## Stackelberg-Cournot-Nash equilibria with Dempster-Shafer uncertainty and α-maxmin preferences

We consider the marginal problem in Dempster-Shafer theory within a finite setting, investigating the structure of the set of bivariate joint belief functions having fixed marginals, by relying on copula theory [3]. Next, we formulate a Kantorovich-like optimal transport problem [6], seeking to minimize the Choquet integral of a given cost function with respect to the set of joint belief functions, by taking the α-maxmin criterion [2, 4] (namely, α-DS-OT), where α ∈ [0, 1]. We show that the particular subcase given by an additive marginal and a non-additive one allows to model a game under ambiguity, through the definition of the Stackelbeg-Cournot-Nash equilibrium with Dempster-Shafer uncertainty and α-maxmin preferences (namely, α-DS-SCNE), which generalizes [1]. Assuming mild regularity conditions, we prove the existence of α-DS-SCNEs and propose an algorithm to approximate an equilibrium based on a suitable entropic formulation of α-DS-OT, inspired to the additive case [5]. REFERENCES [1] B. Acciaio, B.A. Neumann, Characterization of transport optimizers via graphs and applications to Stackelberg-Cournot-Nash equilibria, arXiv, 2306.03843(2023). [2] L. Hurwicz, The Generalized Bayes Minimax Principle: A Criterion for Decision Making Under Uncertainty, Cowles Commission Discussion Paper, 355(1951). [3] R. Malinowski, D. Destercke, Copulas, Lower Probabilities and Random Sets: How and When to Apply Them?, in L.A. García-Escudero et al (eds), Building Bridges between Soft and Statistical Methodologies for Data Science. SMPS 2022, Advances in Intelligent Systems and Computing, vol 1433, Springer, Cham, 2023. [4] D. Petturiti, B. Vantaggi, No-Arbitrage Pricing with α-DS Mixtures in a Market with Bid-Ask Spreads, Proceedings of Machine Learning Research, 215(2023), 401—411. [5] G. Peyré, M. Cuturi, Computational Optimal Transport, Foundations and Trends in Machine Learning, 11(5-6)(2019), 355—607. [6] C. Villani, Optimal Transport: Old and New, Springer, 2009.

Area: CS56 - Conditional knowledge representation and information fusion under coherence (Lydia Castronovo and Giuseppe Sanfilippo)

Keywords: Dempster-Shafer theory, Optimal Transport under ambiguity, Stackelbeg-Cournot-Nash equilibria under ambiguity

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