A Characterization of Stochastically Stable Networks

Jackson and Watts [J. of Econ. Theory 71 (2002), 44-74] have examined the dynamic formation and stochastic evolution of networks. We provide a refinement of pairwise stability, p-pairwise stability, which allows us to characterize the stochastically stable networks without requiring the "tree construction" and the computation of resistance that may be quite complex. When a 1/2-pairwise stable network exists, it is unique and it coincides with the unique stochastically stable network. To solve the inexistence problem of p-pairwise stable networks, we define its set-valued extension with the notion of p-pairwise stable set. The 1/2-pairwise stable set exists and is unique. Any stochastically stable network is included in the 1/2-pairwise stable set. Thus, any network outside the 1/2-pairwise stable set must be considered as a nonrobust network. We also show that the 1/2-pairwise stable set can contain no pairwise stable network and we provide examples where a set of networks is more "stable" than a pairwise stable network.Network formation, Pairwise stability, Stochastic stability

A characterization of stochastically stable networks

Jackson and Watts [J. of Econ. Theory 71 (2002), 44-74] have examined the dynamic formation and stochastic evolution of networks. We provide a refinement of pairwise stability, p-pairwise stability, which allows us to characterize the stochastically stable networks without requiring the "tree construction" and the computation of resistance that may be quite complex. When a 1/2-pairwise stable network exists, it is unique and it coincides with the unique stochastically stable network. To solve the inexistence problem of p-pairwise stable networks, we define its set-valued extension with the notion of p-pairwise stable set. The 1/2-pairwise stable set exists and is unique. Any stochastically stable networks is included in the 1/2-pairwise stable set. Thus, any network outside the 1/2-pairwise stable set must be considered as a non-robust network. We also show that the 1/2-pairwise stable set can contain no pairwise stable network and we provide examples where a set of networks is more "stable" than a pairwise stable network

Variational Principles for Set-Valued Mappings with Applications to Multiobjective Optimization

This paper primarily concerns the study of general classes of constrained multiobjective optimization problems (including those described via set-valued and vector-valued cost mappings) from the viewpoint of modern variational analysis and generalized differentiation. To proceed, we first establish two variational principles for set-valued mappings, which~being certainly of independent interest are mainly motivated by applications to multiobjective optimization problems considered in this paper. The first variational principle is a set-valued counterpart of the seminal derivative-free Ekeland variational principle, while the second one is a set-valued extension of the subdifferential principle by Mordukhovich and Wang formulated via an appropriate subdifferential notion for set-valued mappings with values in partially ordered spaces. Based on these variational principles and corresponding tools of generalized differentiation, we derive new conditions of the coercivity and Palais-Smale types ensuring the existence of optimal solutions to set-valued optimization problems with noncompact feasible sets in infinite dimensions and then obtain necessary optimality and suboptimality conditions for nonsmooth multiobjective optmization problems with general constraints, which are new in both finite-dimensional and infinite-dimensional settings

Weak Minimizers, Minimizers and Variational Inequalities for set valued Functions. A blooming wreath?

In the literature, necessary and sufficient conditions in terms of variational inequalities are introduced to characterize minimizers of convex set valued functions with values in a conlinear space. Similar results are proved for a weaker concept of minimizers and weaker variational inequalities. The implications are proved using scalarization techniques that eventually provide original problems, not fully equivalent to the set-valued counterparts. Therefore, we try, in the course of this note, to close the network among the various notions proposed. More specifically, we prove that a minimizer is always a weak minimizer, and a solution to the stronger variational inequality always also a solution to the weak variational inequality of the same type. As a special case we obtain a complete characterization of efficiency and weak efficiency in vector optimization by set-valued variational inequalities and their scalarizations. Indeed this might eventually prove the usefulness of the set-optimization approach to renew the study of vector optimization

Preparation and toolkit learning

A product set of pure strategies is a prep set ("prep" is short for "preparation") if it contains at least one best reply to any consistent belief that a player may have about the strategic behavior of his opponents. Minimal prep sets are shown to exists in a class of strategic games satisfying minor topological conditions. The concept of minimal prep sets is compared with (pure and mixed) Nash equilibria, minimal curb sets, and rationalizability. Additional dynamic motivation for the concept is provided by a model of adaptive play that is shown to settle down in minimal prep sets.noncooperative games; inertia; status quo bias; adaptive play; procedural rationality

Equivalence of consistency and bilateral consistency through converse consistency

In the framework of (set-valued or single-valued) solutions for coalitional games with transferable utility, the three notions of consistency, bilateral consistency, and converse consistency are frequently used to provide axiomatic characterizations of a particular solution (like the core, prekernel, prenucleolus, Shapley value, and EANSC-value). Our main equivalence theorem claims that a solution satisfies consistency (with respect to an arbitrary reduced game) if and only if the solution satisfies both bilateral consistency and converse consistency (with respect to the same reduced game). The equivalence theorem presumes transitivity of the reduced game technique as well as difference independence on payoff vectors for two-person reduced games. Moulin's complement reduced game, Davis and Maschler's maximum reduced game and Yanovskaya and Driessen's linear reduced game versions are evaluated

Klee sets and Chebyshev centers for the right Bregman distance

We systematically investigate the farthest distance function, farthest points, Klee sets, and Chebyshev centers, with respect to Bregman distances induced by Legendre functions. These objects are of considerable interest in Information Geometry and Machine Learning; when the Legendre function is specialized to the energy, one obtains classical notions from Approximation Theory and Convex Analysis. The contribution of this paper is twofold. First, we provide an affirmative answer to a recently-posed question on whether or not every Klee set with respect to the right Bregman distance is a singleton. Second, we prove uniqueness of the Chebyshev center and we present a characterization that relates to previous works by Garkavi, by Klee, and by Nielsen and Nock.Comment: 23 pages, 2 figures, 14 image