Deformation of Striped Patterns by Inhomogeneities

We study the effects of adding a local perturbation in a pattern forming system, taking as an example the Ginzburg-Landau equation with a small localized inhomogeneity in two dimensions. Measuring the response through the linearization at a periodic pattern, one finds an unbounded linear operator that is not Fredholm due to continuous spectrum in typical translation invariant or weighted spaces. We show that Kondratiev spaces, which encode algebraic localization that increases with each derivative, provide an effective means to circumvent this difficulty. We establish Fredholm properties in such spaces and use the result to construct deformed periodic patterns using the Implicit Function Theorem. We find a logarithmic phase correction which vanishes for a particular spatial shift only, which we interpret as a phase-selection mechanism through the inhomogeneity.Comment: 18 page

On Multicriteria Games with Uncountable Sets of Equilibria

The famous Harsanyi's (1973) Theorem states that generically a finite game has an odd number of Nash equilibria in mixed strategies. In this paper, we show that for finite multicriteria games (games with vector-valued payoffs) this kind of result does not hold. In particular, we show, by examples, that it is possible to find balls in the space of games such that every game in this set has uncountably many equilibria so that uncountable sets of equilibria are not nongeneric in multicriteria games. Moreover, we point out that, surprisingly, all the equilibria of the games cor- responding to the center of these balls are essential, that is, they are stable with respect to every possible perturbation on the data of the game. However, if we consider the scalarization stable equilibrium concept (introduced in De Marco and Morgan (2007) and which is based on the scalarization technique for multicriteria games), then we show that it provides an effective selection device for the equilibria of the games corresponding to the centers of the balls. This means that the scalarization stable equilibrium concept can provide a sharper selection device with respect to the other classical refinement concepts in multicriteria games.

Efficiency First or Equity First?: Two Principles and Rationality of Social Choice

The Pareto efficiency criterion is often in conflict with the equity criteria as no-envy or as egalitarian-equivalence: An allocation x that is Pareto superior to another allocation y can be inferior to y in consideration of equity. This paper formalizes two differnet principles of social choice under possible conflict of efficiency and equity. The efficiency-first principle requires that we should always select from efficient allocations, and when the efficiency criterion is not at all effective as a guide for selection, i.e., when all the available allocations are efficient or there is no efficient allocation, we should apply an equity criterion to choose desirable allocations. The equity-first principle reverses the lexicographic order of application of the two criteria. We examine rationality of the social choice rules satisfying these two principles. It is shown that the degree of rationality varies widely depending on which principle the social choice rules represent. Several impossibility and possibility results as well as a characterization theorem are obtained.