Behind OWA Lead to a Universal and OptimalApproximation

Ordered Weighted Averaging (OWA) operators have been successfully applied in many practical problems. We explain this empirical success by showing that these operators are indeed guaranteed to work (i.e., are universal), and that these operators are the best to use (in some reasonable sense). 1 Aggregation is needed In many areas of science and engineering, we have several estimates x 1 ; : : : ; x n for the same quantity x. These estimates may come from measurements and/or they may come from experts. and we want to combine them into a single (better) estimate y. Several techniques have been successfully used to aggregate diierent estimates. The most widely used idea is to take an arithmetic average y = x 1 + : : : + x n n : In the arithmetic average, we combine all the estimates with equal weights. In some practical situations , it makes sense to give move weight to consistent estimates and less weight to estimates that are far away from the consensus of the majority. For example , in some sports competitions, the lowest and the highest scores are deleted, and the average of the remaining values is taken as the resulting aggregate. In more precise terms, this aggregating operation y = f(x 1 ; where x (1) is the smallest of the n values x 1 ; : : : ; x n , x (2) is the second smallest, etc. Instead of simply ignoring the outstanding estimates (i.e., assigning them 0 weight), we can give them smaller weight depending on their deviation from the others. For example, we can compute the mean x and the standard deviation of the original n estimates , and then combine then with weights proportional to s(jx i ? xj==), where s(z) is a decreasing function. How can we describe diierent possible aggregation techniques? 2 Linearization: what is it and why it is a widely used application tool One of the main tools of applied mathematics is lin-earization; see, e.g., 1]. The need for some tool of this type comes from the fact that the actual dependence y = f(x 1 ; : : : ; x n) between physical quantities can be very complex and thus, very diicult to analyze. However, it is usually smooth (diierentiable). As a result, when we know the approximate values e x 1