Abstract:
We consider the egalitarian welfare aspects of random assignment mechanisms when agents have unrestricted cardinal utilities over the objects. We give bounds on how well different random assignment mechanisms approximate the optimal egalitarian value and investigate the effect that different well-known properties like ordinality, envy-freeness, and truthfulness have on the achievable egalitarian value. Finally, we conduct detailed experiments analyzing the tradeoffs between efficiency with envy-freeness or truthfulness using two prominent random assignment mechanisms --- random serial dictatorship and the probabilistic serial mechanism --- for different classes of utility functions and distributions.

Abstract:
Optimal assignment of classes to classrooms \cite{dickey}, design of DNA microarrays \cite{carvalho}, cross species gene analysis \cite{kolar}, creation of hospital layouts cite{elshafei}, and assignment of components to locations on circuit boards \cite{steinberg} are a few of the many problems which have been formulated as a quadratic assignment problem (QAP). Originally formulated in 1957, the QAP is one of the most difficult of all combinatorial optimization problems. Here, we use statistical mechanical methods to study the asymptotic behavior of problems in which the entries of at least one of the two matrices that specify the problem are chosen from a random distribution $P$. Surprisingly, this case has not been studied before using statistical methods despite the fact that the QAP was first proposed over 50 years ago \cite{Koopmans}. We find simple forms for $C_{\rm min}$ and $C_{\rm max}$, the costs of the minimal and maximum solutions respectively. Notable features of our results are the symmetry of the results for $C_{\rm min}$ and $C_{\rm max}$ and the dependence on $P$ only through its mean and standard deviation, independent of the details of $P$. After the asymptotic cost is determined for a given QAP problem, one can straightforwardly calculate the asymptotic cost of a QAP problem specified with a different random distribution $P$.

Abstract:
We give a conjecture for the expected value of the optimal k-assignment in an m x n-matrix, where the entries are all exp(1)-distributed random variables or zeros. We prove this conjecture in the case there is a zero-cost $k-1$-assignment. Assuming our conjecture, we determine some limits, as $k=m=n\to \infty$, of the expected cost of an optimal n -assignment in an n x n-matrix with zeros in some region. If we take the region outside a quarter-circle inscribed in the square matrix, this limit is thus conjectured to be $\pi^2/24$. We give a computer-generated verification of a conjecture of Parisi for k=m=n=7 and of a conjecture of Coppersmith and Sorkin for $k\leq 5$. We have used the same computer program to verify this conjecture also for k=6.

Abstract:
We consider the problem of minimizing cost among one-to-one assignments of $n$ jobs onto $n$ machines. The random assignment problem refers to the case when the cost associated with performing jobs on machines are random variables. Aldous established the expected value of the smallest cost, $A_n$, in the limiting $n$ regime. However the distribution of the minimum cost has not been established yet. In this paper we conjecture some distributional properties of matchings in matrices. If this conjecture is proved, this will establish that $\sqrt{n}(A_n - E(A_n)) \overset{w}{\Rightarrow} N(0,2)$. We also establish the limiting distribution for a special case of the Random Assignment Problem.

Abstract:
We consider the multi-unit random assignment problem in which agents express preferences over objects and objects are allocated to agents randomly based on the preferences. The most well-established preference relation to compare random allocations of objects is stochastic dominance (SD) which also leads to corresponding notions of envy-freeness, efficiency, and weak strategyproofness. We show that there exists no rule that is anonymous, neutral, efficient and weak strategyproof. For single-unit random assignment, we show that there exists no rule that is anonymous, neutral, efficient and weak group-strategyproof. We then study a generalization of the PS (probabilistic serial) rule called multi-unit-eating PS and prove that multi-unit-eating PS satisfies envy-freeness, weak strategyproofness, and unanimity.

Abstract:
We continue the study of the assignment problem for a random cost matrix. We analyse the number of $k$-cycles for the solution and their dependence on the symmetry of the random matrix. We observe that for a symmetric matrix one and two-cycles are dominant in the optimal solution. In the antisymmetric case the situation is the opposite and the one and two-cycles are suppressed. We solve the model for a pure random matrix (without correlations between its entries) and give analytic arguments to explain the numerical results in the symmetric and antisymmetric case. We show that the results can be explained to great accuracy by a simple ansatz that connects the expected number of $k$-cycles to that of one and two cycles.

Abstract:
The random assignment (or bipartite matching) problem studies the random total cost A_n of the optimal assignment of each of n jobs to each of n machines, where the costs of the n^2 possible job-machine matches has exponential (mean 1) distribution. Mezard - Parisi (1987) used the replica method from statistical physics to argue non-rigorously that EA_n converges to zeta(2) = pi^2/6. Aldous (1992) identified the limit as the optimal solution of a matching problem on an infinite tree. Continuing that approach, we construct the optimal matching on the infinite tree. This yields a rigorous proof of the zeta(2) limit and of the conjectured limit distribution of edge-costs and their rank-orders in the optimal matching.

Abstract:
Assignment games represent a tractable yet versatile model of two-sided markets with transfers. We study the likely properties of the core of randomly generated assignment games. If the joint productivities of every firm and worker are i.i.d bounded random variables, then with high probability all workers are paid roughly equal wages, and all firms make similar profits. This implies that core allocations vary significantly in balanced markets, but that there is core convergence in even slightly unbalanced markets. For the benchmark case of uniform distribution, we provide a tight bound for the workers' share of the surplus under the firm-optimal core allocation. We present simulation results suggesting that the phenomena analyzed appear even in medium-sized markets. Finally, we briefly discuss the effects of unbounded distributions and the ways in which they may affect wage dispersion.

Abstract:
In the Matroid Secretary Problem, introduced by Babaioff et al. [SODA 2007], the elements of a given matroid are presented to an online algorithm in random order. When an element is revealed, the algorithm learns its weight and decides whether or not to select it under the restriction that the selected elements form an independent set in the matroid. The objective is to maximize the total weight of the chosen elements. In the most studied version of this problem, the algorithm has no information about the weights beforehand. We refer to this as the zero information model. In this paper we study a different model, also proposed by Babaioff et al., in which the relative order of the weights is random in the matroid. To be precise, in the random assignment model, an adversary selects a collection of weights that are randomly assigned to the elements of the matroid. Later, the elements are revealed to the algorithm in a random order independent of the assignment. Our main result is the first constant competitive algorithm for the matroid secretary problem in the random assignment model. This solves an open question of Babaioff et al. Our algorithm achieves a competitive ratio of $2e^2/(e-1)$. It exploits the notion of principal partition of a matroid, its decomposition into uniformly dense minors, and a $2e$-competitive algorithm for uniformly dense matroids we also develop. As additional results, we present simple constant competitive algorithms in the zero information model for various classes of matroids including cographic, low density and the case when every element is in a small cocircuit. In the same model, we also give a $ke$-competitive algorithm for $k$-column sparse linear matroids, and a new $O(\log r)$-competitive algorithm for general matroids of rank $r$ which only uses the relative order of the weights seen and not their numerical value, as previously needed.

Abstract:
An assignment problem is the optimization problem of finding, in an m by n matrix of nonnegative real numbers, k entries, no two in the same row or column, such that their sum is minimal. Such an optimization problem is called a random assignment problem if the matrix entries are random variables. We give a formula for the expected value of the optimal k-assignment in a matrix where some of the entries are zero, and all other entries are independent exponentially distributed random variables with mean 1. Thereby we prove the formula 1+1/4+1/9+...+1/k^2 conjectured by G. Parisi for the case k=m=n, and the generalized conjecture of D. Coppersmith and G. B. Sorkin for arbitrary k, m and n.