A game theoretic approach to measuring efficiency

Correspondence with a new mathematical programming definition for efficiency as proposed by Charnes, Cooper and Rhodes (CCR) is established by means of game theoretical models. Contact with all of the CCR results is also maintained so that their results extend to our new game theoretic interpretations. The latter proceeds by means of a family of games related to a linear programming problem. The games-to-programming relations which we establish also open new possibilities for further relations between families of games and linear programs.     

A fuzzy theoretic approach to simulation metamodeling

In this paper, we explore the potential application of fuzzy linear regression in developing simulation metamodels. It should be noted that the basic construct for simulation metamodels involves uncertainties and ambiguities that may be better addressed through fuzzy linear regression application. The solution techniques employed by fuzzy linear regression are very familiar, and the generation of fuzzy outputs may offer a wide range of solution space to the decision maker, thereby reducing the risk of making an incorrect economic decision. A numerical example is presented to show how a possibility distribution is used to capture the vagueness in a dependent variable for a regression metamodel.     

Comment on “A fuzzy soft set theoretic approach to decision making problems”

The algorithm for identification of an object in a previous paper of A.R. Roy et al. [A.R. Roy, P.K. Maji, A fuzzy soft set theoretic approach to decision making problems, J. Comput. Appl. Math. 203(2007) 412–418] is incorrect. Using the algorithm the right choice cannot be obtained in general. The problem is illustrated by a counter-example.     

A graph theoretic approach to matrix inversion by partitioning

LetM be a square matrix whose entries are in some field. Our object is to find a permutation matrixP such thatPM P ^–1 is completely reduced, i.e., is partitioned in block triangular form, so that all submatrices below its diagonal are 0 and all diagonal submatrices are square and irreducible. LetA be the binary (0, 1) matrix obtained fromM by preserving the 0's ofM and replacing the nonzero entries ofM by 1's. ThenA may be regarded as the adjacency matrix of a directed graphD. CallD strongly connected orstrong if any two points ofD are mutually reachable by directed paths. Astrong component ofD is a maximal strong subgraph. Thecondensation D ^* ofD is that digraph whose points are the strong components ofD and whose lines are induced by those ofD. By known methods, we constructD ^* from the digraph,D whose adjacency matrixA was obtained from the original matrixM. LetA ^* be the adjacency matrix ofD ^*. It is easy to show that there exists a permutation matrixQ such thatQA ^* Q ^–1 is an upper triangular matrix. The determination of an appropriate permutation matrixP from this matrixQ is straightforward.This was an informal talk at the International Symposium on Matrix Computation sponsored by SIAM and held in Gatlinburg, Tennessee, April 24–28, 1961 and was an invited address at the SIAM meeting in Stillwater, Oklahoma on August 31, 1961     

On particle transport processes: A potential theoretic approach

This article provides a potential theoretic approach to the study of particle transport stochastic processes (x ( t,w) ,Y (t ?L?:) ) where x i t ) is the temporal evolution of a non-Plarkovian particle motion and y (t) is a Markovian physical process in the medium that governs the scattering or jump of the particle.As opposed to the perturbation technique, our approach immensely enhances the applicatory value of transport processes. We begin with a sample path construction of a transport process and continue with the existence ofcertain invariant measures. Expressing the particle motion x(t) as an additive functional of the transport process (x(t),y(t)), we establish a law of large number and a functional centxal limit theorem,(a Brownian motion approximation),for the non-Markovian particle motion.     

Sharing costs in highways: A game theoretic approach

This paper introduces a new class of games, highway games, which arise from situations where there is a common resource that agents will jointly use. That resource is an ordered set of several indivisible sections, where each section has an associated fixed cost and each agent requires some consecutive sections. We present an easy formula to calculate the Shapley value, and we present an efficient procedure to calculate the nucleolus for this class of games.     

A divisor theoretic approach towards the arithmetic of Noetherian domains

We generalize the divisor theories for Krull and weakly Krull domains to noetherian domains with finitely generated integral closure.     

A fuzzy soft set theoretic approach to decision making problems

The problem of decision making in an imprecise environment has found paramount importance in recent years. A novel method of object recognition from an imprecise multiobserver data has been presented here. The method involves construction of a Comparison Table from a fuzzy soft set in a parametric sense for decision making.     

Realizations of abstract regular polytopes from a representation theoretic view

Peter McMullen has developed a theory of realizations of abstract regular polytopes, and has shown that the realizations up to congruence form a pointed convex cone which is the direct product of certain irreducible subcones. We show that each of these subcones is isomorphic to a set of positive semi-definite hermitian matrices of dimension m over either the real numbers, the complex numbers or the quaternions. In particular, we correct an erroneous computation of the dimension of these subcones by McMullen and Monson. We show that the automorphism group of an abstract regular polytope can have an irreducible character \({\chi}\) with \({\chi \neq \overline{\chi}}\) and with arbitrarily large essential Wythoff dimension. This gives counterexamples to a result of Herman and Monson, which was derived from the erroneous computation mentioned before. We also discuss a relation between cosine vectors of certain pure realizations and the spherical functions appearing in the theory of Gelfand pairs.     

Anticipated utility: A measure representation approach

This paper presents axioms which imply that a preference relation over lotteries can be represented by a measure of the area above the distribution function of each lottery. A special case of this family is the anticipated utility functional. One additional axiom implies this theory. This functional is then extended for the case of vectorial prizes.The author is grateful to Chew Soo Hong, Larry Epstein, Joe Ostroy, Joel Sobel, Peter Wakker, and Menahem Yaari for their comments.     

Burgers and Kadomtsev-Petviashvili hierarchies: A functional representation approach

Functional representations of (matrix) Burgers and potential Kadomtsev-Petviashvili (pKP) hierarchies (and others), as well as some corresponding Bäcklund transformations, can be obtained surprisingly simply from a “discrete” functional zero-curvature equation. We use these representations to show that any solution of a Burgers hierarchy is also a solution of the pKP hierarchy. Moreover, the pKP hierarchy can be expressed in the form of an inhomogeneous Burgers hierarchy. In particular, this leads to an extension of the Cole-Hopf transformation to the pKP hierarchy. Furthermore, these hierarchies are solved by the solutions of certain functional Riccati equations.     

A game theoretic approach to option valuation under Markovian regime-switching models

In this paper, we consider a game theoretic approach to option valuation under Markovian regime-switching models, namely, a Markovian regime-switching geometric Brownian motion (GBM) and a Markovian regime-switching jump-diffusion model. In particular, we consider a stochastic differential game with two players, namely, the representative agent and the market. The representative agent has a power utility function and the market is a “fictitious” player of the game. We also explore and strengthen the connection between an equivalent martingale measure for option valuation selected by an equilibrium state of the stochastic differential game and that arising from a regime switching version of the Esscher transform. When the stock price process is governed by a Markovian regime-switching GBM, the pricing measures chosen by the two approaches coincide. When the stock price process is governed by a Markovian regime-switching jump-diffusion model, we identify the condition under which the pricing measures selected by the two approaches are identical.     

Graph theoretic approach to parallel gene assembly

We study parallel complexity of signed graphs motivated by the highly complex genetic recombination processes in ciliates. The molecular gene assembly operations have been modeled by operations of signed graphs, i.e., graphs where the vertices have a sign + or −. In the optimization problem for signed graphs one wishes to find the parallel complexity by which the graphs can be reduced to the empty graph. We relate parallel complexity to matchings in graphs for some natural graph classes, especially bipartite graphs. It is shown, for instance, that a bipartite graph G has parallel complexity one if and only if G has a unique perfect matching. We also formulate some open problems of this research topic.     

A reverse engineering approach to the Weil representation

We describe a new approach to the Weil representation attached to a symplectic group over a finite or a local field. We dissect the representation into small pieces, study how they work, and put them back together. This way, we obtain a reversed construction of that of T. Thomas, skipping most of the literature on which the latter is based.