Endogenous Threshold Selection with Two-Interval Restricted Tests

We continue to study the endogenous threshold selection game. Two firms have interchangeable products with qualities that are drawn i.i.d. from a common prior. A principal wants to minimize the probability of selecting the product of lower quality by offering a set of threshold tests. The threshold test is that given a threshold, the product can be estimated whether its quality exceeds the threshold. Two firms pick the threshold endogenously to maximize the probability of being chosen by the principal. We consider the scenario where the principal restricts threshold tests to be two-interval. We further characterize properties of Bayes-Nash equilibrium between two firms. Numerically, the minimum probability of incorrect selection is 0.22096 where threshold tests are supported on \([0,0.19]\cup [0.33,0.79]\) . Moreover, our exploration aids in offering valuable insights into signaling games with one receiver and two senders.