Cournot model with piecewise inverse demand Do you know if there exist results for Cournot oligopoly (or duopoly) when the inverse demand is a piecewise continuous?
Something of this type 


$P(Q) = \alpha-\beta Q$ (if $0\leq Q&lt;Q'$)  
$P(Q) = \gamma-\delta Q$ (if $Q' \leq Q \leq Q_{max}$)
In the book Sushko, I. (Ed.). (2002). Oligopoly dynamics: Models and tools. Springer.

there is a chapter,

Puu, T., Gardini, L., &amp; Sushko, I. (2002). Cournot duopoly with kinked demand according to Palander and Wald. (pp. 111-146). 

where the authors examine in detail a specific numerical example, among other variants.  They arrive at the best response functions and find that there exists two Cournot equilibria, for which, as they write


  "...both equilibria, the Cournot points, are locally stable. They hence coexist with each its proper basin of attraction in the space of
  initial conditions."


The authors are concerned with dynamics, hence the mention of "initial conditions". If one wanted to turn their conclusions into the static case with just one move to equilibrium, one should consider in which "basin of attraction" does the zero output vector belongs.