THE BOOK--Playing The Percentages In Baseball

Monday, January 25, 2010

How much to pay for a robot?

Interesting question posed, which I’m going to change the question slightly, for the purposes of this thread: you have a robot that comes out and pitches every 5 days, pitches a complete game each time (regardless of whether it’s in extra innings or not).  He strikes out the first five batters he sees, and gives up a HR to the sixth batter.  He cycles through this all the time (carrying this over into his next starts, so that he’s always giving up 1 run per 5 outs (5.4 runs per 9 innings).  That’s 36 starts, about 324 innings.  How much do you pay this guy?  (Assume your offense scores an average of 4.7 runs per game, but in the normal human distribution of scoring.)

There are two considerations: (a) how often will his team win, and (b) how much do you pay for having perhaps a 9-man (plus one robot) pitching staff?

Part 1
In a nine-inning game, if your offense scores 4 runs for your robot pitcher, you will lose every game.  This will happen 54% of the time.  If your offense scores 7 runs or more, you will win every game.  This will happen 25% of the time.

If your offense scores 5 runs per game (happens 12% of the time), you will go into extra innings the times your robot pitcher gives up 5 runs (which happens 60% of the time), and you lose when your robot pitcher gives up 6 runs in the first 9 innings (happens the other 40% of the time).  Let’s say you win half the time you get into extra innings.  So, that means when you score 5 runs in the first 9 innings, you will win 30% of the time (60% times half).

If your offense scores 6 runs per game (happens 9% of the time), you go into extra innings the 40% of the time your robot pitcher gives up 6 runs, and you will win half those games.  So, when you score 6 runs, you win 80% of the time.

Add it up:
0 to 4 runs: 0 wins
5 runs: .12 x .30 = .036 wins
6 runs: .09 x .80 = .072 wins
7+ runs: .25 wins

Total: .358 wins per game

We’ve just created a robot pitcher that basically guarantees you will win 36% of the time.  In exchange for that, you get one or two extra roster spots. Call me crazy, but you should not pay anything other than zero dollars for that.

Part 2
Now, you want to try something more interesting, how about this: six strikeouts followed by one HR each time.  That means an average of 4.5 runs allowed per 9 innings pitched.  Wow, you must be thinking.  4.5 runs allowed (equivalent to a 4.15 ERA pitcher) with 324 innings?!?!  And guaranteed he won’t get hurt for the three years you sign him?  Wow!!  Sign him up, pay him whatever he wants.

Not so fast.

You are guaranteed a loss when your offense scores 0 to 3 runs: happens 41% of the time.
Guaranteed a win when you score 6+ runs in the first 9 innings: happens 34% of the time.
When you score 4 runs, half the time you lose, half the time it goes to extra innings (and you win half those games); happens 13% of the time
When you score 5 runs, half the time you win, half the time it goes to extra innings (and you win half those games); happens 12% of the time

Add it up:
0 to 3 runs: 0 wins
4 runs: .13 x .25 = .0325 wins
5 runs: .12 x .75 = .090 wins
6+ runs: .34 wins

Total: .4625 wins per game

Now it gets interesting.  Your robot pitcher lets you win 46.5% of the time, and he saves you a roster spot or two.  He never gets hurt, and gets you 324 innings.

Replacement-level theory says to pay him about 0.35 to 0.4MM$ per start, which puts him at 12-14MM$.  The value of going with a 15 position players instead of 13 or 14?  Maybe another million?  Let’s make him 15MM$.  Pretty good.

And this shows you the sweet spot.  At 5.4 runs allowed per game, your robot is worthless.  At 4.5 runs allowed per game, your robot is worth 15MM$.