Sunpar, thanks for your clarifying question. Although Bonifacio probably is replacement level, that wasn't quite what I meant. I actually meant literally players being replaced. In Friday's article, you might see what I mean a little clearer, but some examples would be that the Phillies had Greg Dobbs as the next-best option to making a move at 3B by staying in the organization but chose to replace him with Placido Polanco, the Red Sox had Jed Lowrie in the organization as the next-best option to signing Marco Scutaro at SS who they chose to replace him with instead, the Rays had Dioner Navarro as the next-best option to trading for Kelly Shoppach at C, etc. Friday's article will definitely show you more about what I meant. Thanks for clarifying.

Matt, I liked that piece a lot. I would suggest that if you "are not estimating what teams should pay for wins but what they do pay for wins" that you should create MORP2 that calculates what teams should pay.

I'm a bit confused by the end of this, but it sounds like you're checking how many teams have replacement-level players as defined by the average production of players that are replaced. Won't that number be very close to the number of players that are replaced, i.e., the number of free agent signings each year?

Regardless, it's hard to see exactly what it would prove if each team had 3 or 4 replacement-level starters heading into the offseason. This still means that a team cannot sign 6 low-WARP players for the same production as 3 better players; there is still a scarcity for higher quality players and they will demand a higher price. The only question is HOW non-linear this relationship is. Your graphs seem to imply that the answer is NOT VERY MUCH (presumably because the scarcity seems to provide extra contract security instead of a salary premium).

Still, i think we can look at the theoretical value of this scarcity a little more directly, in contrast to: "We are stuck with the reality that we cannot determine how many extra fans are put in the seats by adding a player with a certain win value, and we are therefore stuck with using the inferred price of free agents that teams have shown us."

In some ways, the increased value of better players can be measured by how far along the win curve their teams are, on average. That is, while marginal revenue is fairly linear in wins for the average team, valuable players make it easier to be further up on the curve, so that marginal wins are higher. By averaging the marginal revenue 4 win players provide (as measured by finding the difference between expected revenue at the actual number of wins - expected revenue for wins less the player's contribution), you'll get a better idea of what this advantage is (which should be greater than 4* the average marginal revenue that 1 win players provide; there's also one extra complication that needs to be taken into account-- the incentive to spend more money on your team when marginal revenue is greater --but this could be corrected for by using salary as a percentage of team payroll and then translating to an average payroll or some other fix). This would leave a similar scatter-plot of MRs and salaries which you could check for linearity.

Hope that was somewhat clear / made sense to anyone but me. I just wouldn't give up on attempting to use MR just because we don't have direct measurements.

@archilochusColubris: The number of replacement level players will outstrip the number of free agents signed, you'll see on Friday. A lot of teams don't sign or trade to improve on replacement level production-- and end up getting it as a result. Think about how many teams have 5th starters with ERA's well into the 5s. The idea is that as we get a sense of who is replaced, we get a sense of what replacement level is.

The example you gave of 6 low-WARP players vs. 3 high-WARP players actually will illustrate the point of exactly how low and high WARP of free agents are. There are only a small handful of free agents that will get you more than 4.0 WARP reliably, while you can certainly see an abundance of guys in the 1.5-2.0 range. I guess it's possible that a team could try to sign 3 guys with 4.0+ WARP, but that's really rare and not common enough to affect the market price, mainly because you'd really need two bidders to push up the price. Maybe the Yankees signed Sabathia, Pettitte, Burnett, and Teixeira last year, but that's a rare year even for them, and they probably not have people competing with them to sign several of them rather than individual teams competing to sign one or two of them (e.g. the Red Sox were not probably also trying to sign three of these guys, probably just Teix and a pitcher at most).

The scarcity is reflected in the already higher price-- it's just that the scarcity comes in linearly.

The important distinction to be made with how far along their teams are along the win curve, on average, is an important one. The average team does not sign free agents-- the team that values the player the most does. Thus, teams that sign free agents are the ones that are likely to be the most competitive. For them, the marginal value of a win is already high since most of them are already in the 85-95 win range.

For your MR measurement, the problem is that we do not have counterfactuals. In other words, even ideally we could only know what the Phillies' revenue was last year given that they won 93 games-- we do not know what their revenue would have been had they not signed Ibanez and won, say, 89 games. Any attempt to solve this empirically would be muddled by the fact that the Phillies clearly thought 89-73 was unpalatable compared to 93-69, so even if we had the revenue of other teams who went 89-73 or even the Phillies in previous years where they won 89 games, those numbers likely understate the difference between 89-73 and 93-69 because in instances where the team won 89 games, they likely didn't have as huge a gap (and didn't want to spend the money), and in instances where a team won 93 games, they likely had a larger gap.

In reality, assuming the very rich individuals and companies who own baseball teams have at least a decent sense of their marginal revenue, we can use the Axiom of Revealed Preference to impute a decent approximation of marginal revenues.

Let me know if I left anything out, and thanks for giving such an incredible amount of thought to this. It's a very challenging job to redo MORP, and I encourage thoughtful comments like these to help guide me along and challenge my assumptions (and change them where necessary!).


@JDSussman: I would say that this is estimating what teams should pay for wins, conditional on the fact that the market is producing the correct price on average. For instance, if Omar Minaya overestimates the win-value of Jason Bay and overpays but Jack Zduriencik sneaks in at gets Chone Figgins at low $/WARP, that would not be an issue in the model as they would cancel out. Where the issue would lie is if winning actually doesn't add much to revenue but Steinbrenner just lets Cashman spend to a certain budget because he can afford it-- rather than because he can make money allowing him to do so competently-- then MORP won't know that this is happening. It assumes that, on average, teams are paying for wins at their marginal benefit.

Matt: interesting article, but I have a question.

If the effect of your predictor variable (say X) is specificied as a quadratic function (b0 + b1(X1) + b2(X^2), you do allow for the possible curvilinearity (parabolic) but without necessarily imposing empirically.

This is true because if in fact the empirical relationship turns out to be strictly linear, then the coefficient b2 would be small and not statistically significant. In that case the linear coefficient (b1) would capture the effect of X on salary.

I agree with Nate that it may be preferable to use the quadratic function as a theoretically more compelling function in this case that would not in fact cause any serious distortion in your MORP estimates if it turns out empirically to be a linear relationship. The cost in degrees in freedom is also negligible.

My analogy is to the price that I paid per meal as a function of the number of stars that a restaurant gets in the Michelin guide. My experience in that case told me that while I might pay $50 for a meal in a 1-star restaurant, and $100 in a 2-star restaurant, it would cost me $200 in a three-star restaurant. The marginal pric of the additional star was a nonlinear function. (This based on a "45 days in Europe" tour some years ago. Your costs may be different!).

Matt, I think you're giving some valuable information here. Even though I have never attended an economics class, I still think I get it.
I may have missed it somewhere in your explanation, but to be certain - I take it you assume that when a team makes a decision to sign a free agent, it is not just in the need to displace a replacement level player, but more specifically to replace a replacement level player at the position of the new signee.

Something that I think Silver was hitting on and is not being captured in the analysis is the economic idea of the subjective theory of value. What is observable on the market is not what people should do but what they actually do. Perhaps there exists an objective way to measure how to achieve a certain number of wins or a certain probability of reaching the playoffs or a certain probability of winning the World series, but there exists NO way to scientifically measure the difference in importance between achieving any of those ends. The only way to perceive how owners and GMs behave in response to those different subjective ends is to allow for other functional forms and accept whichever functional form best fits the data. To do otherwise is to assume some objective trade-off between the desirability of winning a game (as in going from 70 to 71 wins) against getting into the playoffs (the increase in probability by winning 88 and 89 wins). These trade-offs are in every sense implicit in valuing a star player's wins differently than a marginal player's.

I don't think this analysis is economic given the subjective theory of value, which very much is econ 101.