Abstract
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.
Supported by the National Natural Science Foundation of China (Grant No. 62172422).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
For any quality distribution, its quantile is distributed uniformly in [0, 1]. Once we get the results of quantile functions, we can derive the results of the original functions. Thus, it allows us to concentrate on the uniform distribution.
- 2.
We have \(\mathbb {E}[X|X<\theta ]\le \mathbb {E}[X]\le \mathbb {E}[X|X\ge \theta ]\) It indicates that the expected quality of the product passing a test will not be lower than the failed one.
- 3.
For the reason why we do not consider \(a\le b\le c\le d\), if \(a=b\) or \(c=d\), then a point appears in the restricted set. The Bayes-Nash equilibrium may not be unique and symmetric. Because we know that every pair supported on S forms an equilibrium when \(S=\{1-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\}\). And if \(b=c\), the case is equal to one interval and is well studied.
References
Ambrus, A., Takahashi, S.: Multi-sender cheap talk with restricted state spaces. Theor. Econ. 3(1), 1–27 (2008)
Banerjee, S., Kempe, D., Kleinberg, R.: Threshold tests as quality signals: optimal strategies, equilibria, and price of anarchy. In: Feldman, M., Fu, H., Talgam-Cohen, I. (eds.) WINE 2021. LNCS, vol. 13112, pp. 299–316. Springer, Cham (2022). https://doi.org/10.1007/978-3-030-94676-0_17
Bayindir, E.E., Gurdal, M.Y., Ozdogan, A., Saglam, I.: Cheap talk games with two-senders and different modes of communication. Games 11(2), 18 (2020)
Boleslavsky, R., Cotton, C.: Limited capacity in project selection: competition through evidence production. Econ. Theor. 65, 385–421 (2018)
Crawford, V.P., Sobel, J.: Strategic information transmission. Econom. J. Econom. Soc. 50(6), 1431–1451 (1982)
Dughmi, S.: Algorithmic information structure design: a survey. ACM SIGecom Exchanges 15(2), 2–24 (2017)
Dughmi, S., Kempe, D., Qiang, R.: Persuasion with limited communication. In: Proceedings of the 2016 ACM Conference on Economics and Computation, pp. 663–680 (2016)
Dughmi, S., Xu, H.: Algorithmic Bayesian persuasion. In: Proceedings of the Forty-Eighth Annual ACM Symposium on Theory of Computing, pp. 412–425 (2016)
Gentzkow, M., Kamenica, E.: Bayesian persuasion with multiple senders and rich signal spaces. Games Econ. Behav. 104, 411–429 (2017)
Kamenica, E., Gentzkow, M.: Bayesian persuasion. Am. Econ. Rev. 101(6), 2590–2615 (2011)
Li, F., Norman, P.: Sequential persuasion. Theor. Econ. 16(2), 639–675 (2021)
Ogawa, H.: Receiver’s sensitivity and strategic information transmission in multi-sender cheap talk. Int. J. Game Theory 50(1), 215–239 (2021)
Sobel, J.: Giving and receiving advice. Adv. Econ. Econom. 1, 305–341 (2013)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Ren, Z., Liu, Y. (2024). Endogenous Threshold Selection with Two-Interval Restricted Tests. In: Chen, X., Li, B. (eds) Theory and Applications of Models of Computation. TAMC 2024. Lecture Notes in Computer Science, vol 14637. Springer, Singapore. https://doi.org/10.1007/978-981-97-2340-9_7
Download citation
DOI: https://doi.org/10.1007/978-981-97-2340-9_7
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-97-2339-3
Online ISBN: 978-981-97-2340-9
eBook Packages: Computer ScienceComputer Science (R0)