Overtaking optimality in a discrete-time advertising game

Jewaidu Rilwan; Poom Kumam; Idris Ahmed; Jewaidu Rilwan; Poom Kumam; Idris Ahmed

doi:10.3934/math.2022035

AIMS Mathematics

2022, Volume 7, Issue 1: 552-568. doi: 10.3934/math.2022035

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Research article

Overtaking optimality in a discrete-time advertising game

1.
Department of Mathematics, Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand
2.
Center of Excellence in Theoretical and Computational Science (TaCS-CoE), Faculty of Science, King Mongkut's University of Technology Thonburi (KMUTT), 126 Pracha Uthit Rd., Bang Mod, Thung Khru, Bangkok 10140, Thailand
3.
Departments of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan
4.
Department of Mathematical Sciences, Bayero University, Kano 700231, Nigeria
5.
Department of Mathematics and Computer Science, Sule Lamido University, P.M.B 048 Kafin-Hausa, Jigawa State, Nigeria

Received: 01 March 2021 Accepted: 23 September 2021 Published: 14 October 2021
MSC : 34H05, 49N70, 91A23, 91A10

In this paper, advertising competition among $ m $ firms is studied in a discrete-time dynamic game framework. Firms maximize the present value of their profits which depends on their advertising strategy and their market share. The evolution of market shares is determined by the firms' advertising activities. By employing the concept of the discrete-time potential games of González-Sánchez and Hernández-Lerma (2013), we derived an explicit formula for the Nash equilibrium (NE) of the game and obtained conditions for which the NE is an overtaking optimal. Moreover, we analyze the asymptotic behavior of the overtaking NE where the convergence towards a unique steady state (turnpike) is established.
- overtaking optimality,
- advertising game,
- Nash equilibrium
Citation: Jewaidu Rilwan, Poom Kumam, Idris Ahmed. Overtaking optimality in a discrete-time advertising game[J]. AIMS Mathematics, 2022, 7(1): 552-568. doi: 10.3934/math.2022035

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Abstract

In this paper, advertising competition among $ m $ firms is studied in a discrete-time dynamic game framework. Firms maximize the present value of their profits which depends on their advertising strategy and their market share. The evolution of market shares is determined by the firms' advertising activities. By employing the concept of the discrete-time potential games of González-Sánchez and Hernández-Lerma (2013), we derived an explicit formula for the Nash equilibrium (NE) of the game and obtained conditions for which the NE is an overtaking optimal. Moreover, we analyze the asymptotic behavior of the overtaking NE where the convergence towards a unique steady state (turnpike) is established.

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