Learning personalized prices from imperfect evidence

Why transaction records can mislead

A lender looking through past applications sees the price each borrower received and whether they accepted. What it cannot see is how that same borrower would have responded to a different price. Historical comparisons become especially misleading when the original pricing decision used information missing from the analyst’s records.

If a seller charges more when demand is strong, expensive offers may look unusually successful. A model trained on those associations can mistake favorable circumstances for an effect of price itself. Economists call this problem endogeneity: price is entangled with unobserved influences on revenue.

Rui Miao, Zhengling Qi, Cong Shi and Lin Lin propose PRINT—Personalized pRicing using Invalid iNsTrumental variables—to learn pricing rules in a particular class of these situations.

An imperfect lever can still contain information

An instrumental variable is a variable that shifts the price and helps researchers disentangle its effects. A conventional valid instrument must be independent of hidden confounders, after accounting for recorded characteristics, and affect revenue only through price. That last requirement is the exclusion restriction.

But natural candidates in pricing can violate it. An annual percentage rate, or APR, is closely connected to a loan’s constructed price and may also affect acceptance and revenue directly. PRINT permits certain direct effects rather than assuming them away.

This freedom comes with replacement assumptions. The instrument must still be independent of unobserved factors after conditioning on recorded characteristics. More importantly, hidden factors determining how revenue responds to price must be separated, in specified statistical ways, from the historical pricing rule’s conditional price moments.

The intended setting is partial-information pricing: historical decisions may reflect hidden operational conditions without using the same hidden customer-demand information that determines marginal price response. The method does not cover fully informed pricing in which the seller uses that same latent demand information to choose prices.

The lever must also supply enough distinct variation to separate the response coefficients. Merely shifting average prices is insufficient. PRINT exploits changes in higher price moments, including dispersion, and requires the resulting identifying equations to have full rank. Even changing dispersion alone does not guarantee this.

Finding the peak of a revenue curve

The structural model describes expected revenue as a quadratic function of price, with coefficients that can vary across customer profiles and a negative quadratic coefficient. This produces a curved revenue relationship with a well-defined maximum over an allowed price interval. The authors treat this shape as an approximation, not a universal demand law.

PRINT estimates the two functions needed to locate that maximum. It first fits auxiliary regressions using cross-fitting: each observation receives predictions from models trained on other folds. It then uses an adversarial min–max procedure. One component estimates the unknown functions; another searches for violations of the statistical moment restrictions they should satisfy.

The recommended price is the estimated quadratic peak, restricted to the feasible interval. Although the paper develops identification with a structured price–instrument interaction, its subsequent estimator and regret theory focus on the simpler case without that interaction.

The theoretical results bound regret—the expected revenue gap between the learned policy and the best feasible policy under the model. These are conditional guarantees. Their strength depends on identification, estimation and approximation errors, and optimization accuracy. The practical neural-network algorithm is explicitly described as a heuristic, rather than automatically covered by every theoretical condition.

What the numerical studies show

In simulations deliberately constructed to satisfy the identifying restrictions, PRINT’s regret decreases as sample size grows. Two comparison methods that do not correct for the constructed hidden confounding show comparatively flat regret.

The application uses online auto-loan records from July 2002 to November 2004. The researchers deliberately withhold an available competitor-rate variable as an omitted-variable stress test and use APR as the candidate instrument. Policies train on the first 60,000 time-ordered records and are scored on the remaining 148,085.

PRINT achieves about 80.6% of a fitted-model benchmark’s expected revenue and scores 7.2% above the historical policy. Those figures are not measured gains from deploying PRINT. Unobserved outcomes at alternative prices are generated by a fitted logistic acceptance model, which was fitted using all records. The evaluation therefore is not a wholly independent test of that outcome model.

The logistic evaluator also departs from PRINT’s quadratic revenue assumption. The authors acknowledge possible misspecification in the real-data application. The result illustrates performance under one model-based evaluation; it does not establish that APR satisfies the identifying assumptions or that a lender would realize the reported gain.

PRINT’s contribution is consequently narrower than making any imperfect instrument usable: it specifies conditions under which particular imperfect pricing levers can support learning, and makes clear why those conditions matter.