Nobusumi Sagara

Convexity of the lower partition range of a concave vector measure

This paper investigates a class of nonadditive measures on σ-algebras, named concave measures, to establish a Lyapunov-type convexity theorem. To this end, we introduce convex combinations of measurable sets in terms of a nonatomic vector measure and demonstrate the convexity of the lower partition range of a concave vector measure. The main result is applied to a fair division problem along the lines of L.E. Dubins and E.H. Spanier (Amer Math Monthly 68:1–17, 1961).

Existence of efficient envy-free allocations of a heterogeneous divisible commodity with nonadditive utilities

This paper studies the existence of Pareto optimal, envy-free allocations of a heterogeneous, divisible commodity for a finite number of individuals. We model the commodity as a measurable space and make no convexity assumptions on the preferences of individuals. We show that if the utility function of each individual is uniformly continuous and strictly monotonic with respect to set inclusion, and if the partition matrix range of the utility functions is closed, a Pareto optimal envy-free partition exists. This result follows from the existence of Pareto optimal envy-free allocations in an extended version of the original allocation problem.

Concave measures and the fuzzy core of exchange economies with heterogeneous divisible commodities

The main purpose of this paper is to prove the existence of the fuzzy core of an exchange economy with a heterogeneous divisible commodity in which preferences of individuals are given by nonadditive utility functions defined on a @s-algebra of admissible pieces of the total endowment of the commodity. The problem is formulated as the partitioning of a measurable space among finitely many individuals. Applying the Yosida-Hewitt decomposition theorem, we also demonstrate that partitions in the fuzzy core are supportable by prices in L^1.

Representation of preference relations on σ -algebras of nonatomic measure spaces: Convexity and continuity

The purpose of this paper is to show how preference relations on @s-algebras can be represented by means of nonadditive set functions that satisfy appropriate requirements of convexity and continuity on @s-algebras. To this end, we introduce convex combinations of measurable sets, and quasiconcave and concave functions on @s-algebras, which conform with the standard results in convex analysis. We formulate the convexity and the continuity axioms for preference relations on @s-algebras with a metric topology and show the existence of utility functions for convex continuous preference relations. We also show that monotone continuous preference relations are representable by fuzzy measures.

A characterization of [alpha]-maximin solutions of fair division problems

This paper investigates the problem of fair division of a measurable space among a finite number of individuals and characterizes some equity concepts when preferences of each individual are represented by a nonadditive set function on a [sigma]-algebra. We show that if utility functions of individuals satisfy continuity from below and strict monotonicity, then positive Pareto optimality is equivalent to [alpha]-maximin optimality for some [alpha] in the unit simplex and Pareto-optimal [alpha]-equitability is equivalent to [alpha]-maximin optimality. These characterizations are novel in the literature.

A probabilistic representation of exact games on σ-algebras

The purpose of this paper is to establish the intrinsic relations between the cores of exact games on @s-algebras and the extensions of exact games to function spaces. Given a probability space, exact functionals are defined on L^~ as an extension of exact games. To derive a probabilistic representation for exact functionals, we endow them with two probabilistic conditions: law invariance and the Fatou property. The representation theorem for exact functionals lays a probabilistic foundation for nonatomic scalar measure games. Based on the notion of P-convexity, we also investigate the equivalent conditions for the representation of anonymous convex games.

Concave Measures and the Fuzzy Core of Exchange Economies with Heterogeneous Divisible Commodities

The main purpose of this paper is to prove the existence of the fuzzy core of an exchange economy with a heterogeneous divisible commodity in which preferences of individuals are given by nonadditive utility functions defined on a -algebra of admissible pieces of the total endowment of the commodity. The problem is formulated as the partitioning of a measurable space among finitely many individuals. Applying the Yosida–Hewitt decomposition theorem, we also demonstrate that partitions in the fuzzy core are supportable by prices in L1.

Value Functions and Transversality Conditions for Infinite Horizon Optimal Control Problems

This paper investigates infinite horizon optimal control problems with fixed left endpoints with nonconvex, nonsmooth data. We derive the nonsmooth maximum principle and the adjoint inclusion for the value function as necessary conditions for optimality that indicate the relationship between the maximum principle and dynamic programming. The necessary conditions under consideration are extensions of those of [8] to an infinite horizon setting. We then present new sufficiency conditions consistent with the necessary conditions, which are motivated by the useful result by [26] whose sufficiency theorem is valid for nonconvex, nondifferentiable Hamiltonians. The sufficiency theorem presented in this paper employs the strengthened adjoint inclusion of the value function as well as the strengthened maximum principle. If we restrict our result to convex models, it is possible to characterize minimizing processes and provide necessary and sufficient conditions for optimality. In particular, the role of the transversality conditions at infinity is clarified.

A Probabilistic Representation of Exact Games on σ -Algebras

The purpose of this paper is to establish the intrinsic relations between the cores of exact games on σ-algebras and the extensions of exact games to function spaces. Given a probability space, to derive a probabilistic representation for exact functionals, we endow them with two probabilistic conditions: law invariance and the Fatou property. The representation theorem for exact functionals lays a probabilistic foundation for nonatomic scalar measure games. Based on the notion of P-convexity, we also investigate the equivalent conditions for the representation of anonymous convex games.

Continuity of Choquet Integrals of Supermodular Capacities

We prove the continuity of the Choquet integral of supermodular capacities, in L ∞ with respect to the weak*-topology, employing a useful relationship between convex games and their Choquet integrals. The main result is applied to generalized fair division problems, and the existence of Pareto optimal α-allocations is demonstrated for the case of nonadditive measures.