To explain this easily I need to move down to Rock, Paper, Scissors. If you are up against a bot that always throws each 33% of the time with no strategy involved, that is called an un-exploitable strategy, but it's also an un-exploiting strategy. Even if I were to always throw Paper 100% of the time against this bot, we would always on average tie. This is the Nash Equilibrium point.

What you want to do is perhaps start with this random strategy, determine the realtive strength of your opponent, and only move to an exploiting strategy if they are weaker than you. If they throw Rock 40% of the time, you should throw Paper 40% of the time. If you were to enter a RPS tournament playing randomly, it is unlikely that you would reach the end. You need to exploit.

In Poker Cash Games, people just play the Nash and when opponents make flat-out mistakes they yield a little bit of profit. Do this enough with high enough stakes and it works out, even if its extremely boring. But this won't work in Poker Tournaments, where you only get payouts for top 10%, weighted heavily towards winning. So what happens in Poker is what happens in RPS - you attempt to find exploits, and your opponents attempt to find exploits in your play. In turn each of the players are attempting to move the environment of the play towards a position where they know it well and can find those mistakes.

If deviating from the Nash yields better results than the penalty of deviating because in that environment you are stronger than your opponent, you should take it. As this article points out, the same is true in Chess. And as you point out, this is also true of the Eephus pitch.

Much of the velocity on a home run comes from the pitch (equal and opposite reaction) rather than the bat speed. So it’s not that easy to hit a home run on a 50 mph pitch.

This is so incredibly false. They're not throwing +80mph pitches at the MLB home run derby. Batting practice pitches are about 60mph.

You are very incorrect: https://physics.stackexchange.com/questions/18222/how-does-t...

Incoming speed makes significant difference to exit velocity.

If you were going for max distance, you would want 100 mph pitches.

In home run derby, they are optimizing for ability to get a good repeatable swing with enough power to go over the fence. If they were playing in a park with like 440 ft fence, they would need faster pitches.

That's actually not true.

The distance the ball flies has a lot more to do with bat speed than ball speed (something like 5mph for 1mph, if I recall correctly).

Okay, true. But it should be pretty easy to make contact, at least.

Conservation of momentum says otherwise -- the faster the ball is going, the more change in momentum it needs, the harder it needs to be hit.

If you want a ball to bounce further off something, throw it harder.

Yes for elastic shocks, but bat to ball is not an elastic hit (well, barely one, to the point where the physics you refer to don't matter).

Or the farther the bast needs to bounce off th ball.