Threshold Tests as Quality Signals: Optimal Strategies, Equilibria, and Price of Anarchy

We study a signaling game between two firms competing to have their product chosen by a principal. The products have qualities drawn i.i.d. from a common prior. The principal aims to choose the better product, but the quality of a product can only be estimated via a coarse-grained threshold test: for chosen $\theta$, the principal learns whether a product's quality exceeds $\theta$ or not. We study this problem under two types of interactions. In the first, the principal does the testing herself, and can choose tests from a class of allowable tests. We show that the optimum strategy for the principal is to administer different tests to the two products: one which is passed with probability $\frac{1}{3}$ and the other with probability $\frac{2}{3}$. If, however, the principal is required to choose the tests in a symmetric manner (i.e., via an i.i.d.~distribution), then the optimal strategy is to choose tests whose probability of passing is drawn uniformly from $[\frac{1}{4}, \frac{3}{4}]$. In our second model, test difficulties are selected endogenously by the firms. This corresponds to a setting in which the firms must commit to their testing procedures before knowing the quality of their products. This interaction naturally gives rise to a signaling game; we characterize the unique Bayes-Nash Equilibrium of this game, which happens to be symmetric. We then calculate its Price of Anarchy in terms of the principal's probability of choosing the worse product. Finally, we show that by restricting both firms' set of available thresholds to choose from, the principal can lower the Price of Anarchy of the resulting equilibrium; however, there is a limit, in that for every (common) restricted set of tests, the equilibrium failure probability is strictly larger than under the optimal i.i.d. distribution.