یک مسئله باز درباره ماتریس معکوس از سازمان های صنعتی و یک راه حل جزئی

This note formulates a large class of matrices whose inverses form an open research problem. It also provides a partial solution as a starting point to tackle the problem in future studies.

The formation and stability of a coalition structure (or a partition) is an important and unsettled issue in both sciences and social sciences, such as artificial intelligence (Monderer and Tennenholtz [5]) and game theory (Shubik [8]). An obstacle in resolving this issue is that one needs to obtain the analytical expressions of equilibrium payoffs for an arbitrary coalition structure, which often requires one to invert matrices whose inverses are unknown. This note derives a large class of such matrices from industrial organization whose inverses yield the strategic equilibria (or Nash equilibria [6]) in most linear oligopoly models. Such inverses form an open problem, which includes the analytical expression of the inverse for a general symmetric matrix.As a starting point to find a complete solution in future studies, the paper provides a partial solution by inverting a non-trivial subset of the derivedmatrices. This partial solution, previously unavailable in the literature, will be useful to other scholars in their future studies. 1 The rest of the note is organized as follows: Section 2 defines and Section 3 derives the open problem, Sections 4 and 5 provide a partial solution and an application, Section 6 concludes, and the appendix provides proofs.

The above analysis has formulated a large class ofmatrices whose inverses yield the strategic equilibria in most linear oligopolymodels. A complete understanding of these matrices remains unknown and their properties form a long list of open problems: the analytical expressions of their inverses, their eigenvalues and eigenvectors, their rank correction properties, and how these properties are determined by the partitions. The author hopes that readerswill bemotivated and challenged to investigate these open problems, in particular, to obtain more partial solutions or calculate more nontrivial sets of the inverses based on a particular set of models in industrial organization or networks, or on a particular set of partitions, or a particular set of parameters, which might someday lead to a complete answer in the future.