It turns out people are really very dedicated to the idea of replacement level. I think that's great. I've admired the beauty of the idea since I read Keith Woolner's description of VORP. And the debate about replacement level has made me stumble across some really fascinating uses of algebra in sabermetrics. Some of them are inspired, some of them are bizarre, some of them are just parlor tricks, but they're all pretty neat. So let's do it to it--algebra, that is!

**Table of Contents**

Slope-Intercept and Marginal Wins

The OBP/WHIP Trick

The OPS/Runs Trick

Appetites for Risk

Discussion Question of the Day

**Slope-Intercept and Marginal Wins**

You remember slope-intercept form of a line? The one from junior high, where for some idiotic reason m stood for slope (still looking for a good answer on that one)? It does a great job of describing linear relationships in terms of--you guessed it!--the slope (rise over run) and y-intercept (the value of the dependent variable when the independent variable is zero).

Well, Tom Tango has dusted off the ol' slope-intercept form and applied it to wins above average, in an effort to prove that marginal wins have linear value. That is to say, he is trying to prove that every marginal win is worth the same amount of money, whether it is one that gets you from 50 to 51 wins or one that gets you from 80 to 81 wins (he might make an exception for making the playoffs, but you would have to ask him that). Nevertheless, in general wins are purchased linearly, he argues. Here's an excerpt of his reasoning:

y = mx + b

, where y is the salary, m is the slope (multiplier), x is the wins above average, and b is the intercept (i.e., replacement level)

Importantly, y is the salary minus the major league minimum (marginal salary). Using this technique, he finds that replacement level is a pitcher who has an expected .410 winning percentage or a position player who is 2.25 wins below average.

Have you ever thought about how similar OBP allowed and WHIP are for a pitcher? If so, you're not alone. They in fact express similar information (how many baserunners has the pitcher allowed?), but are not directly equivalent (nor are they simply the inverse of one another).

But, you may be thinking, there must be an easy way to convert one into the other, mustn't there? There indeed is! Kerry Whisnant has got us covered!

The full solution is a little messy, but the following formulas give a very good approximation:

OPB = WHIP/(3 + WHIP) + .016 or

WHIP = 3*(OBP - .016)/(1.016 - OBP).

Whisnant also has a nice reference table at the link for those without a calculator. You can also see the steps ("show your work!").

WHIP to OBP, you say? A mere parlor trick! "And the LORD hardened the heart of Pharaoh, and he hearkened not unto them." Well, then, how about the plague of OPS?!

Walk Like a Sabermetrician has figured a way, with some strange alchemy, to transmute lead into gold OPS into runs and OPS allowed into runs allowed. Behold!

Runs = (.496*OPS - .182)*(AB - H)

Which is a neat trick even if the correlation isn't terribly high. You can of course follow the steps at the link. The author is especially careful to note that it breaks down rapidly at the extremes:

This is the part where I set off the secret beacon in the Statue of Liberty and perform a mind-wipe, and you forget everything you just read and never, ever actually use the Reynolds estimate, okay?

The question of how to value risk is a tricky one. Some very smart financial analysts thought they had it all figured out, and are still trying very hard to get better. But some mystery, it seems, may always remain. I like to chalk it up to the difference between uncertainty and risk. But however you slice, it applies just as equally to fantasy baseball as it does to the transactions of big league teams. So I think Eriq Gardner's comments are especially insightful:

You may believe you are making [a trade offer] that serves the rational interests of your trading partner, but are you selling an equity to someone who prefers a lottery ticket? Are you marketing a low-probability chance to win millions to a team less fearful of losses? Understand someone’s tolerance for seeking or avoiding risk or aptitude for measuring gains and losses and you may begin to get a sense about how to trade with them.

Given the current financial crisis, I think this point carries added weight. Some sports teams, faced with large losses, may be cutting costs to hedge against the risk of future losses. Other teams, with implicit backing, may have bigger than ever appetites for risk.

*The New Yorker* recently sat down with sports economist Andrew Zimbalist, and he had some things to say about how teams are reacting in the current economy:

[I]n the seventies and eighties the owners got the lion’s share of revenue from selling tickets and a little bit more from selling national television contracts through the league. Today, teams are increasingly earning revenue from other sources—corporate sponsorships, signage at the ballpark, catering at the ballpark, the Internet, and so on. All of these new sources of revenue lean more heavily on higher-income groups and discretionary spending, and are much more affected by forces in the macroeconomy.

And, to repeat something that’s been said a lot over the last six months, this recession is different. It can’t be sloughed off like the previous ones.

It certainly seems to be affecting teams' decisions, and many have wondered if it might not have played a role in Toronto's decision to waive Alex Rios and the $60 million he was owed over the remainder of his five-year contract. Even still, you can be left with the feeling that he made a mistake. I think that Keith Law put the folly of the Rios waiving best:

But this raises a much bigger question about general manager J.P. Ricciardi's tenure in Toronto, which increasingly has been marked not by bad baseball decisions, but by bad financial ones.

The decision to give away Rios indicates that the Jays' front office believes the contract extension the team gave Rios -- heavily back-loaded, as is every deal Ricciardi has handed out -- exceeded his current market value. In other words, the Blue Jays screwed up and overpaid him. (Either that, or they just screwed up by giving him away for nothing.)

Whatever your assumptions, somewhere along the way, it had to have been Ricciardi's fault.

**Discussion Question of the Day**

If you're curious, you can follow Jeff, with a little help from me, in a 2010 fantasy baseball mock draft. I feel a little like the stock clerk at KMart who has to bring out the Christmas decorations in August.

Pursuant to Tango's algebra lesson above, can you think of an argument (not an assertion!) for why marginal wins/marginal cost might not be linear in the aggregate? Let's exclude for the moment the marginal win that would put a team into the playoffs and just look at all marginal wins. Is there a reason to believe a four win player isn't worth twice as much as a two win player? That a six win player isn't worth three times as much?

I'm curious to hear what you have to say.