As a championship amateur baseball center fielder I have some perspective on how to play this. 

What you have to be thinking in this situation is that even if you have a big arm you're probably not going to throw the runner out at the plate on a single up the middle even cheating in some (aka play a little shallower than normal.)

Therefore, your best bet is to play deep because if the batter muscles up and hits one over your head you're *really* screwed. Not only is the tie run going to score (which you were hoping to cut off by playing shallow), but the batter is going to have at LEAST a double and possibly a triple if they're a fast runner. 

This is exactly what happened to the Astros center fielder tonight and led to a Rangers win. Nice. 

But don't fret, Astros center fielder. In my youth I would have optimistically played it the same way. And this is why baseball, more than any other sport, requires extra smarts and makes it > all other sports.