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THE EXPECTED RUN MATRIX At various points in this book we have referred to the Expected Run Matrix. As we indicated in the Preface, there are eight possible configurations of the baserunners in baseball (two possibilities for each of the three bases), and three possibilities for the number of outs in an inning. Thus, at any given point in an inning, the game can be classified as being in exactly one of twenty-four possible states. For example, at the beginning of each inning, the state is: nobody on, nobody out. Later, we might be in the state: runners on second and third, two outs. It is quite useful to associate with each of these twenty-four states, an estimate of the expected number of runs that an average team will score in the remainder of the inning. For the state (nobody on, nobody out), that value is simply the average number of runs scored per inning, which for the years 1985−2011 was 0.510. In contrast, if runners are on second and third with two outs, then the expected number of runs scored in the inning is 0.378. Obviously, these values are approximations of unknown quantities, since in a real game there are dozens of variables for which we haven’t accounted . Nevertheless, knowing the expected run values for each state has proven to be an extremely useful framework for sabermetric analysis. It is important to note that the Expected Run Matrix is not static. That is, the values can change considerably during different eras, when, for whatever reason, the run scoring environment is different. In Table 19, we present the Expected Run Matrix for 1982−1985 alongside the corresponding matrix for 1997−1999. Note how increasing the number of outs always decreases the run expectation, while increasing the numbers of baserunners always increases run expectation. To illustrate, observe that with a runner on first and nobody out, a successful sacrifice bunt that advances the runner to second would result in a net 140 Appendix loss of 0.866 – 0.667 = 0.199 expected runs in the early 1980s, but a decrease of 0.939 – 0.707 = 0.232 expected runs in the late 1990s. The matrix is one methodology that is used for detecting the average incremental impact of different strategies and batter outcomes during the course of a game. Table 19. Expected Run Matrix for 1982−1985 and 1997−1999 1982−1985 1997−1999 Bases \ Outs 0 1 2 0 1 2 Bases empty 0.482 0.254 0.098 0.552 0.296 0.114 Runner on first 0.866 0.505 0.215 0.939 0.571 0.242 Runner on second 1.115 0.667 0.321 1.197 0.707 0.343 Runners on first and second 1.459 0.894 0.421 1.540 0.968 0.457 Runner on third 1.361 0.963 0.368 1.427 0.999 0.376 Runners on first and third 1.734 1.187 0.479 1.879 1.214 0.527 Runners on second and third 2.006 1.400 0.598 2.044 1.430 0.594 Bases loaded 2.294 1.568 0.749 2.364 1.634 0.784 ...

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