Large Ranking Games with Diffusion Control

Stefan Ankirchner (), Nabil Kazi-Tani (), Julian Wendt () and Chao Zhou ()
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Stefan Ankirchner: Institute for Mathematics, University of Jena, 07743 Jena, Germany
Nabil Kazi-Tani: Institut Elie Cartan de Lorraine, Université de Lorraine, L’Unité de Formation et de Recherche Mathématiques Informatique et Mécanique (UFR MIM), 57073 Metz Cedex 03, France
Julian Wendt: Institute for Mathematics, University of Jena, 07743 Jena, Germany
Chao Zhou: Department of Mathematics and Risk Management Institute, National University of Singapore, 119076, Singapore

Mathematics of Operations Research, 2024, vol. 49, issue 2, 675-696

Abstract: We consider a symmetric stochastic differential game where each player can control the diffusion intensity of an individual dynamic state process, and the players whose states at a deterministic finite time horizon are among the best α ∈ ( 0 , 1 ) of all states receive a fixed prize. Within the mean field limit version of the game, we compute an explicit equilibrium, a threshold strategy that consists of choosing the maximal fluctuation intensity when the state is below a given threshold and the minimal intensity otherwise. We show that for large n , the symmetric n -tuple of the threshold strategy provides an approximate Nash equilibrium of the n -player game. We also derive the rate at which the approximate equilibrium reward and the best-response reward converge to each other, as the number of players n tends to infinity. Finally, we compare the approximate equilibrium for large games with the equilibrium of the two-player case.

Keywords: Primary: 91A15; secondary: 91A06; 91A10; 91A16; 93E20; diffusion control; game; rank-based reward; mean field limit; oscillating Brownian motion (search for similar items in EconPapers)
Date: 2024
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Persistent link: https://EconPapers.repec.org/RePEc:inm:ormoor:v:49:y:2024:i:2:p:675-696

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