Strong Berge and Pareto Equilibrium Existence for a Noncooperative Game

In this paper, we study the main properties of the strong Berge equilibrium which is also a Pareto efficient (SBPE) and the strong Nash equilibrium (SNE). We prove that any SBPE is also a SNE, we prove also existence theorem of SBPE based on the KyFan inequality. Finally, we also provide a method for computing SPBE.Strong Berge equilibrium, Pareto efficiency, strong Nash equilibrium, Ky Fan inequality

Constrained School Choice

Recently, several school districts in the US have adopted or consider adopting the Student-Optimal Stable mechanism or the Top Trading Cycles mechanism to assign children to public schools. There is evidence that for school districts that employ (variants of) the so-called Boston mechanism the transition would lead to efficiency gains. The first two mechanisms are strategy-proof, but in practice student assignment procedures typically impede a student to submit a preference list that contains all his acceptable schools. We study the preference revelation game where students can only declare up to a fixed number of schools to be acceptable. We focus on the stability and efficiency of the Nash equilibrium outcomes. Our main results identify rather stringent necessary and sufficient conditions on the priorities to guarantee stability or efficiency of either of the two mechanisms. This stands in sharp contrast with the Boston mechanism which has been abandoned in many US school districts but nevertheless yields stable Nash equilibrium outcomes.school choice, matching, stability, Gale-Shapley deferred acceptance algorithm, top trading cycles, Boston mechanism, acyclic priority structure, truncation

Constrained school choice

Recently, several school districts in the US have adopted or consider adopting the Student-Optimal Stable Mechanism or the Top Trading Cycles Mechanism to assign children to public schools. There is clear evidence that for school districts that employ (variants of) the so-called Boston Mechanism the transition would lead to efficiency gains. The first two mechanisms are strategy-proof, but in practice student assignment procedures impede students to submit a preference list that contains all their acceptable schools. Therefore, any desirable property of the mechanisms is likely toget distorted. We study the non trivial preference revelation game where students can only declare up to a fixed number (quota) of schools to be acceptable. We focus on the stability of the Nash equilibrium outcomes. Our main results identify rather stringent necessary and sufficient conditions on the priorities to guaranteestability. This stands in sharp contrast with the Boston Mechanism which yields stable Nash equilibrium outcomes, independently of the quota. Hence, the transition to any of the two mechanisms is likely to come with a higher risk that students seek legal actionas lower priority students may occupy more preferred schools

A Crash Course in Implementation Theory

These lectures are meant to familiarize the audience with some of the fundamental results in the theory of implementation and provide a quick progression to some open questions in the literature

Games implementing the stable rule of marriage problems in strong Nash equilibria

In a marriage problem, we introduce a condition called "exclusive matchability (EM)": the condition mainly says that each pair of a man and a woman can choose to be a matching pair regardless of others' actions. This condition is essential to strong Nash implementation of the stable rule. We show that any mechanism which satisfies exclusive matchability implements the stable rule in strong Nash equilibria.