# A sad story

**It is funny but mostly sad.**

In 1991, in a paper entitled** Large Games and Competitive Markets 2: Applications**, I introduced a section entitled "*Applications to Hedonic Pricing and Monotonicity*". (See here for a scanned copy.) Much of that section is devoted to a discussion of the 1991 Engl-Scotchmer paper (with citation of course) and the differences between what they do and my own research. Indeed, I write that

- "Engl-Scotchmer state their main result in an appealing way." [Wooders (1991, revised 1992)], page 78

(I was trying to do the right thing and, in spite of Scotchmer's conduct, to be gracious). I also write, referring to E&S as "The authors":

- "The authors perhaps anticipate results similar to mine when they state that they see no reason why their results could not be extended to the case where V might not be homogeneous, since the limiting per capita payoff function is concave. (This limiting concavity, as noted by Aumann (1987) with regard to Wooders (1983), is a consequence of the per-capita boundedness and superadditivity conditions of that paper and underlies much of the work of this paper and "Games=Markets".) They suggest that in the absence of homogeneity a limiting homogeneous and concave function could be defined which approximates the limiting behaviour of large games. A limiting function having the required properties is the utility function defined in Section 4 of "Games=Markets" and in Wooders (1988b) for the case with for the case with a compact metric space of attributes, or, taking attributes as the arguments, the utility function U defined in this section." [Wooders (1991, revised 1992), pages 78-79].

In 1991 E&S used homogeneity of the payoff function. As I suggested would be possible, in the published version of their paper they use the same conditions as in my earlier work, *superadditivity and per capita boundedness*.

E&S (1996) rule out boundaries. When this is done, then per capita boundedness is equivalent to small group effectiveness (that all of almost all gains to collective activities can be realized by groups of players bounded in size (Theorem 4 of Games=Markets, *Econometrica* 1994). Note also that the utility function mentioned above, which Scotchmer claims is hers and Engl's (which means, I suppose, that I "misappropriated" it), is also in Winter-Wooders (1990) and in the 1988 CORE DP version of my 1994 *Econometrica* Games =Markets paper,.

My results, dating back to Shubik and Wooders (1982) and earlier papers (including Wooders 1983, published in *Journal of Mathematical Economics*), use only per capita boundedness (with superadditivity of course) of the characteristic function of a game and in fact, given the similarity of the E&S model to that of my earlier work and the fact that their model is essentially a TU private goods economy, it was clear that they did not require homogeneity of the payoff function. There is no reason why E&S would need more restrictive assumptions than my (prior) assumptions.

In next versions of their paper, the authors did indeed drop the homogeneity assumption on V and but replaced it with a uniform convergence condition. In his letter to Shafer, after noting their concern about my related papers referencing theirs (As he wrote, I cited them!) Engl notes that they can drop uniform convergence. As their published paper shows, they indeed dropped this condition and used the conditions of my earlier work.

In her response to my comments on their paper, Scotchmer writes that, after hearing their presentation, I introduced a section into my paper entitled *Applications to Hedonic Pricing and Monotonicity*. She is clearly right on that. I introduced the section -- the reader can see for himself by clicking on the link to the paper. But from reading Scotchmer's comments, one might think that I took their concept for my own (and she does not cite the published Bonn Working Paper so that the reader could see for himself). In fact, I introduced the section to discuss their work and compare it to mine, even referring to their "appealing" result.

For the reader unfamiliar with the hedonic pricing literature, hedonic pricing (as pricing of unobservable characteristics, as the term is usually used) has a long history and is quite distinct from the so-called hedonic pricing in E&S where all characteristics of relevance are observable.