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.
Analysis of random instances of the linear sum assignment problem was first carried by @cite_24 , and was subsequently improved by @cite_13 @cite_25 @cite_22 . For a survey on the topic and other related literature see @cite_0 .
@cite_22 @cite_24 @cite_0 @cite_13 @cite_25
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Given an n by n matrix X, the assignment problem asks for a set of n entries, one from each column and row, with the minimum sum. It is shown that the expected value of this minimum sum is less than 3, independent of n, if X consists of independent random variables uniformly distributed from 0 to 1. This paper analyzes the most efficient algorithms for the Linear Min-Sum Assignment Problem and shows that they derive from a common basic procedure. For each algorithm, we evaluate the computational complexity and the average performance on randomly-generated test problems. Efficient FORTRAN implementations for the case of complete and sparse matrices are given. Analysis of random instances of optimization problems provides valuable insights into the behavior and properties of problem's solutions, feasible region, and optimal values, especially in large-scale cases. A class of problems that have been studied extensively in the literature using the methods of probabilistic analysis is represented by the assignment problems, and many important problems in operations research and computer science can be formulated as assignment problems. This paper presents an overview of the recent results and developments in the area of probabilistic assignment problems, including the linear and multidimensional assignment problems, quadratic assignment problem, etc. We present randomized algorithms that solve Subset Sum and Knapsack instances with n items in O*(20.86n) time, where the O*(·) notation suppresses factors polynomial in the input size, and polynomial space, assuming random read-only access to exponentially many random bits. These results can be extended to solve Binary Linear Programming on n variables with few constraints in a similar running time. We also show that for any constant k≥ 2, random instances of k-Sum can be solved using O(nk-0.5(n)) time and O(logn) space, without the assumption of random access to random bits. Underlying these results is an algorithm that determines whether two given lists of length n with integers bounded by a polynomial in n share a common value. Assuming random read-only access to random bits, we show that this problem can be solved using O(logn) space significantly faster than the trivial O(n2) time algorithm if no value occurs too often in the same list.