Lanchester duopoly model revisited: advertising competition under time inconsistent preferences

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.