On sharing of risk and resources (Q2906481)

The author considers a general model \(M_1\) described by four parameters \(I\), \({\mathcal X}\), \((e_i)_{i\in I}\) and \(({\mathcal U}_i)_{i\in I})\), where: (1) \(I\) is a finite set of agents; (2) \( {\mathcal X}\) is a topological linear space (with possible interpretation its elements as the agents' commodity bundles, contingent claims, etc.); (3) \(e_i\in {\mathcal X}\) is an initial endowment of agent \(i\); (4) \({\mathcal U}_i \rightarrow {\mathcal R} \cup \{-\infty\}\) is the utility function of agent \(i\), possibly non-concave or non-smooth. For this model the so-called \textit{shadow prices} are introduced. The first result of the paper gives a characterization of the set of efficient profile allocations \((x_i)_{i\in I}\in {\mathcal X}^I\) (defined by \(\sum_{i\in I} x_i =\sum_{i\in I} e_i\)) in terms of shadow prices. In the next step, a generalization \(M_2\) of the model \(M_1\) is considered, where all the elements \(x_i\) in \({\mathcal X}\) are random variables on the same probability space, and a cooperative game \(G\) corresponding to \(M_2\) is defined. Next, for such a game \(G\) its core is characterized. The remaining results discuss the model \(G_2\) under some ``smooth'' assumptions on the utility functions.NEWLINENEWLINEFor the entire collection see [Zbl 1241.00017].