I would argue that preventing the running game s the one with most impact on winning and losing. This is the case since the difference between the best and worst catchers (at the major league level) in regards to the first two aspects listed above largely measured by passed balls and errors, just don't result in a "large" number of bases saved by the best catchers. And the third aspect has been studied and Woolner notes that "the current state of evidence supports the theory that catchers exercise minimal (at best) influence on how well their pitchers perform versus opposing batters." As an aside I should note that concepts such as Catchers ERA do seem to have some currency with major league teams as I see it referenced on the Royals pre-game notes given to members of the media before each game.
So in terms of preventing stolen bases Woolner calculated what he called XSB (Extra Stolen Bases Allowed) and XSBR (Rate of XSB) by using the matrix of expected run values of the various base and out combinations along with play by play data from Retrosheet. After running the numbers he calculated that Mike Piazza had 4 of the 11 worst seasons in the period 1972-2003 with the following XSB values:
1999 49.33
2000 50.21
2001 50.34
2002 60.65
That means that in those years Piazza gave up on average over 50 more bases than an average catcher would have given the same opportunities. Conversely the best catchers in the period saved around 25 bases per year with Benito Santiago in 1989 garnering the top spot. So the spread here is around 75 bases and of course the difference between the best and worst catchers in terms of passed balls and errors is nowhere near 75. Couple that with the fact that the bases gained or saved in these scenarios are second and third and often taken in stategic situations, their importance only increases. Piazza is simply not a good defensive catcher and has never been one. He certainly should have been moved to first base a long time before now.
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