# Linear Program

The Player Rating Formula and the ensuing Runner, Pitcher, and Catcher ratings allow us to generate a model that will deliver a binary decision making tool that considers both the quality of the base runner and the quality of the pitcher/catcher tandem. With each Runner rating associated to a player's name, we can show how each individual player compares based on their data from the 2015 season. To capture the essence of a stolen base, we offset the Runner rating by the summed average Pitcher and Catcher ratings, therefore creating an offense/defense equality. Additionally, to further expand our tool, we developed the equation for both Pitcher and Catcher ratings one standard deviation superior and one standard deviation inferior than the league average. Consider this the 'above average' pitchers/catchers and the 'below average' pitchers/catchers.

The output Xi represents the difference between the offensive and combined defensive ratings. When Xi is positive, the tool will output a 1. When Xi is negative, the tool will output a 0. Further details on the binary output are provided under the Discussions tab.

With the output of a 0 or 1, we now have a simple tool that offers more clarity on whether or not to attempt a stolen base.

After we generated the output data for all three Pitcher and Catcher percentiles (-Sigma, Average, +Sigma), we show that the base runners fall within four tiers. The tiers are divided based on the standard deviation data provided under Player Rating Formula.

The primary function of our binary output is to synthesize the situational Runner, Pitcher, and Catcher ratings into a more easily understandable form. Instead of calculating the probability of success for a stolen base attempt, which would necessarily factor in a considerable number of external inputs that contribute to the event of a stolen base, we focus on comparing the average quality of the runner against the average quality of the pitcher/catcher duo. The value lies in comparing the ability of the runner based on statistical data against the ability of the pitcher/catcher duo. When the linear program outputs a 1, this translates to the average effort from the runner will beat the average effort from the pitcher/catcher duo. When the program outputs a 0, this translates to the average effort from the pitcher/catcher duo will be stronger than the average attempt from the runner. To be clear, the binary output does not comment on the probability of success, and the output of a 1 does not indicate that the base runner will successfully steal second base. Similarly, an output of 0 does not indicate that the base runner will be thrown out if he attempts a steal. For example, a below average base runner could attempt a steal against an above average pitcher/catcher duo, and because of a poor pitch or a poor throw, which would be atypical but possible, the runner could be successful. By focusing on the average effort, we can thus provide a subtle recommendation that provides the manager with a tool for deciding whether or not to attempt a stolen base. Furthermore, when the output is 1, we consider the expected value of the stolen base attempt to be positive, and that over the long run the accumulation of positive expected value stolen base attempts will provide a significant advantage for the team.

By working with one standard deviation above and below the average defensive rating, we hope to capture a more comprehensive view of the league, offering managers the ability to tailor the output to their needs. Stolen base prevention is not consistent across the league, and therefore by differentiating our model into three tiers we offer another level of information for the manager. Lastly, our goal is to provide an elementary decision making tool, rooted in the organic statistics of the game, that will offer an ability to view a particular situation with simplified data available.

To demonstrate the utility of our decision making tool, we outline three base running scenarios where our binary output can be effective.