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The D-backs certainly appear to be attempting to win this one with pitching given a lower half of the order with goes Fuentes, Ahmed, Mathis and the pitcher's spot. There's an argument to be made that Greinke is close to being the second-best hitter among those players. Going by career OPS+, we have Ahmed at 61, Mathis (52), Fuentes (42) and Grienke (51). Admittedly, Zack has not been the hitter we have to expect this season, being just 1-for-22: in each of the previous three seasons, he has batted better than .200, with an overall line of .212/.246/.313. That compares favorably to Mathis's line over the same period: .203/.252/.297.
Our pitchers' hitting in general has been close to league average in terms of a triple slash: the D-backs sit at .127/.191/.137, for a .328 OPS, while league average has been .126/.160/.159 for a .320 OPS. So, a few more walks, but less power. Indeed, Arizona's seven walks by their pitchers is best in the league: there are seven teams in the NL who have combined for that many bases on balls by their hurlers. But so far we have only one extra-base hit (a double off the bat of Robbie Ray), and have only driven in one run (by Taijuan Walker), the latter tied for fewest in the league.
I'm not sure if Greinke's lackluster performance at the plate is the result of team direction, telling him to rein in his swing. You may remember, he hit the disabled list in July last season, suffering from an oblique strain which he apparently picked up at the plate. He ended up having to miss about six weeks as a result, and I can imagine Torey Lovullo telling Zack that his value to the team is more as a pitcher, and not to do anything that could potentially jeopardize taking the mound. The way Greinke has pitched over his last seven starts - an ERA of 2.32 and a K:BB ratio of 64:7 in only 50.1 innings - I wouldn't blame the team.