Elite Evaluator Rents

The elite evaluator story discussed in my last post is this: evaluators vary in the perceived average quality of the applicants they endorse. So applicants seek the highest ranked evaluator willing to endorse them. To keep their reputation, evaluators can’t consistently lie about the quality of those they evaluate. But evaluators can charge a price for their evaluations, and higher ranked evaluators can charge more. So evaluators who, for whatever reason, end up with a better pool of applicants can sustain that advantage and extract continued rents from it.

This is a concrete plausible story to explain the continued advantage of top schools, journals, and venture capitalists. On reflection, it is also a nice concrete story to help explain who resists prediction markets and why.

For example, within each organization, some “elites” are more respected and sought after as endorsers of organization projects. The better projects look first to get endorsement of elites, allowing those elites to sustain a consistently higher quality of projects that they endorse. And to extract higher rents from those who apply to them. If such an organization were instead to use prediction markets to rate projects, elite evaluators would lose such rents. So such elites naturally oppose prediction markets.

For a more concrete example, consider that in 2010 the movie industry successfully lobbied the US congress to outlaw the Hollywood Stock Exchange, a real money market just then approved by the CFTC for predicting movie success, and about to go live. Hollywood is dominated by a few big studios. People with movie ideas go to these studios first with proposals, to gain a big studio endorsement, to be seen as higher quality. So top studios can skim the best ideas, and leave the rest to marginal studios. If people were instead to look to prediction markets to estimate movie quality, the value of a big studio endorsement would fall, as would the rents that big studios can extract for their endorsements. So studios have a reason to oppose prediction markets.

While I find this story as stated pretty persuasive, most economists won’t take it seriously until there is a precise formal model to illustrate it. So without further ado, let me present such a model. Math follows.

Let a unit quantity of applicants have quality x uniformly distributed over the range x in [0,1]. An evaluator i claims that its endorsed applicants have a quality of at least xi, and later suffers prohibitive penalties if such claims are ever found to be wrong. Thus an evaluator who chooses limit xi can actually only endorse applicants for whom x ≥ xi. There are N evaluators i in [1,N] who are endowed with different prior reputations that restrict their choice of limit xi. Evaluator i must choose xi, in [0,2i-N) because observers just won’t believe that they could attract applicants of quality x ≥ 2i-N.

An evaluator who charges price p to accurately endorse the set of applicants in the range [a,b] gains profit p*(a-b); evaluators have no other costs or revenue. Applicants who pay price p to to be endorsed as having quality x ≥ a gain net value V = a – p because of how they are treated by later observers. This value is not larger due to adverse selection in later observer process.

The order of play is as follows. First, evaluators choose sequentially in order of increasing index i. Each i chooses both price pi and quality limit xi simultaneously. After evaluators have chosen, then applicants, knowing all the pi  and xi and their own quality x, simultaneously each choose an evaluator. Finally evaluators choose whether or not to endorse each of their applicants. (We get the same results if applicants don’t know their x, and can repeatedly apply to evaluators until one endorses them.) Let i=0 correspond to paying nothing and getting no endorsement, with x0 = p0 = 0.

A simple (and maybe unique) equilibrium of this game is: each evaluator i chooses  pi = xi =2i-N-1, each applicant applies to the highest i such that their x ≥ xi , and then all applicants are accepted. (Applicants with x< x1 “apply” for no endorsement and get it.) All applicants get exactly zero net value, and evaluator i endorses 2i-N-1 applicants, gaining profit 22(i-N-1).

Note that higher ranked evaluators endorse more applicants, and gain more profits. “Big” goes with “high.” And evaluators take all the gains in this world; applicants get nothing.

Proof: For xi,pi to beat offer xi-1,pi-1, need max pi given xi to satisfy xi – pi ≥ xi-1 – pi-1, gives xi – pi = xi-1 – pi-1 and pi = pi-1 + xi – xi-1. Assume xi = ci+(1-ci)(xi-1 – pi-1). Gives the correct xN+1 = 1 with cN+1 = 1, and substituting these into profit πi =pi(xi+1 – xi) gives πi =(xi + pi-1  – xi-1)(ci+1+(1-ci+1)(xi-1 – pi-1) – xi). Maximizing πi with respect to xi gives first order condition xi = ci+1/2 +(1-ci+1/2)(xi-1 – pi-1), which confirms assumption with ci = ci+1/2. Combined with cN+1 = 1 and x0 – p0 = 0 gives xi = pi = ci = 2i-N-1.