IT'S NOT THE ECONOMY, STUPID!: MATCHING THEORY AND THE VOLATILE MARKET FOR HITTERS

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This past offseason has surprised a
lot of people.  On one hand, Mark
Teixeira, Derek Lowe, CC Sabathia, and Rafael Furcal had no trouble finding
contracts that we would have expected before the recession hit.  On the other hand, Adam Dunn, Bobby Abreu,
Pat Burrell, and Orlando Hudson all received contracts that surprised
many.  The typical reason that most
people give for these weak contracts is the economy.  In this article, I will explain why
microeconomics, and not the macroeconomic outlook, is best suited to explain
this outcome.  I do not believe that any
pitchers received lower contracts than expected this offseason, and the hitters
who did receive weak contracts played similar positions.  Specifically, many played corner outfield
positions.  Each hitter on the free agent
market can fill only one or two positions, and many teams do not have openings
at these positions.  Each pitcher on the
free agent market, however, can fulfill a role that is valuable to all 30
teams.  Therefore, the ratio of suppliers
to consumers for pitching services is always more stable.  Later in this article, I will discuss some
statistical evidence that hints even stronger at this conclusion.


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The main evidence that the current economic downturn is not
the primary cause for the weakened free agent market is that so many players
received good deals.  I mentioned four at the beginning of the
article.  Many people have pointed out the Yankees as the deviant not
playing along, but this is not realistic.  To understand this, it is
important to remember the market structure in which the Yankees signed these
players.  When a baseball player reaches unrestricted free agency, teams
compete for his services in an auction format.  While teams may not know
the details of each other’s offers with certainty, it is well reported when an
offer is made, and teams can approximate each other’s offers as well as we
can.  To win an auction, you need to offer more than the second highest
bidder.  The Yankees had competition for each of these players.  The
fact that they won at such high levels demonstrates that more than the Yankees
are responsible for those high salaries.  The salaries those players received
were very similar to the kinds of salaries many predicted before the downturn.

The reason that a number of players received lower salaries
via free agency than would have been expected is that the second highest bids
for their services were not very high.  I believe that the reason can be
explained using a little bit of matching theory and a little bit of auction
theory.

As I mentioned, every team needs a good starting
pitcher.  There are always several starting pitchers available on the free
agent market, and always many teams that inquire as to their
availability.  The number of replacement level (or at least close to
replacement level) pitchers that would have to fill rotations without free
agents is very high.  However, few teams have openings at any given
position.  Think of your favorite team.  How many positions could
they have significantly upgraded this offseason?  I planned on cracking a
joke at this point, which started with “Unless your team is the…” but I cannot
even figure out who would finish this sentence.

At this point, you might be thinking to yourself that the
lack of demand for hitters might depress their prices–to the contrary. 
The supply at any position is also relatively low.  Sometimes the price of
a player will be very high instead.  The important thing to understand is
that the price of hitting talent is naturally going to be variable, as the
ratio of suppliers and consumers of labor at these positions will vary wildly,
adjusting the bargaining positions of these hitters by a lot each year.

This fact may seem counterintuitive at first.  Most of
us understand that pitchers’ performances are difficult to predict, mostly
because of defense that they cannot control.  However, if you look at what
teams expect on average out of a pitcher, you can easily put a price on that
expectation.  In fact, the price will vary more for hitters even if the
prediction is sharper.

The value that a team is willing to pay for a player is the
Marginal Revenue Product of their labor.  This is represented by the
following formula:

MRP
of a Player to a Team = Dollar Value of a Win to a Team * Wins Added by that
Player to that Team.

The value of a win to different teams will vary–teams on the
cusp of making the playoffs in large markets with fickle fans have high
valuation for a win.  Below average teams with no chance of making the
playoffs in small markets with loyal fans have low valuation for a win.

Wins added by a player to a team is the variable that is
important.  As Sabermetricians, we tend to approximate this by comparison
to someone called a “Replacement Player”.  I have yet to exactly identify
who this guy is.  It seems that different sources have different estimates
of what a replacement level player should be, and how good he is.  In
reality, a replacement level to a team at an individual position depends on the
situation.  If the Cardinals signed a first basemen, he would need to have
a very high value.  While one could argue that they would trade Pujols,
they would be in a vulnerable situation and would not be able to get fair value
for him.  Therefore, the wins added by Mark Teixeira to the Cardinals
would be very low.  In fact, very few teams had high values for Bobby
Abreu, Adam Dunn, Manny Ramirez, Pat Burrell, Milton Bradley, and Raul Ibanez
this winter.  Three of those six guys–all above average corner
outfielders–got very weak contracts, which were much lower than many
expected.  The reason was that so few teams had many wins added by that
player to that team. 

Many have tried to make the argument that the first
term–”dollar value of a win to a team”–is what has gone down this winter, but
this is belied by so many large deals.  What was particularly weak? 
The market for corner outfielders was, and therefore so were the wins those
players would add to those teams.  Orlando Hudson got a weak deal as
well.  This is not surprising either.  For a large set of those teams
with reasonably high values for a win–the teams Baseball Prospectus predicts
will be over .500 this year–only the Dodgers had a replacement level second
basemen (the exception could perhaps be the Indians, but their spending habits
suggest that perhaps they have a low enough value for wins that it was not
worth filling the spot anyway, and they are probably trying to grow Asdrubal
Cabrera even if his projection is low).  Of course, Hudson would not get a good deal.  To
anti-paraphrase James Carville, It’s not the economy, stupid!

The equation above says that this will determine the amount
that a team is willing to pay for a player, but remember that the team that
signs the player ends up paying approximately what the second highest bidder is
willing to pay them.  This does not change the argument at all,
however.  Higher volatility in value of the each player to each team will
yield this result anyway.

The argument that I made about pitchers sounded good, but I
was not quite ready to take my own logic at face value, and pat myself on the
back.  Rather, I decided to go through Baseball Prospectus’ Depth Charts
for 2009 and take a look at the VORP projections for the worst of the five
starters that BP listed and the first spot starter listed as well.  In
fact, the average VORP for the worst of those five starters was 5.3.  More
impressively, the standard deviation was only 6.6.  The maximum value was
16.5 and the minimum was -7.5.  The top spot starter averaged a VORP of
5.5 with a standard deviation of 4.5, a max of 14.6 and a min of -1.9. 
This is even after the market sorted itself out!  Nearly all teams even
now have the same replacement level.  Clearly, all teams have a very
similar wins added value for CC Sabathia.  The Yankees just had a higher
value for those wins due to the nature of their market and the nature of their
division.

Even still, I was skeptical that I could gather any
statistical proof to show this.  My plan was to test the projected 2009
wins value per dollar for each player with more than six years service team who
was signed to a major league contract for 2009.  However, more than
anything else, I would expect that Sabermetricians do a relatively poor job at
projecting pitching compared to teams.  The famous “TINSNAAP” claim of
years past was never really true for scouts.  Statistical projections
never really show that a guy is going to turn the corner slowly but had high upside
when he got a hold of things, or that a guy with hot minor league numbers was
going to struggle against hitters who could lay off his breaking ball. 
Scouts can do that, though.  My realistic hope was that the standard
deviation would be ambiguously similar, and that would be a claim that my point
was accurate.  However, I was pleasantly surprised that my guesses were
not only accurate, but statistically significant–they survived an F-test. 
Even including relief pitchers, I was able to get significant results most of
the time.

I found 235 players who were currently signed to major league
contracts for 2009.  I approximated their salaries using pro-rated signing
bonuses where the signing bonus was not explicit.  I did not include any
performance bonuses.  Obviously, teams sign players to more complicated
contracts–and many times the value of a player is not at all highest in the
year where his pay is the highest (take a look at Vernon Wells’ contract)–but
this was meant more as an approximation.  I gathered their WAR according
to the CHONE projections as well as their WARP/WXRL for hitters/pitchers
according to the Baseball Prospectus PECOTA projections.  I used WAR/$1MM
(and WARP or WXRL/$1MM) rather than dollars per win, because many players had
high salaries and were not projected to even really play this year and doing
dollars per win let certain outliers affect the outcomes too much.  It was
more reasonable to use wins per dollar.  (I actually used wins per dollar
above league minimum..

The standard deviation for all 235 players with major league
contracts according to the CHONE projections were as listed for the following
groups:

All
Pitchers: .451
All
Hitters: .490
Starting
Pitchers: .205

Using an F-test, starting pitchers show significantly lower
variance in the projected wins per dollar in their salaries than hitters. 
This result held true when I separated contracts into under $2MM, between
$2-5MM, and over $5MM.  Each subset of data passed an F-test.

Then, I checked the Baseball Prospectus PECOTA projections.

All
Pitchers: .786
All
Hitters: .484
Starting
Pitchers: .231

Clearly, starting pitchers against showed statistically
significantly lower variance in the projected wins per dollar in their
salaries.  However, pitchers overall add higher variances.  Breaking
that down by salary level:

Under
$2MM, PECOTA:

All
Pitchers: 1.807

All
Hitters: .723

Starting
Pitchers: .470

 

Between
$2-5MM, PECOTA:

All
Pitchers: .142

All
Hitters: .245

Starting
Pitchers: .188

 

Over
$5MM, PECOTA:

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All
Pitchers: .156

All
Hitters: .161

Starting
Pitchers: .162

It
seems that the relief pitchers are very volatile in the wins per dollar in
their salaries.  This could be that only a few teams need closers at a
time, or it could be that PECOTA’s WXRL does a poor job of valuing closers (or
at least, values them very differently than baseball teams).  Looking at
the data more closely, it is actually mostly Trevor Miller and Tom Gordon who
are both making $500K this year only ($100K above league minimum, which is what
I computed everything relative to), but PECOTA projects to be significantly
above replacement level, that explains this odd data.  Otherwise, the
results are statistically significant when comparing all hitters to all
pitchers, even if the closer position has a lot of the same properties as the
market for an individual type of position player.

That
brings us to the question of what teams should do about it.  Certainly,
knowing about this market structure can help teams understand how to bid for
free agents and anticipate what players will get.  If knowing in advance
that the market is going to overflow in 2009-2010 for a certain position,
perhaps corner outfielders again, a team out of contention might try to trade a
corner outfielder at the trading deadline even if they plan on competing in
2010 and that player they have is under contract through 2010.  That team
may be able to get that player cheaper.  Certainly, there are many
adjustments teams can make anticipating the market.  Even further than
that, teams can think about whose free agent years to buy out as well–those for
whom the market will good.

I
am sure there are may other ways that this can help teams.  They can
adjust their decisions this way, but they can also work on their assumptions as
well and anticipate other similar patterns.  Baseball’s free agent market
is a labor market with frictions.  Much research has been done on these
kinds of markets in general, and this is probably only the tip of the iceberg
on its applications to baseball free agency.

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