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### A Markov Chain Model of Baseball

Eric Kuennen

Department of Mathematics

University of Wisconsin Oshkosh

Used as a project for an undergraduate Stochastic Modeling course

Presented at: Joint Mathematics Meetings

Washington, D.C.

January 6, 2009

Markov Chain Model for Baseball

- View an inning of baseball as a stochastic process with 25 possible states.
- There are 8 different arrangements of runners on the bases: (bases empty, runner on 1st, runner on 2nd, runner on 3rd, runners on 1st and 2nd , runners on 1st and 3rd, runners on 2nd and 3rd , bases loaded) and three possibilities for the number of outs (0 outs, 1 out, 2 outs), for a total of 24 non-absorbing states.
- The 25th state (3 outs) is an absorbing state for the inning.

Transition Probabilities

- A Markov Chain is a stochastic process in which the next state depends only on the present state. In other words, future states are independent of past states.
- Let Pijdenote the probability the next state is j, given the current state is i.
- Form the Transition Matrix T = [Pij].

w = probability of a walk

s = probability of a single

d = probability of a double

t = probability of a triple

h = probability of a home run

out = probability of an out

Methods of Analysis

Theoretical Calculations with Maple

- Expected Run Values for each state
- Steady State Probability Vector
- Expected Value of a given play in a given state or in general

Expected Run Values

- Let vi be the expected number of runs scored starting in state i
- Students use Maple’s linear algebra package to solve for the vector v

Sacrifice Bunting

Is it ever advantageous to sacrifice bunt?

Stealing Bases

How successful does a base-stealer need to be on average in order for it to be worth-while to attempt to steal second base with a runner on first and no outs?

Methods of Analysis

Experimental Simulations with Minitab

- Students write a Minitab macro that uses a random number generator to simulate the step by step evolution of the Markov Chain
- Large-scale simulations are used to estimate Expected Run Values and perform situational strategy analyses

Two Simulated Innings

First Inning

1. Single

2. Out

3. Double

4. Single

5. Out

6. Single

7. Out

Second Inning

8. Single

9. Homerun

10. Out

11. Out

12. Single

13. Out

Sacrificing with the game on the line

In the ninth inning, your team needs one run to win or tie. Suppose the first batter reaches first. Should you bunt?

Mean number of runs scored:

0.909

Probability of scoring at least one run:

0.390

Mean number of runs scored:

0.665

Probability of scoring at least one run:

0.406

Reference

- Sokol, J.S. (2004) “AnIntuitive Markov Chain Lesson From Baseball,” Informs Transactions on Education. 5 pp. 47-55.

Please contact me for:

- Sample Maple Worksheet
- Sample Minitab Macro
- Project Assignment Handout

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