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 (real-valued) qualities, which are drawn i.i.d. from a common prior. The principal aims to choose the better of the two products, but the quality of a product can only be estimated via a coarse-grained threshold test: given a threshold \(\theta \), the principal learns whether a product’s quality exceeds \(\theta \) or fails to do so.

We study this selection problem under two types of interactions. In the first, the principal does the testing herself, and can choose tests optimally 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 interaction model, test difficulties are selected endogenously by the two firms. This corresponds to a setting in which the firms must commit to their testing (quality control) procedures before knowing the quality of their products. This interaction model naturally gives rise to a signaling game with two senders and one receiver. 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.

1. 1.

   This is the more common view of signaling in the economics community: a signaling scheme is interpreted as a device (physical or otherwise) that maps relevant states of the world to observable signals. Fixing a device constitutes committing to a signaling scheme. In contrast, recent works in computer science apply signaling/persuasion to scenarios such as communications where it is less clear whether the sender has the ability to commit to a mapping.

2. 2.

   Lerner and Tirole [13] do briefly discuss a multi-firm setting, but only consider one extremely limited example.

3. 3.

   However, we note that in addressing the same real-world scenario, Gill and Sgroi [11] instead consider a model where the owner knows the state before choosing the certifier; see the full version [3] for details.

4. 4.

   We adopt the convention that the cumulative distribution function (cdf) of a probability measure on \(\mathbb {R} \) is defined by setting F(x) to be the measure of the set \((-\infty ,x]\) under the distribution.

5. 5.

   Alternatively, the setting may be such that the agents naturally have the choice of test difficulty, such as in external certification of product quality [10, 11, 13] or students’ selection of which classes to attempt [18]. In these settings, it is still frequently assumed that agents are not aware of their private quality value when they make their choice of difficulty; see for example [13] for a model of certification and [16] for a model of contracting between students and employers.

6. 6.

   There are naturally other objectives in between these two extremes.

7. 7.

   Recall that the Kendall \(\tau \) distance between two rankings is the number of inversions between the rankings, i.e., the number of pairs of elements that are in different order.

8. 8.

   Recall from Sect. 2.2 that this assumption is without loss of generality.

9. 9.

   As will be established in Theorem 9.

10. 10.

    Recall that we write \(I(F) = \mathbb {E}\left[ {I(\mathcal {T}_{F})}\right] \).

11. 11.

    Of course, if the principal can choose different sets for different firms, then she can choose \(S_X = \{\frac{1}{3} \}\) and \(S_Y = \{\frac{2}{3}\}\), which would implement the optimal strategy for her. The more interesting question is to find one set S to restrict all firms to, which naturally corresponds to prescribing standards for quality control.

12. 12.

    We thank Nicole Immorlica for suggesting this interpretation.

We would like to thank Odilon Camara, Peter Frazier, Moshe Hoffman, Nicole Immorlica, Jonathan Libgober, Erez Yoeli, and Christina Lee Yu for useful discussions and pointers.

Authors and Affiliations

1. Cornell University, Ithaca, NY, 14853, USA

   Siddhartha Banerjee & Robert Kleinberg

2. University of Southern California, Los Angeles, CA, 90089, USA

   David Kempe

Editors and Affiliations

1. Tel Aviv University, Tel Aviv, Israel

   Michal Feldman

2. Shanghai University of Finance and Economics, Shanghai, China

   Hu Fu

3. Technion – Israel Institute of Technology, Haifa, Israel

   Inbal Talgam-Cohen

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