Uniqueness of Nash allocations for the Eisenberg–Noe aggregation

Determine whether, in the sensitive systemic risk measure built from the Eisenberg–Noe aggregation function with the natural per-bank decomposition, the (A,(Λ_i^{EN})_{i∈[N]})-Nash allocation rule is always unique for every financial network with strictly positive obligations to society, every stress scenario X ∈ L^∞(R^N), and every acceptance set A of a coherent risk measure that is continuous from above. In particular, prove or refute that multiple Nash allocations cannot occur for Λ^{EN}(x,m)=∑_{i∈[N]}(π_{i0} p_i(x+m)−γ ar p_{i0}), where p(·) is the Eisenberg–Noe clearing vector and Λ_i^{EN}(x,m)=π_{i0} p_i(x+m)−γ ar p_{i0}.

Background

The paper introduces Nash allocation rules for allocating systemic risk capital and provides general existence and, under certain contraction conditions, uniqueness results. For the Eisenberg–Noe (EN) aggregation function, the authors construct a natural decomposition that attributes to each bank its payment to society minus a threshold, and they show existence of Nash allocations along with sufficient conditions for uniqueness.

Corollary 4.6 gives two settings guaranteeing uniqueness: when each bank’s fraction of obligations to society is sufficiently large, and when the stress scenario is deterministic. Beyond these, the authors report that they could not find any counterexample exhibiting multiple Nash allocations for the EN aggregation, suggesting the possibility that uniqueness might hold more generally.