Lanchester duopoly model revisited: Advertising competition under time inconsistent preferences

Abstract

In this paper, we study a finite time horizon advertising dynamic game under the assumption of time-inconsistent preferences. Specifically, we consider two types of discounting, heterogeneous discounting and hyperbolic discounting. In the case of heterogeneous discounting, the relative importance of the final function will increase/decrease as the end of the planning horizon approaches compared with current payoffs. Whereas when agents discount future payoffs hyperbolically, their discount rates diminish rapidly in earlier stages and then slowly in the long term. We compute time-inconsistent and time-consistent feedback Nash equilibrium strategies, and compare them with those of the standard discounting case. Our results reveal that heterogeneous discounting would lead to some last-minute changes, i.e., some adapting behaviours in the last years in accordance with their increasing/decreasing valuations of the final state. Under some circumstances, the change can be so radical that the pre-commitment solution takes the contrary path of time-consistent strategies. Concerning the competition under hyperbolic discounting, the temporal evolution of advertising efforts show a quite different nature. Different strategies exhibit disparity in the beginning, and encounter in the neighbourhood in the end, which is contrary to the heterogeneous discounting. Besides, a strong commitment power might induce over investment.

Keywords:

• Advertising
• Lanchester model
• time inconsistency
• heterogeneous discounting
• hyperbolic discounting
• differential game

Acknowledgements

We are very grateful to the editor and the two anonymous reviewers for their insightful comments and suggestions.

Disclosure statement

The authors declare that they have no conflict of interest.

Notes

1 Defined as “the preference for immediate utility over delayed utility” (Frederick et al., 2002).

2 For more detailed discussion, we refer to the Section 4 of Sorger (1989).

3 The corresponding $\overline{\rho }$ is computed from ${\int }_t^{\infty }{e}^{-\overline{\rho }(s-t)}ds={\int }_t^{\infty }\theta (s-t)ds.$

4 In the sensitivity analysis provided in Section 4.3, we will see that a higher instantaneous discount rate implies a lower investment.

5 Due to space constraints, we do not present the sensitivity analysis of all parameters in Table 3, but all the results will be mentioned in the text.

Additional information

Funding

The second author’s research is supported by Ministerio de Economía, Industria y Competitividad, Gobierno de España [ECO2013-48248-P, ECO2017-82227-P (AEI/FEDER, UE)].