It is well-known that fractional-order discrete-time systems have a major advantage over their integer-order counterparts, because they can better describe the memory characteristics and the historical dependence of the underlying physical phenomenon. This paper presents a novel fractional-order triopoly game with bounded rationality, where three firms producing differentiated products compete over a common market. The proposed game theory model consists of three fractional-order difference equations and is characterized by eight equilibria, including the Nash fixed point. When suitable values for the fractional order are considered, the stability of the Nash equilibrium is lost via a Neimark-Sacker bifurcation or via a flip bifurcation. As a consequence, a number of chaotic attractors appear in the system dynamics, indicating that the behaviour of the economic model becomes unpredictable, independently of the actions of the considered firm. The presence of chaos is confirmed via both the computation of the maximum Lyapunov exponent and the 0-1 test. Finally, an entropy algorithm is used to measure the complexity of the proposed game theory model.

中文翻译：

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Caputo分数差分算子对新型博弈模型的影响

众所周知，分数阶离散时间系统相对于整数阶离散时间系统具有主要优势，因为它们可以更好地描述基础物理现象的存储特征和历史依赖性。本文提出了一种具有有限理性的新型分数阶三元博弈，其中三个生产差异化产品的公司在一个共同市场上竞争。所提出的博弈论模型由三个分数阶差分方程组成，并具有八个平衡点，其中包括纳什不动点。当考虑分数阶的合适值时，Nash平衡的稳定性会因Neimark-Sacker分歧或翻转分歧而失去。结果，许多混乱的吸引子出现在系统动力学中，表明经济模型的行为变得不可预测，与所考虑的公司的行为无关。通过最大李雅普诺夫指数的计算和0-1测试可以确认混沌的存在。最后，使用熵算法来衡量所提出的博弈论模型的复杂性。