A Systemic Optimal Risk Transfer Equilibrium (SORTE) was introduced in "Systemic Optimal Risk Transfer Equilibrium" for the analysis of the equilibrium among financial institutions or in insurance-reinsurance markets. A SORTE conjugates the classical B"uhlmann's notion of an equilibrium risk exchange with a capital allocation principle based on systemic expected utility optimization. In this paper we extend such notion to the case in which the value function to be optimized has two components, one being the sum of the single agents' utility functions, the other consisting of a truly systemic component. The latter could be either enforced by an external regulator or be agreed on by the participants in the market. Technically, the extension of SORTE to the new setup requires developing a theory for multivariate utility functions and selecting at the same time a suitable framework for the duality theory. Conceptually, this more general framework allows us to introduce and study a Nash Equilibrium property of the optimizer. We prove existence, uniqueness, Pareto optimality and the Nash Equilibrium property of the newly defined Multivariate Systemic Optimal Risk Transfer Equilibrium.
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