The Power of Substitution: A Rejoinder on the Linearity of Dollars and WAR


Two weeks ago, I posted a piece examining one of the more controversial aspects of player valuation, that being whether or not the market pays for production (WAR) in a linear or non-linear manner. Specifically, I examined all MLB Opening Day payrolls 2009 to 2011, focusing on players with 6 or more years service time in order to see whether high payroll teams paid more per WAR on players in this service class than smaller payroll teams, and I found no evidence to support the hypothesis that teams pay for production in a non-linear manner. I also attempted to frame the subject by briefly outlining some of the theoretical reasons for the and against the linearity of relationship between dollars and WAR, including one reason I find compelling, which I called "the substitution effect". Shortly after, jessef posted a piece disagreeing with my conclusions, criticizing some of the methodology and limitations of the data, but largely focused on the theory underpinning both views: that non-linear valuation has an "underlying guiding simple that it's beautiful"; whereas linear valuation rests upon a "faulty" premise. I addressed the former criticisms here, but I committed to addressing the theoretical criticisms separately in detail upon reflection. What follows is that (belated) rejoinder.

At the outset, I think it's useful to make an important distinction. Broadly put, there are two methods of investigating an issue: descriptive - describing the world as it is; and normative - describing the world as one thinks it ought to be. At its core, my study was descriptive, looking at the empirical evidence of what teams actually do. I took a null hypothesis of linear WAR (widely used in objective baseball analysis based on previous analysis), tested the evidence and found no statistically significant evidence for the alternate hypothesis of non-linear valuation; and therefore no reason to reject the null hypothesis. In providing some background on the topic, I introduced a theory I find to be a compelling explanation (and in hindsight, did not fully explain it), but it had no impact whatsoever on testing the data. Perhaps it would have been best to leave that out entirely, but to consequently suggest that the empirical study and conclusions were consequently "inherently flawed" on this basis is fundamentally disingenuous - one has nothing to do with the other. One is looking at the world as it is, the other proposing how the world should be. It has become apparent that some of this is a matter of different emphasis - I am more interested in what is occurring in the market, jessef more interested in how the market should function; nonetheless, the descriptive and normative must be evaluated separately, for they are not mutually exclusive. It is possible, for example, that teams should pay for WAR in a non-linear manner, but that they have not actually done so.

Notwithstanding that difference in emphasis, I do agree that the normative argument about how teams should pay for production is critically important. Indeed, if there is a disconnect between what teams should do and what what they are doing, this represents a market inefficiency to be exploited. I have done a lot of thinking about how teams should pay for wins in the market, weighed the various factors involved, and I've come to a conclusion that is ultimately based on a somewhat novel, but I believe compelling, application of the economic concept of substitution. At the same time, normative arguments are by their nature fraught with peril since they tend to be quite sensitive to underlying assumptions. I think the greatest error would be being overly overconfident in a theory, and not being willing to modify or discard in the face of evidence to the contrary. With that said, I present my in-depth view of the normative case below.

What is Substitution? How Does it Apply to Baseball?

Substitution is a basic principle in economics which holds that when the price of a good rises, consumers will change their behaviour, reduce their consumption of that good (that is, the quantity demanded), and reallocate their spending to other substitute goods. A simple example would be that if bad weather destroyed the crop of Florida oranges, the price of orange juice would rise and consumers would react by buying less orange juice (indeed, they have to - the reason that the price increased is that the supply decreased). Instead, they would likely buy more other juices and possibly other beverages, since they are substitute goods. To take the analysis one step further, a second order effect would be that the price of those other substitutes might rise a little bit, to represent the increased demand and therefore a new equilibrium. In this sense, a market dynamically reacts re-balance supply and demand among interchangeable goods through the price mechanism.

How does this apply to baseball? At a core level, each player is unique, and has his own unique market for his services. However, baseball is all about runs, and through this mechanism, the value of all players are tied to each other. A pitcher who prevents 30 runs above replacement is equally as valuable as the DH who creates 30 runs above replacement at the plate, and so there value will be generally linked. Of course, the market is not perfect - if both are free agents in a given offseason, they might both get different salaries since the particular supply/demand fundamentals might favour one over the other. However, a market populated by well informed agent should be reasonably efficient in aggregate at paying equal salaries for equal production.

Therefore, purely from the standpoint of baseball production, players are in an economic sense substitute goods. To reiterate, this means that as the price of one rises, the price of others will generally rise as well, and that as these prices rise, teams will change their behaviour in how they accumulate talent. If we broadly divide players into tiers - stars, above average regulars, average regulars, below average regulars and replacement level players, we will find that replacement level player salaries will be anchored near the minimum salary, and that if the free market salary for the highest tier of players rises then the price of other tiers will rise as well (regardless if valuation is linear or non-linear. A rising tide lifts all boats. One important distinction needs to be reiterated - this is confined to players with six or more years service time, as players with less service time have to ability to have their salaries determined in the free market.

When thinking about price changes in the market, it is necessary to remember what drives them - supply and demand changes. Focusing just on the supply of players among these different tiers, it is important to remember what the distribution of talent looks like. In a population of people, we assume a normal distribution of skill talent as a particular tasks. But MLB players are a small subset of the population, and even a replacement level player is many standard deviations above average. We are dealing with the extreme right tail, and that means that there will be very few star players, a greater number of above average players, an even greater number of average players, etc. As a result of this pyramid distribution, it is inevitable that teams will end up with a large number of replacement players, since they are the most populous group. If you reduced the number of teams or roster spots, this would still be the case, as it just redefines the replacement level - the fundamental distribution of talent remains intact. Therefore, there will always be a number of teams looking for below average players to replace replacement level players, some teams looking for average players to replace below average players, and so on. Obviously, the higher you go, the more impractical it becomes. Bringing in a star level player may directly replace an above average player, but that above player is likely to replace someone else lesser. This is referred to as the chaining effect.

It is my theory that due to this nature of this substitution between classes of players and the talent distribution in baseball, baseball production (WAR) should be paid for in the market in an essentially linear manner in terms of dollars ($). It is commonly held that a star player (5 fWAR) is more valuable than two average players (2.5 fWAR), and therefore should be paid more than twice as much as each 2.5 fWAR player. It is certainly true that the 5 fWAR player is more valuable, but the relative scarcity of even average MLB players means that the additive production of two 2.5 fWAR players is the same as the 5 fWAR player in terms of a market wide (across MLB) equilibrium. In the end, this market wide equilibrium will drive the pricing of this production, and should do so in a manner that results in linear $/WAR. In terms of the value of their production relative to the ultimate league replacement level, two 2.5 fWAR players should be paid as much as one 5 fWAR player. Below, I examine other theories that have been advanced to explain why the relationship should or shouldn't be linear, and how they do not supplant these fundamental characteristics. I then turn to some anecdotal evidence of market behaviour consistent with substitution. Finally, I advance a supporting argument for a linear relationship independent of the other arguments advanced here.

An Overview of the Theories Involved

I tried briefly going over some of the theories at play in my previous article, and didn't do a very good job. Nonetheless, it's important to trace some of the factors that are involved in the debate, so I'll go into more detail here. To start with, we have the premise that more productive players, who are better at creating runs with finite plate appearances and preventing runs in the innings they pitch (and generally pitch more innings, displacing inings from poorer players), should theoretically be paid more. This is uncontroversial and implies a non-linear relationship between dollars and WAR.

Next, we also have a theory that I neglected to mention, but that jessef identified as the critical argument in favour of a linear valuation model, whereby signing more productive players to large contracts involves a large amount of risk, and so teams (especially smaller payroll teams for whom a commitment will constitute a larger percentage of their payroll) will not to be willing to pay the salary implied by a strict application of the increased productivity. Dave Cameron reiterated this in a Fangraphs article in November 2011 on the same subject examining early data from the 2011 offseason:

The reality of the MLB free agent market is that premium players do not get a substantial bump that reflects that teams substantially value a high WAR concentration from a single roster spot. In fact, most of the evidence (including the Kenny Williams quote from the piece I wrote in February) suggests the opposite, that most teams are more interested in risk avoidance by spreading their production out over multiple roster spots.

I call this the insurance theory, because it essentially involves transferring risk between players and teams at a cost, which is what an insurance contract does. jessef refers to this as "risk averse spending" and suggests that you can just discount the projected WAR, to essentially get a risk adjusted WAR which is ultimately used in the valuation. Intuitively, this makes appears to make sense, but it ignores the fundamental nature of risk, that is the variance. You can't just use an average discount factor, because if you only have one of these huge contracts, the danger is that is completely blows up on you, not that you lose the average % of the contract. This is precisely why the reason the player wants to transfer the risk in the first place rather than keeping it and being paid more for it! Due to this dynamic, I do not believe this insurance factor, combined with the productivity factor, has to imply a linear valuation. Indeed, I believe that taken to a logical conclusion, the insurance effect alone could imply non-linear valuation as well. An example will illustrate exactly how this works.

We have a fictitious player, Player A, a 27 year old free agent pitcher. Based on his past performance and clean injury history, it is estimated that he will be worth $25 million per year over each of the next six years, if he were to sign successive one year free agent contracts (ignore how we get those numbers, and just assume it would be the results of a formal auction for his services, and ignore inflation - these are real values). In other words, his fair market value to teams is $150 million over six years. However, there is the risk that the pitcher could be injured at any time, including catastrophic injuries that end his career and then he can't earn anymore. At the stage he becomes a free agent, Player A faces all of this risk himself, but he wants to security and so he wants a team to assume it. The team is willing to do so, but for a price - in other words, they won't pay the entire $150 million, since they have to account for the expected value of injuries. Teams are better suited to assume risk since they hold a portfolio of players and contracts that helps spread the injury risk out; whereas for the player it is a one shot deal. So Player A and Team B agree to a 6 year, $125 million contract, which transfers the entire risk over 6 years to the team from the player - at a cost of $25 million. However, for a team which only has one of these type of players, it's unacceptable level of risk, since if that player gets hurt a significant potion of their budget will be dead money. So many teams insure some or all of the contract against the risk of injury, again, at a price (which is essentially from the player, since the player accepted a discount from going year-to-year to transfer the risk). Insurance contracts include the cost of the risk (which will involve premiums in the case of specialized contracts like $100 million dollar pitchers), and have administrative costs built in.

Larger payroll teams, however, tend to have numerous larger contracts, which changes the analysis. The Yankees for example, have $100 million commitments to Alex Rodriguez, Mark Teixeira and CC Sabathia. Moreover, they also tend to have higher franchise values and larger cash flows, meaning that a large loss is less catastrophic and the need for insurance is lower. Therefore, to some extent, they can diversify the risk of these large contracts due to having more of them, and can afford to assume more risk, thus reducing the need for costly insurance. Essentially, this would allow larger payroll teams to offer higher salaries and guaranteed dollars for premium players than smaller market teams. If anything, I believe this would be more likely to promote a non-linear valuation. In any event, the productivity effect and the insurance effect are separate effects, and should ideally be analyzed separately. Our tools for doing to so are very limited, since we don't exactly how teams value risk or about the insurance contracts they sign. I think it is a mistake to assume that just because we observe a linear $/WAR relationship, and we have two factors which may work in opposite directions (and I'm not entirely sure that this is the case), that they are netting off against each other to result in a linear $/WAR model. We really don't know enough to say that this is causation, rather than correlation.

Why Roster and Playing Time Constraints Don't Prevent the Substitution Effect

Finally, we come to the theory that represents the crux of the argument jessef put forth, which is basically that due to roster and playing time constraints, one player who produces a given amount of WAR is necessarily more valuable than two players who together produce that same amount of WAR:

[N]eeding two players means that a team uses playing time that would otherwise be going to a player who is presumably above replacement level....The underlying guiding principle here is so simple that it's beautiful: WAR production becomes increasingly valuable (due to the fact that good teams don't field replacement-level players). Thus, teams should be willing to pay more for players projected to produce more (after adjustments for injury risk, etc. are made).

The important thing to notice here is the fundamental assumption underlying this view - the idea that the second player displaces another who is above replacement, and therefore the net addition is smaller than adding the single player. Cameron frames from a slightly different angle in his article:

[G]etting consolidated value creates a higher potential maximum than if everyone on the team was of equal value. If you have a +5 win player and a +0 win player, you can theoretically replace the +0 win guy and end up in a better position than if you have two +2.5 win players who you don’t really want to replace.

Basically, this is an argument about the fungibility of players, that once again is intuitively logical on its face. If you have two 2.5 WAR players and want to upgrade by 2.5 WAR within those two, you need to get a 5 WAR player, which is not easy to come by. If you have a 5 WAR player and a 0 WAR player, and want the same 2.5 WAR upgrade, finding a 2.5 WAR player is much easier because there are more of them.

My problem with this assumption is that it underestimates the degree to which teams can shift players around positionally - essentially the chaining effect. If you sign a 3 WAR CF and you already have a 2.5 WAR CF, it's not just a 0.5 WAR upgrade, because that CF could replace a corner outfielder who might a 1 WAR player, which means a 1.5 WAR upgrade there to add to the initial 0.5 WAR. And the the replaced corner outfielder might become the 4th outfielder (reduced playing time, so say 0.5 WAR) who replaces a replacement level reserve, and then we're up to a 2.5 WAR upgrade. A similar argument can be made with pitchers . There's a little bit of leakage in these scenarios, but for the most part, the majority of the value acquired is preserved and not made redundant.

Moreover, teams can also address needs by trading amongst themselves, whereby teams with an excess of above replacement level players can package them for a different combination of value. By definition, a player above replacement level has value, and even if a team lacks replacement level projected players, they can trade upgrades to other teams for whom that player represents an upgrade. Therefore, the chaining effect occurs not only on an intra-team basis, but also on a inter-team basis. For example, if you have two 2.5 WAR players who only play 1B and DH, and you sign a 3.5 WAR player who can also do the same, it's a 1 WAR upgrade. But you can take the replaced player, find a team with a need, and perhaps trade for a league average SS where you might have a replacement player (or any other position). This all serves to keep a general equilibrium within different segments of the market, and to keep the relationship between WAR and dollars linear. The prevalence of replacement level players across MLB essentially guarantees this to be the case.

As a thought experiment, consider a team that projected to have league average players at every position, 25 men deep. They're a .500 team, and seemingly the only way to move the needle through free agency is to sign premium player - a 3 WAR player doesn't really help. But that team can take a couple of those league average regulars and flip them to another team with maybe a 4.5 WAR player and a 0 WAR player. The league average team gets their upgrade at one position, and the other teams gets slightly more total production. Then the league average team signs the 3 WAR player (rather than potentially "overbidding" and paying in a non-linear manner for the 5 WAR player. They still end up with a net 2.5 WAR upgrade. Once again, we do see some leakage in value, but it's fairly small, such that the linearity is essentially preserved.

As for the argument that good teams have very few replacement level players who can easily be replaced by better players, I went through the 6 teams who spent the most on players who had six or more years service time (who were all good teams, partially because they spent a lot to try and buy WAR in the market). It's pretty clear there's a lot of replacement type production. Not all of this is necessarily foreseeable, so it represents a "negative surprise" and means that you can't replace these guy ahead of time; at the same time, having that depth of players above replacement level is the best guard against this.

Essentially, what this argument boils down to is that the roster and playing time constraints are not binding. Theoretically they exist, but in practice, due to the scarcity of greater than replacement level MLB talent and injuries, teams end up filling rosters and playing time with easily upgradable talent.

Some Anecdotal Evidence

At the beginning of the Winter Meetings, Brian Cashman - GM of the Yankees - had a problem. His 2012 rotation options under contract were as follows (2012 ZiPS projections):

1) CC Sabathia - 218 IP, 3.55 ERA, 126 ERA+ (96 is average for starters),

2) Ivan Nova - 178.1 IP, 4.44 ERA, 100 ERA+

3) Phil Hughes - 112.2 IP, 4.84 ERA, 92 ERA+

4) Freddy Garcia - 128 IP, 4.85 ERA, 92 ERA+

5) Hector Noesi - 103 IP, 5.24 ERA, 85 ERA+

6) AJ Burnett - 159.1 IP, 5.34 ERA, 84 ERA+ (much lower peripherals, the ERA-FIP split is almost a run)

For the team with the largest payroll and the consequent playoff expectations, this was a rotation that screamed out for improvement. Though anchored by a veritable ace, beyond that were were essentially 3 pitchers who projected to be league average, another whose peripherals projected at league average, and a reserve who projected at replacement level in the event of injuries.

In other words, this would seem to the prototypical situation where a big market team has a plethora of average type options but, due to the playing time constraint, the only way to significantly improve is to get a well-above average player to displace Burnett. The best options available in the free agent market were C.J. Wilson (120 ERA+), Yu Darvish (125 ERA+) and Mark Buerhle (109 ERA+). The first two in particular projected to offer the Yankees an upgrade of 3-4 WAR (depending if you use ERA or FIP for Burnett), at a cost ranging from 15-18 million per year over 4 or 5 years.

But of course, we know the Yankees did neither. Instead, in a late January strike, they traded for Michael Pineda (157.2 IP, 106 ERA+) and signed Hiroki Kuroda (156 IP, 101 ERA+) for $10M, which displaced both Burnett (traded) and Garcia (who seems likely to move to the bullpen and give rotation depth). Some of this may have represented (self-imposed) fiscal pressures - wariness to commit to a 5 year contract for a pitcher, and the ability to acquire a cost controlled pitcher - but that's a question of how many WAR a team buys, not how much they pay. Interestingly, there was a pitcher similar to Pineda on the free agent market in Edwin Jackson (197.2 IP, 105 ERA+). Let's compare the following rotations:

Original: Sabathia/Nova/Hughes/Garcia/Burnett - 796.1 IP, 101.5 ERA+ (weighted by innings)

Hypothetical one big improvement: Sabathia/Wilson/Nova/Hughes/Garcia - 837.1 IP, 109.2 ERA+

Actual: Sabathia/Pineda/Kuroda/Nova/Hughes - 822.67 IP, 107.1 ERA+

Modified Actual: Sabathia/Jackson/Kuroda/Nova/Hughes - 862.67 IP, 106.9 ERA+

We can see that all three scenarios upgrade the rotation to well above average, and result in broadly similar quality rotations. This doesn't take into account the depth beyond the 5th starter - with one upgrade in Wilson, Freddy Garcia remains in the rotation, whereas in the latter two scenarios he's in the bullpen and available as an injury replacement. The difference narrows further of the 6th best starting pitcher projections are added in.

This is the essence of substitution at work - two smaller upgrades on the order of 1-2 WAR rather than one 3-4 WAR upgrade at a similar price. Granted, this is only one example, but I believe it's quite telling that it represents the biggest spending team - the one that, in theory, would have the least likelihood of being able to do this type of upgrade.

Let's also look briefly at what another team did - the St. Louis Cardinals. They could have decided to retain Pujols, a player who projects at something like 6-7 WAR, for 10 years and $250M. Instead, they let him walk, moved Berkman to first and backfilled that gap with Carlos Beltran at half the AAV and one fifth the years. That left plenty of money to re-sign Yadier Molina, and extend Carpenter for two years (replacing a 2012 club option). Would the money had been there to keep Carpenter and extend Molina had Pujols been on the books? Maybe - and I stress maybe - one, but almost certainly not both.

This is a more complicated example, since it spans multiple years and so we can't easily break out the difference in production and money, but it's basically the same principle at play. They could have kept one phenomenal player, instead they keep a number of players who are probably each a couple WAR over what could easily be found on the open market. It's a example of the general point that, within short and longer term budgetary constraints, teams have ample choices regarding how to retain and add production and I just don't really see a binding playing time constraint. This ability to choose among different tiers of production keeps the relative costs in balance - much as substitution works in the broader economy to keep the prices of substitutable goods appropriate in relation to each other.

A Further Reason for Linear $/WAR:

As I stated above, I think the actual evidence of what teams do suggests they do pay in linear manner for WAR. More importantly, I have never seen any evidence that suggests it's not linear. If we assume that teams should pay in a non-linear manner, but that they are not paying in a linear manner, it implies means that the market is collectively inefficient, and this represents an opportunity to exploit. This is counter to the general economic theory that there are not generally free lunches in markets, and when they do exist and actors become aware of them, they will rapidly disappear. As Dave Cameron wrote in the article referenced above:

You can make a case that teams are currently being too risk-averse, and that this is an inefficiency that could be exploited, but the people currently in charge would probably argue that they have a significantly better handle on the actual risks of having all of your eggs in one basket than us outsiders do. The fact that not even the well capitalized and extremely well run franchises of the northeast have begun to pay those kinds of premiums should suggest to us that there are legitimate reasons why the market isn’t supporting the theory, and that the focus on consolidated value is missing key elements that affect actual roster construction

To be clear, Cameron is talking about a different theory than I am advocating to ultimately explain why WAR should be linear, but the same idea hold generally. Teams have highly specialized, highly skilled management teams with huge incentives to seek out advantages, And it's not like this debate is secret - anyone with access to Fangraphs is aware of it. So, while I am not a adherent to the idea of perfectly efficient markets, I do however believe that markets are generally efficient enough that we assume there should not be large, persistent inefficiencies that are sitting there to be plucked - low hanging fruit essentially.

Of course, this presumption deserves some scrutiny, given that inefficiencies in valuing baseball production have previously (and recently) existed - this is the premise of Moneyball. Even once the methods to properly value skills like on base and power became available, there was widespread resistance to them, which still exists to some extent. So could it still be happening, just in a different way today? It's possible, but I would argue that the changing of the guard in the past 10 years means it's very unlikely. Part of the reason those inefficiencies existed was due to the insular nature of baseball management - most of the management ranks came from inside the game, steeped in the traditional ways of evaluating and valuing talent and production and generally without formal management education. Also, until the collapse of the reserve system 35 years ago, baseball was not big business and the stakes were simply not very large. Neither of those are true today. Baseball is big business, with guaranteed dollars running into the hundreds of million on individual contracts. Moreover, the management ranks have been shaken up by outsiders with backgrounds in which searching for inefficiencies is the name of the game. This was most famously chronicled in The Extra 2% about Andrew Friedman and his Wall Street background, but similarly applies to Theo Epstein, Jon Daniels, Alex Anthopolous, Jeff Luhlow, Ben Cherington, and others. I find it much harder to believe that there is a massive, low hanging fruit sitting in front of these guys.


I believe there is a strong theoretical basis for substitution in the market for baseball players resulting in a linear relationship between dollars and WAR, essentially due to the lack of a binding playing time constraint. Some strong anecdotal evidence supports some of the key assumptions behind the theory. Finally, given a body of evidence suggesting that teams actually have been paying linearity, a belief in a non-linear relationship requires the existence of a massive market inefficiency and systemic irrationality by current team managements. Notwithstanding the above, I think it's important to once again stress that much of what is involved here is just theory that rests on significant assumptions. I think they're reasonable, but I remain open to new information and evidence to the contrary which could cause a revision on this analysis. Maybe that sounds like a huge cop-out, but for me that's just a reflection of the inherent uncertainty in the world. I think the most foolhardy thing of all is to assume our own infallibility and stick to dogmas in the face of new information.

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