Tuesday, November 27, 2018

A flawed argument that marginal offense and defense have equal value

Last post, I argued that a defensive run saved isn't necessarily equally as valuable as an extra offensive run scored.  

But I didn't realize that was true right away.  Originally, I thought that they had to be equal.  My internal monologue went like this:

Imagine giving a team an extra run of offense over a season.  You pick a random game, and add on a run, and see if that changes the result.  Maybe it turns an extra-inning loss into a nine-inning win, or turns a one-run loss into an extra-inning game.  Or, maybe it just turns an 8-3 blowout into a 9-3 blowout.

(It turns out that every ten games, that run will turn a loss into a win ... see here.  But that's not important right now.)

But, it will always be the same as giving them an extra run of defense, right?  Because, it doesn't matter if you turn a 5-4 loss into a 5-5 tie, or into a 4-4 tie.  And it doesn't matter if you turn an 8-3 blowout into a 9-3 blowout, or into a 8-2 blowout.  

Any time one more run scored will change the result of a game, one less run allowed will change it in exactly the same way!  So, how can the value of the run scored possibly be different from the value of the run allowed?

That argument is wrong.  It's obvious to me now why it's wrong, but it took me a long time to figure out the flaw in this argument.

Maybe you're faster than I was, and maybe you have an easier explanation than I do.  Can you figure out what's wrong with this argument?  

(I'll answer next post if nobody gets it.  Also, it helps to think of runs (or goals, or points) as Poisson, even if they're not.)

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