Genetic Algorithms for Pareto Optimization in Bayesian Cournot Games Under Incomplete Cost Information

This paper develops a practical computational framework for the Bayesian Cournot model with bilateral incomplete cost information, where each player is uncertain about the opponent’s marginal cost, drawn from a continuous compact interval [c_∗,c^∗] with 0 <∞. The infinite dimensionality of the functional strategy spaces (mappings from types to production quantities) renders analytical closed-form solutions infeasible in this continuous-type setting. To overcome this challenge, we restrict the strategy spaces to finite-dimensional differentiable sub-manifolds—specifically, one-parameter families of oscillatory functions (cosine, sine, and mixed forms). After suitable affine Q-rescaling to map the oscillatory range into the production interval [0,Q], and with parameter ranges satisfying α, β > (π/2)/c_∗, these curves ensure near-exhaustivity: the joint production map (α,β)→ (xα(s),yβ(t)) covers [0,Q]^2 densely for every fixed cost pair (s,t), thereby recovering (up to density and closure) the full ex-post payoff space. We introduce the ex-post payoff mapping Φ(s,t,x,y) = (e_s(x,y)(t), f_t(x,y)(s)), which collects every realizable payoff pair once nature draws the types and players select their strategies. The image of Φ defines the general payoff space of the game, and its non-dominated points constitute the general ex-post Pareto frontier—all efficient realized outcomes across type strategy realizations, without dependence on private probability measures over types. Using multi-objective genetic algorithms, we numerically approximate this frontier (and selected collusive compromises) within the restricted but representative sub-manifolds. The resulting frontiers are computationally accessible, robust to parameter variations, and validated through hypervolume convergence, sensitivity analysis, and comparisons with NSGA-II, PSO and scalarization methods. The findings are significant because they provide decision-makers in oligopolistic markets (e.g., electric vehicles) with viable, implementable production policies that explore efficient trade-offs under genuine cost uncertainty, without requiring explicit forecasts of the opponent’s type distribution—a limitation of traditional expected-utility approaches. By focusing on ex-post efficiency, the method reveals belief independent compromise solutions that may guide tacit coordination or collusive outcomes in real-world strategic settings.