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- W2469842011 abstract "This paper studies the approximation of pseudo-Boolean functions by linear functions and more generally by functions of (at most) a specified degree. Here a pseudo-Boolean function means a real valued function defined on {0, 1} n, and its degree is that of the unique multilinear polynomial that expresses it; linear functions are those of degree at most one. The approximation consists in choosing among all linear functions the one which is closest to a given function, where distance is measured by the Euclidean metric on R 2n. A characterization of the best linear approximation is ob- tained in terms of the average value of the function and its first derivatives. This leads to an explicit formula for computing the approximation from the polynomial expression of the given function. These results are later generalized to handle approximations of higher degrees, and further results are obtained regarding the interaction of approximations of different degrees. For the linear case, a certain constrained version of the approximation problem is also studied. Special attention is given to some important properties of pseudo-Boolean functions and the extent to which they are preserved in the approximation. A separate section points out the relevance of linear approximations to game theory and shows that the well known Banzhaf power index and Shapley value are obtained as best" @default.
- W2469842011 created "2016-07-22" @default.
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- W2469842011 date "2016-12-31" @default.
- W2469842011 modified "2023-09-23" @default.
- W2469842011 title "7. Applications to Game Theory" @default.
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- W2469842011 doi "https://doi.org/10.1515/9781400880928-008" @default.
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