Let [gamma][subscript]X = (X[subscript]sp(1)',X[subscript]sp(2)',...,X[subscript]sp(n)') and [gamma][subscript]Y = Y[subscript]sp(1)',Y[subscript]sp(2)',...,Y[subscript]sp(n)') be two groups of stochastically increasing rv's, which can represent, say, the increasing strengths of the members of two chess teams or two tennis teams, etc. Let [pi] = ([pi][subscript]1,[pi][subscript]2,...,[pi][subscript]n) be a permutation of (1,2,...,n). Then the statistic S([pi]) = [sigma][subscript]spi=1n I(Y[subscript]sp(i)' > X[subscript]sp(i)') measures the performance of [gamma][subscript]Y over [gamma][subscript]X under the permutation or matching [pi], where I(y > x) is an indicator function. We are interested in the relationship between [pi] and ES([pi]) = [sigma][subscript]spi = 1n P(Y[subscript]sp(i)' > X[subscript]sp([pi][subscript] i)'), especially in comparing ES([pi]) when [pi] = (1,2,..., n), corresponding to ordered matching, and when [pi] is randomly given. The P(Y[subscript]sp(i)' > X[subscript]sp(i)') are of interest in themselves. A class of special matchings called 'fair matchings' is emphasized. We say a matching [pi] is fair if ES([pi]) = n[over] 2 when [gamma][subscript]X ~ [gamma][subscript]Y. Simple matching and symmetric matching, which are fair under certain conditions, are also defined. The problems are investigated under two models, i.e., the order statistics model and the linear preference model;In the order statistics model, we assume that X[subscript]sp(i)' and Y[subscript]sp(i)' have the same marginal distributions as X[subscript](i) and Y[subscript](i), the i-th order statistics in two random samples of size n from F(x) and G(x), respectively. In this case, ES([pi]) = [sigma][subscript]spi = 1n P(Y[subscript](i) > X[subscript]([pi][subscript] i)), and when [pi] is randomly given ES([pi]) = [sigma][subscript]spi =1n P(Y[subscript]i > X[subscript]i), where (X[subscript]1,X[subscript]2,...,X[subscript]n) and (Y[subscript]1,Y[subscript]2,...,Y[subscript]n) are random samples from F(x) and G(x), respectively. If G(x) = F(x - [mu]), where [mu] ≥ 0, then it is shown that [sigma][subscript]spi = 1n P(Y[subscript](i) > X[subscript](i)) ≥ [sigma][subscript]spi = 1n P(Y[subscript]i > X[subscript]i). Moreover, we have [sigma][subscript]spi = 1n P(Y[subscript](i) > X[subscript](i)) ≥ ES([pi]), for any simple matching [pi]. If F(x) is a distribution function of a symmetric rv, then this inequality holds also for any symmetric matching [pi];In the linear preference model, we assume that X[subscript]sp(i)' ~ F(x - [lambda][subscript](i)) and Y[subscript]sp(i)' ~ F(x - [mu][subscript](i)), for i = 1,2,..., n, where F(x) is a unimodal distribution function, [lambda][subscript](1) ≤ [lambda][subscript](2) ≤ ... ≤ [lambda][subscript](n), and [mu][subscript](1) ≤ [mu][subscript](2) ≤ ... ≤ [mu][subscript](n). If U(x) is the cdf of Z[subscript]1 - Z[subscript]2, where Z[subscript]1 and Z[subscript]2 are iid with cdf F(x), the expectation of S([pi]) can be written as E[S([pi])] = [sigma][subscript]spi = 1n U([mu][subscript](i) - [lambda][subscript]([pi][subscript] i)). Under certain conditions, we get similar results to those in the order statistics model. We also obtain some other miscellaneous results about ordered matching, as well as maximization and rearrangement properties of ES([pi]).;In both models, the case with ties permitted is also considered.