Abstract
We analyzed a dynamic duopoly game where players adopt specific preferences. These preferences are derived from Cobb–Douglas utility function with the assumption that they depend on past choices. For this paper, we investigated two possible cases for the suggested game. The first case considers only focusing on the action done by one player. This action reduces the game’s map to a one-dimensional map, which is the logistic map. Using analytical and numerical simulation, the stability of fixed points of this map is studied. In the second case, we focus on the actions applied by both players. The fixed points, in this case, are calculated, and their stability is discussed. The conditions of stability are provided in terms of the game’s parameters. Numerical simulation is carried out to give local and global investigations of the chaotic behavior of the game’s map. In addition, we use a statistical measure, such as entropy, to get more evidences on the regularity and predictability of time series associated with this case.
Highlights
As it is known in economy that the Cobb–Douglas utility function is emanated from empirical production studies
The model begins by recalling the typical Cobb–Douglas utility function that is maximized subject to a budget constraint
This paper studied the dynamics of a game with players using quantities as their decision variables, and their preferences depended on past choices
Summary
As it is known in economy that the Cobb–Douglas utility function is emanated from empirical production studies. In Reference [4], a heterogeneous duopoly game constructed based on an isoelastic demand function derived from Cobb–Douglas utility function was analyzed. It was used in studying the dynamic characteristics that arise in a fish stock harvested game consisting of two competitors with cost functions derived from it in Reference [5]. The main novelty of our discussed models with respect to other models that exist in literature includes multi-stability of different types of periodic cycles in the numerical simulation This assumption yields a nonlinear, two-dimensional discrete dynamic map describing our models in this manuscript.
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