We reconsider the Cournot duopoly problem in light of the theory for long
memory. We introduce the Caputo fractional-order difference calculus to
classical duopoly theory to propose a fractional-order discrete Cournot duopoly
game model, which allows participants to make decisions while making full use
of their historical information. Then we discuss Nash equilibria and local
stability by using linear approximation. Finally, we detect the chaos of the
model by employing a 0-1 test algorithm

Kwon, Yekyung