 
Summary: BASINS OF ATTRACTION IN A COURNOT DUOPOLY MODEL
OF KOPEL
DOUGLAS R. ANDERSON, NICHOLAS G. MYRAN, AND DUSTIN L. WHITE
Abstract. For a nonlinear Cournot duopoly map of Kopel, we show that
a circle, lines, and rectangles play a key role in determining the basins of
attraction in the case of three nontrivial Nash equilibria.
1. A Nonlinear Cournot Duopoly Map of Kopel
In [1, p. 191] we found the discrete map
xn+1 = (1  A)xn + AByn(1  yn)
yn+1 = (1  A)yn + ABxn(1  xn) (1)
that models a nonlinear Cournot duopoly game [2] with normalized outputs x, y
[0, 1] for the duopolists with adjustment coefficient A [0, 1] and parameter B > 0,
a measure of the positive external influence one player exerts on the other player.
The righthand sides represent the reaction functions of the players given initial
beliefs (x0, y0), and determine the subsequent trajectory of beliefs (xn, yn). Game
(1) was introduced by Kopel [3], and initially analyzed in terms of fixed points
(often called Nash equilibria or equilibrium beliefs) and chaos control by Agiza [4].
Bischi and Kopel [5] then provided an extensive economic rationale for the model
together with an analysis of parameter space in which three nontrivial consistent
beliefs, corresponding to Nash equilibria in the product quantity space, coexist,
