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.