I've been playing for a few years with some success, but I haven't seen nearly as many seasons as many of you. I'm curious how veteran players balance the value of a short passing game vs. a medium passing game. I'm looking forward to hearing different opinions on this one. Thanks!

I think this is very dependent on formations and talent but I would say you see way more medium pass teams and it's by a significant margin.

For what type of offenses I play, I haven't found any advantage to calling a short pass instead of a medium pass.

I have found that Deep passes are almost never completed vs teams equal or better than yours.
(Bring back Daryle Lamonica - The Mad Bomber ! :)
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From what I see...the vast majority of my opponents throw medium. I so see quite a bit of short. Rarely do I see a deep, long, or very short. That information comes from teams primarily playing Pro Set and some I-Formation. I see a little bit of Trips and not much of any other formations.

I'm absolutely NOT a passing guru (just ask any of my opponents)...but there can be advantages in the short passing game vs medium. You just have to find them........personnel, formation, playbook settings

From what I've seen, with all else even:

shorter passes have higher completion %, give up less sacks, and force the defense to tackle to keep it a short gain. Long passes are completed less %, give up far more sacks, but do go for big gains. Medium is decent in all aspects and doesn't have any huge pro's or cons. Decent completion %, you'll give up a few sacks, completions usually go around 8 ish yards.

Someone might throw short if their OL can't hold up a rush, or if they think their pass receivers can out maneuver the defense after the catch.