Local monotonicity of power: Axiom or just a property?

This paper discusses whether Local Monotonicity (LM) should be regarded as a property of the power distribution of a specific voting game under consideration, indicated by a power measure, or as a characteristic of power per se. The latter would require reasonable power measures to satisfy a corresponding LM axiom. The former suggests that measures which do not allow for a violation of LM fail to account for dimensions of power which can cause nonmonotonicity in voting weight. Only if a measure is able to indicate nonmonotonicity, it can help design voting games for which power turns out to be monotonic. The argument is discussed in the light of recent extensions of traditional power indices.     

CHARACTERIZATION OF OPTIMAL STOPPING REGIONS OF AMERICAN ASIAN AND LOOKBACK OPTIONS

A general framework is developed to analyze the optimal stopping (exercise) regions of American path-dependent options with either the Asian feature or lookback feature. We examine the monotonicity properties of the option values and stopping regions with respect to the interest rate, dividend yield, and time. From the ordering properties of the values of American lookback options and American Asian options, we deduce the corresponding nesting relations between the exercise regions of these American options. We illustrate how some properties of the exercise regions of the American Asian options can be inferred from those of the American lookback options.     

The Neglect of Monotone Comparative Statics Methods

Monotone methods enable comparative static analysis without the restrictive assumptions of the implicit-function theorem. Ease of use and flexibility in solving comparative static and game-theory problems have made monotone methods popular in the economics literature and in graduate courses, but they are still absent from undergraduate mathematical economics courses and textbooks. In this article, the authors illustrate the generality of monotone comparative statics relative to the implicit function approach. For example, to sign the effect of a discrete policy shift on a choice variable, the marginal returns will increase with the policy parameter. They also apply monotone methods in game theory settings. As mathematical economics courses and majors gain popularity, incorporating monotone methods into curriculum and textbooks would provide a modern treatment of comparative static analysis.     

Virtual Nash implementation with admissible support

A social choice correspondence (SCC) is virtually implementable if it is ?$?$-close (in the probability simplex) to some (exactly) implementable correspondence [Abreu, D., Sen, A., 1991. Virtual Implementation in Nash Equilibrium. Econometrica 59, 997–1021] proved that, without restriction on the set of alternatives receiving strictly positive probability at equilibrium, every SCC is virtually implementable in Nash Equilibrium. We study virtual implementation when the supports of equilibrium lotteries are restricted. We provide a necessary and sufficient condition, imposing joint restrictions on SCCs and admissible supports. Next, we discuss how to construct supports, and we underline an important difficulty. Finally, we study virtual implementation when the support is restricted to the efficient or individually rational alternatives.     

Axiomatizations of Dutta-Ray’s egalitarian solution on the domain of convex games

We show that on the domain of convex games, Dutta-Ray’s egalitarian solution is characterized by core selection, aggregate monotonicity, and bounded richness, a new property requiring that the poorest players cannot be made richer within the core. Replacing “poorest” by “poorer” allows to eliminate aggregate monotonicity. Moreover, we show that the egalitarian solution is characterized by constrained welfare egalitarianism and either bilateral consistency à la Davis and Maschler or, together with individual rationality, by bilateral consistency à la Hart and Mas-Colell.