Non-Linear Cost-Per-Win Analysis

A month ago I wrote the classic Cost Per Marginal Win post, comparing each team's payroll above league minimum (about $13MM) to their win total above replacement (about 48 wins).  It's a nice approach, but actually only addresses a very specific question: "How much are teams spending per marginal win?"  While interesting, we often want to answer questions about front office comptetency. 

The main flaw in a straight cost-per-win analysis is that the first extra dollar spent on payroll is much more efficient than the 100-millionth extra dollar spent.  Florida, for example, barely spent more than they had to.  That money went to very small raises to good players in their first three years of team control (Dan Uggla made $27,000 more than the league minimum), first-year arbitration salaries (Alfredo Amazega made just under $1MM), and cheap, cost-efficient free agents (Jorge Cantu provided 2.8 WAR for $500K).  Players in those categories earn anywhere from literally nothing per marginal win to, say, $500K per marginal win.

On the other hand, the New York Yankees, with their $209MM payroll, spent the majority of their money on free agents.  Free agents are not good bargains, earning about $4.5MM per marginal win.  And while there's a good argument to be made for the Yankees mismanaging their money, there's no way they can spend that much money on payroll and not spend a lot of it on free agents.  There's only so much money to spend on cost-controlled and arbitration-eligible players.  If you want to spend more, you have to sign free agents.

To summarize, most teams need to spend more in order to win more games, but the more you spend, the less efficient each dollar is.  Starting at the cheapo end of the scale, here are some estimates of where extra money goes as more is spent, at what rate it's spent, and how good the team can expect to be:

• The London McScrooges, minimum payroll: Money is spent exclusively on players in their first three years of service, at the league-minimum rate of $390,000 with minimal raises, if any at all.  Some replacement-level free agents might be signed.  The cost per win isn't much above zero.  Success isn't common, although the farm system should contain enough talent that this team is definitely better than replacement-level.  66 wins.
• 1997 Tampa Bay Rays, $25MM: Extra money is spent on first-year arbitration players at about $1.8MM per marginal win (40% of FA rates).  Payroll isn't much more than with the Scrooges, but the extra money buys quite a few wins.  71 wins.
• 1998 Colorado Rockies, $55MM : Extra money is spent on second and third year arbitration players at $2.7MM to $3.6MM per marginal win (60% and 80% of FA rates).  In addition, teams might buy out some free agent years at similar rates while buying out arbitration years (see the recent Evan Longoria and Dustin Pedroia contracts).  These teams are still below league-average payroll, but have a non-zero shot at a playoff spot given a strong crop of young players.  76 wins.
• 2000s St. Louis Cardinals, $90MM: Extra money is spent on lower-level free agents who may come at a discount.  Teams take what they can get to fill holes without spending the full price of $4.5MM per free agent marginal win.  Big name free agents rarely fit the criteria.  At this point, teams are spending about the league average and winning a league average number of games.  81 wins.
• New York Mets, $120MM.  Extra money is spent on bigger name free agents.  Holes are filled by Derek Lowe- and Adam Dunn-types instead of Jon Garland- and Garret Anderson-types.  This spending isn't very efficient ($4.5MM per marginal win compared to the overall average of $2.7MM per marginal win) but going from 81 wins to 86 wins leads to a significant increase in playoff probability.  86 wins.
• Boston Red Sox to New York Yankees, $160M to $210MM.  Extra money is spent on the best of the best.  Because there are a limited number of players who will contribute on the field, it's worth overspending on star players.  Obviously, there's a huge payroll gap between the Red Sox and Yankees.  If the two teams were equally intelligent, you'd expect about 91 wins from a payroll the size of Boston's and 96 wins from New York's $200MM+ payroll.  That the Sox have kept pace with the Yankees for the last five years is a testament to Theo Epstein and company.

So what we need now is a model of how the cost-per-win changes the more money a team spends.  Thankfully, David Gassko and Tom Tango have done some work in this regard, and I'm going to borrow their result:

ExpWins = (P + 2*L) / (P + 5*L) * 162

P is team payroll and L is league-average payroll.  The model is basically saying "Given a front office with average ability, how many wins would you expect out of a team that spends P on their payroll?"   Here's the effective cost-per-win at various payroll levels given the model:

Once we know how many wins to expect given a team's payroll, we can compare it to how many wins a team actually had to judge how effectively they spent their money:

Obviously, the Rays were the cream of the crop last year.  They spent very little money and won a lot of games.  The strength of this system shines when looking at the second team on the list, however.  The Angels spent a lot of money, but still won many more games than expected, something not recognized by a straight cost-per-win analysis, which ranked them tenth in efficiency.

Salary data from USA Today's Opening Day payroll database