Entropic selection of Nash equilibrium

This study argues that Nash equilibria with less variations in players' best responses are more appealing. To that regard, a notion measuring such variations, the entropic selection of Nash equilibrium, is presented: For any given Nash equilibrium, we consider the cardinality of the support of a player's best response against others' strategies that are sufficiently close to the behavior specified. These cardinalities across players are then aggregated with a real-valued function on whose form we impose no restrictions apart from the natural limitation to nondecreasingness in order to obtain equilibria with less variations. We prove that the entropic selection of Nash equilibrium is non-empty and admit desirable properties. Some well-known games, each of which display important insights about virtues / problems of various equilibrium notions, are considered; and, in all of these games our notion displays none of the criticisms associated with these examples. These examples also show that our notion does not have any containment relations with other associated and well-known refinements, perfection, properness and persistence

Aggregate efficiency in random assignment problems

We introduce aggregate efficiency (AE) for random assignments (RA) by requiring higher expected numbers of agents be assigned to their more preferred choices. It is shown that the realizations of any aggregate efficient random assignment (AERA) must be an AE permutation matrix. While AE implies ordinally efficiency, the reverse does not hold. And there is no mechanism treating equals equally while satisfying weak strategyproofness and AE. But, a new mechanism, the reservation-1 (R1), is identified and shown to provide an improvement on grounds of AE over the probabilistic serial mechanism of Bogomolnaia and Moulin (2001). We prove that R1 is weakly strategyproof, ordinally efficient, and weak envy--free. Moreover, the characterization of R1 displays that it is the probabilistic serial mechanism updated by a principle decreed by the Turkish parliament concerning the random assignment of new doctors: Modifying the axioms of Hasimoto, et. al. (2012) characterizing the probabilistic serial mechanism to satisfy this principle, fully characterizes R1

Tenacious selection of Nash equilibrium

We propose a complexity measure and an associated refinement based on the observation that best responses with more variations call for more precise anticipation. The variations around strategy profiles are measured by considering the cardinalities of players' pure strategy best responses when others' behavior is perturbed. After showing that the resulting selection method displays desirable properties, it is employed to deliver a refinement: the tenacious selection of Nash equilibrium. We prove that it exists; does not have containment relations with perfection, properness, persistence and other refinements; and possesses some desirable features

Sticky matching in school choice

We introduce the notion of sticky-stability in order to accommodate appeal costs in real-life school-choice systems. When appealing is costly, students (or parents) may not find it worth appealing, even if their priorities are violated in their preferred schools. In order to incorporate this into the school-choice setting, we consider the vector of parameters, namely the profile of students' stickiness degrees resulting from students' cost-benefit analyses and consisting of each student's least rank difference associated with appeals he/she finds beneficial. Then, sticky-stability rules out only appeal-causing priority violations. Consequently, we introduce the following two mechanisms, both of which elicit these parameters from students: "efficiency-improving deferred-acceptance mechanism" (EIDA) and "efficiency-corrected deferred-acceptance mechanism (ECDA). We show that both are sticky-stable and dominate stable matchings in terms of efficiency. Furthermore, the latter is efficient within the class of sticky-stable mechanisms. In a complete information setting, both are manipulable while the EIDA is immune to manipulations via stickiness degree misreporting. However, if students have limited information, then the EIDA becomes robust to manipulations, whereas the ECDA continues to be vulnerable, but with a diminished scope