Stackelberg-Cournot-Nash equilibria under ambiguity and $$\alpha $$-maxmin preferences

Abstract A Kantorovich-like optimal transport problem is formulated by seeking to minimize the Choquet integral of a given cost function with respect to the $$\alpha $$ α -mixture of joint belief functions with given marginals, and their dual plausibility functions. This extension of the classical probabilistic problem allows to deal with misspecified marginal distributions by relying on a parameter $$\alpha \in [0,1]$$ α ∈ [ 0 , 1 ] that acts like a pessimism index. We show that the particular subcase given by a marginal belief function and a marginal probability measure can be used to model a game under ambiguity, through the definition of the Stackelberg-Cournot-Nash equilibrium with Dempster-Shafer uncertainty and $$\alpha $$ α -maxmin preferences. An algorithm is provided for approximating an equilibrium based on a suitable entropic formulation of the defined optimal transport problem. Finally, a distinguished application is found in a market with a finite set of types of investors having misspecified distribution. Investors’ choices result in a purchase portfolio on a finite set of assets which, in turn, depends on asset prices fixed by the market maker and the interactions among investors.