### Formalizing Posner’s economics of privacy argument

I’d like to take a more formal look at Posner’s economics of privacy argument, in light of other principles in economics of information, such as those in Shapiro and Varian’s Information Rules.

By “formal”, what I mean is that I want to look at the mathematical form of the argument. This is intended to strip out some of the semantics of the problem, which in the case of economics of privacy can lead to a lot of distracting anxieties, often for legitimate ethical reasons. However, there are logical realities that one must face despite the ethical conundrums they cause. Indeed, if there weren’t logical constraints on what is possible, then ethics would be unnecessary. So, let’s approach the blackboard, shall we?

In our interpretation of Posner’s argument, there are a number of applicants for a job, $i \in I$, where the number of candidates is $n = \left\vert{I}\right\vert$. Let’s say each is capable of performing at a certain level based on their background and aptitude, $x_i$. Their aptitude is sampled from an underlying probability distribution $x_i \sim X$.

There is an employer who must select an applicant for the job. Let’s assume that their capacity to pay for the job is fixed, for simplicity, and that all applicants are willing to accept the wage. The employer must pick an applicant $i$ and gets utility $x_i$ for their choice. Given no information on which to base her choice, she chooses a candidate randomly, which is equivalent to sampling once from $X$. Her expected value, given no other information on which to make the choice, is $E[X]$. The expected welfare of each applicant is their utility from getting the job (let’s say it’s $1$ for simplicity) times their probability of being picked, which comes to $\frac{1}{n}$.

Now suppose the other extreme: the employer has perfect knowledge of the abilities of the applicants. Since she is able to pick the best candidate, her utility is $\max x_i$. Let $\hat i = arg\max_{i \in I} x_i$. Then the utility for applicant $\hat i$ is $1$, and it is $0$ for the other applicants.

Some things are worth noting about this outcome. There is more inequality. All expected utility from the less qualified applicants has moved to the most qualified applicant. There is also an expected surplus of $(\max x_i) - E[X]$ that accrues to the totally informed employer. One wonders if a “safety net” were to be provided those who have lost out in this change; if it could be, it would presumably be funded from this surplus. If the surplus were entirely taxed and redistributed among the applicants who did not get the job, it would provide each rejected applicant with $\frac{(\max x_i) - E[X]}{n-1}$ utility. Adding a little complexity to the model we could be more precise by computing the wage paid to the worker and identify whether redistribution could potentially recover the losses of the weaker applicants.

What about intermediary conditions? These get more analytically complex. Suppose that each applicant $i$ produces an application $y_i$ which is reflective of their abilities. When the employer makes her decision, her expectation of the performance of each applicant is $P(x_i \vert y_i) \propto P(y_i \vert x_i)P(x_i)$

because naturally the employer is a Bayesian reasoner. She makes her decision by maximizing her expected gain, based on this evidence: $arg\max E[P(x_i \vert y_i)] =$ $arg\max \sum_{x_i} x_i p(x_i \vert y_i) =$ $arg\max \sum_{x_i} x_i p(y_i \vert x_i) p(x_i)$

The particulars of the distributions $X$ and $Y$ and especially $P(Y \vert X)$ matter a great deal to the outcome. But from the expanded form of the equation we can see that the more revealing $y_i$ is about $x_i$< the more the likelihood term $p(y_i \vert x_i)$ will overcome the prior expectations. It would be nice to be able to capture the impact of this additional information in a general way. One would think that providing limited information about applicants to the employer would result in an intermediate outcome. Under reasonable assumptions, more qualified applicants would be more likely to be hired and the employer would accrue more value from the work.

What this goes to show is how ones evaluation of Posner's argument about the economics of privacy really has little to do with the way one feels about privacy and much more to do with how one feels about the equality and economic surplus. I've heard that a similar result has been discovered by Solon Barocas, though I'm not sure where in his large body of work to find it.