Modeling the interactions between a reinsurer and several insurers, or between a central management branch (CB) and several subsidiary business branches, or between a coalition and its members, are fascinating problems, which suggest many interesting questions. Beyond two dimensions, one cannot expect exact answers. Occasionally, reductions to one dimension or heuristic simplifications yield explicit approximations, which may be useful for getting qualitative insights. In this paper, we study two such problems: the ruin problem for a two-dimensional CB network under a new mathematical model, and the problem of valuation of two-dimensional CB networks by optimal dividends. A common thread between these two problems is that the one dimensional reduction exploits the concept of invariant cones. Perhaps the most important contribution of the paper is the questions it raises; for that reason, we have found it useful to complement the particular examples solved by providing one possible formalization of the concept of a multi-dimensional risk network, which seems to us an appropriate umbrella for the kind of questions raised here.
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