### Credit scores and information economics

The recent Equifax data breach brings up credit scores and their role in the information economy. Credit scoring is a controversial topic in the algorithmic accountability community. Frank Pasquale, for example, writes about it in The Black Box Society. Most of the critical writing on the subject points to how credit scoring might be done in a discriminatory or privacy-invasive way. As interesting as those critiques are from a political and ethical perspective, it’s worth reviewing what credit scores are for in the first place.

Let’s model this as we have done in other cases of information flow economics.

There’s a variable of interest, the likelihood that a potential borrower will not default on a loan, $X$. Note that any value sampled from this $x$ will vary within the interval $[0,1]$ because it is a value of probability.

There’s a decision to be made by a bank: whether or not to provide a random borrower a loan.

To keep things very simple, let’s suppose that the bank gets a payoff of $1$ if the borrower is given a loan and does not default and gets a payoff of $-1$ if the borrower gets the loan and defaults. The borrower gets a payoff of $1$ if he gets the loan and $0$ otherwise. The bank’s strategy is to avoid giving loans that lead to negative expected payoff. (This is a gross oversimplification of, but is essentially consistent with, the model of credit used by Blöchlinger and Leippold (2006).

Given a particular $x$, the expected utility of the bank is:

$x (1) + (1 - x) (-1) = 2x - 1$

Given the domain of $[0,1]$, this function ranges from -1 to 1, hitting 0 when $x = .5$.

We can now consider welfare outcomes under conditions of now information flow, total information flow, and partial information flow.

Suppose the bank has no insight into $x$ besides a prior expectation $X$. Then the expected value of the bank upon offering the loan is $E[2x+1]$. If it is above zero, the bank will offer the loan and the borrower gets a positive payoff. If it is below zero, the bank will not offer the loan and both the bank and potential borrower will get zero payoff. The outcome depends entirely on the prior probability of loan default and is either rewards borrowers or not depending on that distribution.

If the bank has total insight into $x$, then the outcomes are different. The bank can use the option to reject borrowers for whom x is less than .5, and accept those for whom x is greater than .5. If we see the game as repeated over many borrowers whose chances of paying off their loan are all sampled from $X$. Then the additional knowledge of the bank creates two classes of potential borrowers, one that gets loans and one that does not. This increases inequality among borrowers.

It also increases the utility of the bank. This is perhaps best illustrated with a simple example. Suppose the distribution $X$ is uniform over the unit interval $[0,1]$. Then the expected value of the bank’s payoff under complete information is

$\int_{.5}^{1} 2x - 1 dx = 0.25$

which is a significant improvement over the expected payoff of 0 in the uninformed case.

Putting off an analysis of the partial information case for now, suffice it to say that we expect partial information (such as a credit score) to lead to an intermediate result, improving bank profits and differentiating borrowers with respect to the bank’s choice to loan.

What is perhaps most interesting about this analysis is the similarity between it and Posner’s employment market. In both cases, the subject of the variable of interest $X$ is a person’s prospects for improving the welfare of the principle decision-maker upon their being selected, where selection also implies benefit to the subject. Uncertainty about the prospects leads to equal treatment of prospective persons and reduced benefit to the principle. More information leads to differentiated impact to the prospects and benefit to the principle.

References

Blöchlinger, A., & Leippold, M. (2006). Economic benefit of powerful credit scoring. Journal of Banking & Finance, 30(3), 851-873.