attempt it.  If the estimated probability is not higher than the break-even point, then taking the next base should not be attempted.

Although this still considers two average teams playing each other, it does consider the specific capabilities of the baserunner and the defensive players.

This winter, I took a different approach that allowed me to consider the individual characteristics of the players on a team.  This involved a technique called the Monte Carlo method.

I wrote a computer program to simulate seven offensive innings with a particular lineup of players each with specific probabilities of producing different outcomes when they bat. I used the average probabilities for the season for each player.  But each player's average probabilities were different.

I ran the simulation many times and calculated the average runs scored for seven innings.  Then I was able to do statistical analysis to find a confidence interval on the average number of runs.  With this, I was able to determine if the differences in the simulation results were significant or if they might have been caused by random elements of the simulation.

I used the simulation to evaluate if one batting order was significantly better than another batting order.  I found some interesting results.  In most cases, the choice of which individual players are placed in the batting order makes a difference.  However, there is considerable leeway in which spot in the batting order they can be placed.  In a few cases, the same players may be in the batting order but the placement of the players in the order makes a significant difference.

I should also be able to consider the potential value of a successful bunt based on the individual characteristics of the players on the team.

The limitation of this approach is that it does not consider the defensive capabilities of the players on the team  The inclusion of defensive capabilities in the simulation may have to be postponed until a future date.

Purpose of the Men's Fastpitch Softball Analysis Blog

The purpose of this blog is to present analysis of Men's Fastpitch Softball that is comparable to the analysis found in the field of sabermetrics for Major League Baseball.

Most of the raw data is collected from observations of the International Softball Congress World Tournament held every year in the third week of August and data posted on Intermediate Level Men's Fastball League websites across Ontario, Canada.