Abstract
The classical Hotelling game is played on a line segment whose points represent uniformly distributed clients. The n players of the game are servers who need to place themselves on the line segment, and once this is done, each client gets served by the player closest to it. The goal of each player is to choose its location so as to maximize the number of clients it attracts.
In this paper we study a variant of the Hotelling game where each client v has a tolerance interval, randomly distributed according to some density function f, and v gets served by the nearest among the players eligible for it, namely, those that fall within its interval. (If no such player exists, then v abstains.) It turns out that this modification significantly changes the behavior of the game and its states of equilibria. In particular, it may serve to explain why players sometimes prefer to “spread out,” rather than to cluster together as dictated by the classical Hotelling game.
We consider two variants of the game: symmetric games, where clients have the same tolerance range to their left and right, and asymmetric games, where the left and right ranges of each client are determined independently of each other. We characterize the Nash equilibria of the 2-player game. For \(n\ge 3\) players, we characterize a specific class of strategy profiles, referred to as canonical profiles, and show that these profiles are the only ones that may yield Nash equilibria in our game. Moreover, the canonical profile, if exists, is uniquely defined for every n and f. In the symmetric setting, we give simple conditions for the canonical profile to be a Nash equilibrium, and demonstrate their application for several distributions. In the asymmetric setting, the conditions for equilibria are more complex; still, we derive a full characterization for the Nash equilibria of the exponential distribution. Finally, we show that for some distributions the simple conditions given for the symmetric setting are sufficient also for the asymmetric setting.
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.
There are several reasons this phenomenon is increasingly prevalent online. First, exposure to content is curated by algorithms according to each user’s personal preferences. Second, on social media, users are more likely to share with their network content that agrees with their own opinion. Third, it has become increasingly easier to join private discussion groups that consist of like-minded individuals.
- 2.
There are at most \(n-2\) points which are at equal distances from the nearest player on the right and on the left, and given that there is a continuum of clients in total, modifying \(A_v(\mathbf {s})\) in those points does not affect player utilities.
References
Ahlin, C., Ahlin, P.D.: Product differentiation under congestion: hotelling was right. Econ. Inq. 51(3), 1750–1763 (2013)
Ben-Porat, O., Tennenholtz, M.: Shapley facility location games. In: Devanur, N.R., Lu, P. (eds.) WINE 2017. LNCS, vol. 10660, pp. 58–73. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-71924-5_5
Brenner, S.: Location (Hotelling) Games and Applications. Wiley Encyclopedia of Operations Research and Management Science. Wiley, New York (2010)
Cohen, A., Peleg, D.: Hotelling games with multiple line faults. CoRR, abs/1907.06602 (2019). http://arxiv.org/abs/1907.06602
d’Aspremont, C., Gabszewicz, J.J., Thisse, J.-F.: On hotelling’s “stability in competition”. Econometrica: J. Econometric Soc. 47, 1145–1150 (1979)
De Palma, A., Ginsburgh, V., Thisse, J.-F.: On existence of location equilibria in the 3-firm Hotelling problem. J. Ind. Econ. 36, 245–252 (1987)
Downs, A.: An economic theory of political action in a democracy. J. Polit. Econ. 65(2), 135–150 (1957)
Eaton, B.C., Lipsey, R.G.: The principle of minimum differentiation reconsidered: some new developments in the theory of spatial competition. Rev. Econ. Stud. 42(1), 27–49 (1975)
Eiselt, H.A.: Equilibria in competitive location models. In: Eiselt, H., Marianov, V. (eds.) Foundations of Location Analysis. International Series in Operations Research Management Science, vol. 155, pp. 139–162. Springer, Heidelberg (2011). https://doi.org/10.1007/978-1-4419-7572-0_7
Eiselt, H.A., Laporte, G., Thisse, J.-F.: Competitive location models: a framework and bibliography. Transp. Sci. 27(1), 44–54 (1993)
Feldman, M., Fiat, A., Obraztsova, S.: Variations on the hotelling-downs model. In: 13th AAAI Conference on AI (2016)
Feldotto, M., Lenzner, P., Molitor, L., Skopalik, A.: From hotelling to load balancing: approximation and the principle of minimum differentiation. In: Proceedings of 18th AAMAS, pp. 1949–1951 (2019)
Garimella, K., De Francisci Morales, G., Gionis, A., Mathioudakis, M.: Political discourse on social media: echo chambers, gatekeepers, and the price of bipartisanship. In: Proceedings of 2018 WWW, pp. 913–922 (2018)
Hotelling, H.: Stability in competition. Econ. J. 39(153), 41–57 (1929)
Kohlberg, E.: Equilibrium store locations when consumers minimize travel time plus waiting time. Econ. Lett. 11(3), 211–216 (1983)
Osborne, M.J., Pitchik, C.: Equilibrium in hotelling’s model of spatial competition. Econometrica: J. Econometric Soc. 55, 911–922 (1987)
Peters, H., Schröder, M., Vermeulen, D.: Hotelling’s location model with negative network externalities. Int. J. Game Theory 47(3), 811–837 (2018)
Peters, H.J.M., Schröder, M.J.W., Vermeulen, A.J.: Waiting in the queue on Hotelling’s main street (2015)
Shen, W., Wang, Z.: Hotelling-Downs model with limited attraction. In: Proceedings of 16th AAMAS, pp. 660–668 (2017)
Acknowledgments
The authors would like to thank Shahar Dobzinski and Yinon Nahum for many fertile discussions and helpful insights, and the anonymous reviewers for their useful comments. This research was supported in part by a US-Israel BSF Grant No. 2016732.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Cohen, A., Peleg, D. (2019). Hotelling Games with Random Tolerance Intervals. In: Caragiannis, I., Mirrokni, V., Nikolova, E. (eds) Web and Internet Economics. WINE 2019. Lecture Notes in Computer Science(), vol 11920. Springer, Cham. https://doi.org/10.1007/978-3-030-35389-6_9
Download citation
DOI: https://doi.org/10.1007/978-3-030-35389-6_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-35388-9
Online ISBN: 978-3-030-35389-6
eBook Packages: Computer ScienceComputer Science (R0)