# Introducing Grid WAR: Rethinking WAR for Starting Pitchers

Traditional methods of computing WAR (wins above replacement) for pitchers are based on an invalid mathematical foundation. Consequently, these metrics, which produce reasonable values for many pitchers, can be substantially inaccurate for some. Specifically, FanGraphs and Baseball Reference compute a pitcher's WAR as a function of his performance averaged over the entire season. This is wrong because WAR must be a convex function of the number of runs allowed by the pitcher in a game. Hence we propose a new way to compute WAR for starting pitchers: Grid WAR (GWAR). The idea is to compute a starter's GWAR for each of his individual games, and define a starter's seasonal GWAR as the sum of the GWAR of each of his games. We show that GWAR is indeed a convex function in the number of runs allowed during a game. As such, GWAR accounts for a fundamental baseball principle that not all runs allowed have the same impact in determining the outcome of a game: for instance, the difference in GWAR between allowing 1 run in a game instead of 0 is much greater than the difference in GWAR between allowing 6 runs in a game instead of 5. Moreover, Jensen's inequality implies that, by ignoring the convexity of WAR, current implementations of WAR undervalue certain pitchers, particularly those who allow few runs (specifically, 0 or 1 run) in many games. It also unfairly penalizes pitchers who are credited with a large number of runs in a short outing. These flaws are corrected by GWAR.

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