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We consider the problem of maximizing welfare when allocating m items to n players with subadditive utility functions. Our main result is a way of rounding any fractional solution to a linear programming relaxation to this problem so as to give a feasible solution of welfare at least 1/2 that of the value of the fractional solution. This approximation ratio of 1/2 is an improvement over an Ω(1/ log m) ratio of Dobzinski, Nisan, and Schapira [Proceedings of the 37th Annual ACM Symposium on Theory of Computing (Baltimore, MD), ACM, New York, 2005, pp. 610-618]. We also show an approximation ratio of 1 - 1/e when utility functions are fractionally subadditive. A result similar to this last result was previously obtained by Dobzinski and Schapira [Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms (Miami, FL), SIAM, Philadelphia, 2006, pp. 1064-1073], but via a different rounding technique that requires the use of a so-called "XOS oracle." The randomized rounding techniques that we use are oblivious in the sense that they only use the primal solution to the linear program relaxation, but have no access to the actual utility functions of the players. [ABSTRACT FROM AUTHOR]