# A One-Parameter Markov Chain Model for Baseball Run Production

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1 for Winthrop University April 13, 2013 s s and

2 : A is an ideal candidate for mathematical modelling, as it has these features: a relatively small number of configurations, a relatively small number of events that can happen, a discrete nature. These factors also suggest using a discrete-time stochastic process as a model. For the last 50 years, intermittent attempts have been made to model baseball using a type of stochastic process called a Markov chain, and to derive measures of players offensive abilities from an analysis of that model. s s and

3 s A Markov chain is a mathematical model for movement between states. A process starts in one of these states and moves from state to state. The moves between states are called steps or transitions. The chain can be said to move between states and to be at a state or in a state after a certain number of steps. The state of the chain at any given step is not known, but the probability that the chain is at a given state after n steps depends only on the state of the chain after n 1 steps, and the probabilities that the chain moves from one state j to another state i in one step. s s and

4 The Transition Matrix P These probabilities are called transition probabilities for the Markov chain. The transition probabilities are placed in a matrix called the transition matrix P for the chain by entering the probability of a transition from state j to state i at the (i, j)-entry of P. So if there were m states named 1, 2,... m, the transition matrix would be the m m matrix P = From: 1 j m To:. 1 p ij i m s s and

5 The Big Idea Use a Markov chain to model run production in baseball. The states will be different possible offensive configurations. The transition probabilities will be calculated from actual data. s s and

6 States of the Transient States Bases Occupied Outs Bases Occupied Outs None 0 1st,2nd 1 1st 0 1st,3rd 1 2nd 0 2nd,3rd 1 3rd 0 1st,2nd,3rd 1 1st,2nd 0 None 2 1st,3rd 0 1st 2 2nd,3rd 0 2nd 2 1st,2nd,3rd 0 3rd 2 None 1 1st,2nd 2 1st 1 1st,3rd 2 2nd 1 2nd,3rd 2 3rd 1 1st,2nd,3rd 2 s s and Absorbing States Left on Base Outs Left on Base Outs

7 Transition Probabilities Only the following events are allowed for the player at bat: single, double, triple, home run, walk, hit batsman, out. No base stealing, errors, sacrifices, sacrifice flies, double plays, triple plays, or other ever-more-bizarre events are allowed. For the following analysis to work, each batter must have the same probability of singling, doubling, etc. Each batter is identical. s s and

8 Outcomes of Batting Events Event Walk or Hit Batsman Single Double Triple Home Run Out Outcome The batter advances to first base. Runners advanced only if forced. The batter advances to first base. Runners on first base or third base advance one base. A runner on second base advances to third base with probability 1 p and scores with probability p. (This parameter is chosen to minimize model error.) The batter advances to second base. All runners advance two bases. The batter advances to third base. All runners advance three bases. The batter scores. All runners score. No runners advance. The number of outs increases by one. s s and

9 Calculation of Transition Probabilities The transition probabilities are calculated from the above rules and from aggregate data. For example, the data from the 2012 MLB season is: Batting Event Occurrences Probability Walk or Hit Batsman Single Double Triple Home Run Out Example: The probability of a transition from runner on first, 0 out to runners on first and second, 0 out is the probability of a walk or a hit batsman or a single, which is = s s and

10 The Transition Matrix P The transition probabilities are calculated as above and placed in the transition matrix P. We order the states listing absorbing first, then transient. The probability of a transition from an absorbing state to itself is 1. The probability of a transition from an absorbing state to any other state is 0. The probabilty of a transition from a transient state is governed by the above probabilities. s s and [ ] I S P = O Q

11 The Fundamental Matrix M The fundamental matrix for the Markov chain is M = (I Q) 1. Suppose that the chain starts at state i. Then The expected number of visits to state j before being absorbed is the (j, i)-element in M. The expected number of steps until absorption is the sum of the (j, i)-elements of M, where j ranges over all transient states. In terms of baseball, the sum of the i th column of M is the expected number of batters that will come to bat during the remainder of the half-inning starting from state i. s s and The sum of the first column of M is the expected number of batters E(B) that will bat in a half-inning.

12 The Matrix A = SM The matrix A = SM is important to our analysis. Suppose that the chain starts at the transient state i. Then The (j, i)-element of the matrix A = SM is the probability that the Markov chain is eventually absorbed at state j. In the baseball model, the i th column of A contains the probabilities that 0, 1, 2, or 3 runners are left on base at the end of the half-inning starting at state i. Thus the first column of A can be used to compute the expected number of runners E(L) left on base: s s and 0 P(0 left) + 1 P(1 left) + 2 P(2 left) + 3 P(3 left)

13 The Conservation Law of Every batter who comes to the plate makes an out, scores, or is left on base. Let B be the number of batters who come to the plate in an half-inning, R be the number of (earned) runs scored, and L be the number left on base. Taking expected values, B = 3 + R + L, or R = B L 3 s s and E(R) = E(B) E(L) 3 and we can calculate E(B) and E(L), thus E(R).

14 Does this Work? We set p equal to that value that would predict exactly the number of earned runs scored in MLB since This value is p = For the 2012 MLB season, the model predicts that earned runs will score in a single half-inning. Since 43,355 innings were played during the season, the model predicts that 19,059.6 earned runs would score during the season. In reality, 19,302 earned runs scored during the season, or per half-inning. The error in the model was earned runs for the season; the relative error was 1.26%. s s and

15 Does this Work? Earned Runs Relative Year Scored Prediction Error Error % % % % % % % % % % % % % Total % s s and

16 Earned Runs Scored ( ) and Predictions ( ) s s and

17 Player Rating Consider the same player batting over and over. Use the player s batting statistics to generate transition probabilities. Run model to find the expected number of earned runs the player would score in a single half-inning or in a nine-inning game. s s and

18 Example: MLB 2012 The 10 MLB players with the highest batting averages in the 2012 season were studied using this model. The p value was set at the value that would produce the exact number of earned runs for the entire 2012 MLB season. This value is p = Markov Runs Name Created 27 OPS Posey Cabrera McCutchen Trout Beltre Braun Mauer Jeter Molina Fielder s s and

19 MLB 2012 Rankings Markov Runs Name Created 27 OPS Posey Cabrera McCutchen Trout Beltre Braun Mauer Jeter Molina Fielder s s and

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