Popular Matchings with One-Sided Bias

Let $G = (A \cup B,E)$ be a bipartite graph where the set $A$ consists of agents or main players and the set $B$ consists of jobs or secondary players. Every vertex has a strict ranking of its neighbors. A matching $M$ is popular if for any matching $N$, the number of vertices that prefer $M$ to $N$ is at least the number that prefer $N$ to $M$. Popular matchings always exist in $G$ since every stable matching is popular. A matching $M$ is $A$-popular if for any matching $N$, the number of agents (i.e., vertices in $A$) that prefer $M$ to $N$ is at least the number of agents that prefer $N$ to $M$. Unlike popular matchings, $A$-popular matchings need not exist in a given instance $G$ and there is a simple linear time algorithm to decide if $G$ admits an $A$-popular matching and compute one, if so. We consider the problem of deciding if $G$ admits a matching that is both popular and $A$-popular and finding one, if so. We call such matchings fully popular. A fully popular matching is useful when $A$ is the more important side -- so along with overall popularity, we would like to maintain ``popularity within the set $A$''. A fully popular matching is not necessarily a min-size/max-size popular matching and all known polynomial-time algorithms for popular matching problems compute either min-size or max-size popular matchings. Here we show a linear time algorithm for the fully popular matching problem, thus our result shows a new tractable subclass of popular matchings.Comment: A preliminary version of this paper appeared in Proc. of the 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020), 70:1--70:18, 202

Popular Matchings with One-Sided Bias

Let G = (A ? B,E) be a bipartite graph where A consists of agents or main players and B consists of jobs or secondary players. Every vertex has a strict ranking of its neighbors. A matching M is popular if for any matching N, the number of vertices that prefer M to N is at least the number that prefer N to M. Popular matchings always exist in G since every stable matching is popular. A matching M is A-popular if for any matching N, the number of agents (i.e., vertices in A) that prefer M to N is at least the number of agents that prefer N to M. Unlike popular matchings, A-popular matchings need not exist in a given instance G and there is a simple linear time algorithm to decide if G admits an A-popular matching and compute one, if so. We consider the problem of deciding if G admits a matching that is both popular and A-popular and finding one, if so. We call such matchings fully popular. A fully popular matching is useful when A is the more important side - so along with overall popularity, we would like to maintain "popularity within the set A". A fully popular matching is not necessarily a min-size/max-size popular matching and all known polynomial time algorithms for popular matching problems compute either min-size or max-size popular matchings. Here we show a linear time algorithm for the fully popular matching problem, thus our result shows a new tractable subclass of popular matchings

College admissions and the role of information : an experimental study

We analyze two well-known matching mechanisms—the Gale-Shapley, and the Top Trading Cycles (TTC) mechanisms—in the experimental lab in three different informational settings, and study the role of information in individual decision making. Our results suggest that—in line with the theory—in the college admissions model the Gale-Shapley mechanism outperforms the TTC mechanisms in terms of efficiency and stability, and it is as successful as the TTC mechanism regarding the proportion of truthful preference revelation. In addition, we find that information has an important effect on truthful behavior and stability. Nevertheless, regarding efficiency, the Gale-Shapley mechanism is less sensitive to the amount of information participants hold

Finding large stable matchings

When ties and incomplete preference lists are permitted in the stable marriage and hospitals/residents problems, stable matchings can have different sizes. The problem of finding a maximum cardinality stable matching in this context is known to be NP-hard, even under very severe restrictions on the number, size, and position of ties. In this article, we present two new heuristics for finding large stable matchings in variants of these problems in which ties are on one side only. We describe an empirical study involving these heuristics and the best existing approximation algorithm for this problem. Our results indicate that all three of these algorithms perform significantly better than naive tie-breaking algorithms when applied to real-world and randomly-generated data sets and that one of the new heuristics fares slightly better than the other algorithms, in most cases. This study, and these particular problem variants, are motivated by important applications in large-scale centralized matching schemes

College admissions and the role of information : an experimental study

We analyze two well-known matching mechanisms—the Gale-Shapley, and the Top Trading Cycles (TTC) mechanisms—in the experimental lab in three different informational settings, and study the role of information in individual decision making. Our results suggest that—in line with the theory—in the college admissions model the Gale-Shapley mechanism outperforms the TTC mechanisms in terms of efficiency and stability, and it is as successful as the TTC mechanism regarding the proportion of truthful preference revelation. In addition, we find that information has an important effect on truthful behavior and stability. Nevertheless, regarding efficiency, the Gale-Shapley mechanism is less sensitive to the amount of information participants hold.Experiments, Information, Matching

College Admissions and the Role of Information: An Experimental Study

We analyze two well-known matching mechanisms\the Gale-Shapley, and the Top Trading Cycles (TTC) mechanisms\in theexperimental lab in three different informational settings, and study the role of information in individual decision making. Our results suggest that\in line with the theory\in the college admissions model the Gale-Shapley mechanism outperforms the TTC mechanisms in terms of efficiency and stability, and it is as successful as the TTC mechanism regarding the proportion of truthful preference revelation. In addition, we find that information has an important effect on truthful behavior and stability. Nevertheless, regarding efficiency, the Gale-Shapley mechanism is less sensitive to the amount of information participants hold.

Social status in economic theory: a review.

Social distinction or status is an important motivation of human behaviour. This paper provides a selective survey of recent advances in the economic analysis of the origins and consequences of social status. First, a selection of empirical research from a variety of scientific disciplines is discussed to underpin the further theoretical analysis. I then consider the origins and determinants of tastes for status, discuss the endogenous derivation of such a preferences for relative standing and assess the different formalisations these preferences. Subsequently, the consequences of preferences for status are studied for a variety of problems and settings. The last section discusses a number of implications of status concerns for normative economics and public policy.

Popular Edges with Critical Nodes

In the popular edge problem, the input is a bipartite graph G = (A ? B,E) where A and B denote a set of men and a set of women respectively, and each vertex in A? B has a strict preference ordering over its neighbours. A matching M in G is said to be popular if there is no other matching M\u27 such that the number of vertices that prefer M\u27 to M is more than the number of vertices that prefer M to M\u27. The goal is to determine, whether a given edge e belongs to some popular matching in G. A polynomial-time algorithm for this problem appears in [Cseh and Kavitha, 2018]. We consider the popular edge problem when some men or women are prioritized or critical. A matching that matches all the critical nodes is termed as a feasible matching. It follows from [Telikepalli Kavitha, 2014; Kavitha, 2021; Nasre et al., 2021; Meghana Nasre and Prajakta Nimbhorkar, 2017] that, when G admits a feasible matching, there always exists a matching that is popular among all feasible matchings. We give a polynomial-time algorithm for the popular edge problem in the presence of critical men or women. We also show that an analogous result does not hold in the many-to-one setting, which is known as the Hospital-Residents Problem in literature, even when there are no critical nodes