Tiered Random Matching Markets: Rank Is Proportional to Popularity

We study the stable marriage problem in two-sided markets with randomly generated preferences. Agents on each side of the market are divided into a constant number of "soft" tiers, which capture agents\u27 qualities. Specifically, every agent within a tier has the same public score, and agents on each side have preferences independently generated proportionally to the public scores of the other side. We compute the expected average rank which agents in each tier have for their partners in the man-optimal stable matching, and prove concentration results for the average rank in asymptotically large markets. Furthermore, despite having a significant effect on ranks, public scores do not strongly influence the probability of an agent matching to a given tier of the other side. This generalizes the results by Pittel [Pittel, 1989], which analyzed markets with uniform preferences. The results quantitatively demonstrate the effect of competition due to the heterogeneous attractiveness of agents in the market

British male students continue to fall behind in secondary school achievement

It is common knowledge that boys fall behind in school performance, and UK policy makers have addressed this issue in the past decade. In fact, they seem committed to narrowing gender gaps of any kind. This paper asks whether actual progress has been made in reducing the degree to which boys fall behind, and also whether gender differences in subject preference have changed in the period 2001 to 2013. Using an analysis of British secondary-education exam data and a comparison with data from the Programme for International Student Assessment (PISA), it is concluded that no progress has been made: Boys attained fewer top grades in nearly all school subjects. Further, boys and girls continue to choose elective school subjects along traditional interest lines. The problem of boys falling behind is obscured by the finding that grades of all children have risen considerably in this period. However, a comparison of Mathematics and English exam grades with PISA data suggests that this rise is due to grade inflation, not real improvement. The paper closes with recommendations for solutions

British male students continue to fall behind in secondary education

It is common knowledge that boys fall behind in school performance, and UK policy makers have addressed this issue in the past decade. In fact, they seem committed to narrowing gender gaps of any kind. This paper asks whether actual progress has been made in reducing the degree to which boys fall behind, and also whether gender differences in subject preference have changed in the period 2001 to 2013. Using an analysis of British secondary-education exam data and a comparison with data from the Programme for International Student Assessment (PISA), it is concluded that no progress has been made: Boys attained fewer top grades in nearly all school subjects. Further, boys and girls continue to choose elective school subjects along traditional interest lines. The problem of boys falling behind is obscured by the finding that grades of all children have risen considerably in this period. However, a comparison of Mathematics and English exam grades with PISA data suggests that this rise is due to grade inflation, not real improvement. The paper closes with recommendations for solutions

Love and Money: A Theoretical and Empirical Analysis of Household Sorting and Inequality

This paper examines the interactions between household matching, inequality, and per capita income. We develop a model in which agents decide whether to become skilled or unskilled, form households, consume and have children. We show that the equilibrium sorting of spouses by skill type (their correlation in education) is increasing as a function of the skill premium. In the absence of perfect capital markets, the economy can converge to different steady states, depending upon initial conditions. The degree of marital sorting, wage inequality, and fertility differentials are positively correlated across steady states and negatively correlated with per capita income. We use household surveys from 34 countries to construct several measures of the skill premium and of the degree of correlation of spouses' education (marital sorting). For all our measures, we find a positive and significant relationship between the two variables.

Stable Matching: Choosing Which Proposals to Make

Why does stable matching work well in practice despite agents only providing short preference lists? Perhaps the most compelling explanation is due to Lee (Lee, 2016). He considered a model with preferences based on correlated cardinal utilities. The utilities were based on common public values for each agent and individual private values. He showed that for suitable utility functions, in large markets, for most agents, all stable matchings yield similar valued utilities. By means of a new analysis, we strengthen this result, showing that in large markets, with high probability, for \emph{all} but the bottommost agents, all stable matches yield similar valued utilities. We can then deduce that for \emph{all} but the bottommost agents, relatively short preference lists suffice. Our analysis shows that the key distinction is between models in which the derivatives of the utility function are bounded, and those in which they can be unbounded. For the bounded derivative model, we show that for any given constant $c\ge 1$, with $n$ agents on each side of the market, with probability $1 - n^{-c}$, for each agent its possible utilities in all stable matches vary by at most $O((c\ln n/n)^{1/3})$ for all but the bottommost $O((c\ln n/n)^{1/3})$ fraction of the agents. When the derivatives can be unbounded, we obtain the following bound on the utility range and the bottommost fraction: for any constant $\epsilon>0$, for large enough $n$, the bound is $\epsilon$. Both these bounds are tight. In the bounded derivative model, we also show the existence of an $\epsilon$-Bayes-Nash equilibrium in which agents make relatively few proposals. Our results all rely on a new technique for sidestepping the conditioning between the matching events that occur over the course of a run of the Deferred Acceptance algorithm. We complement these theoretical results with a variety of simulation experiments.Comment: 52 page