Characteristic function (convex analysis)

Abstract: We study here the problem of modifying the (Shapley) value of a characteristic function game so as to take into account the possibility that some players — because of personal or political affinities — may be more likely to act together than others. We shall use y[v] to denote the usual value of the game v.

Abstract: Automatic phase-picking algorithms are designed to detect a seismic signal on a single trace and to time the arrival of the signal precisely. Because of the requirement for precise timing, a phase-picking algorithm is inherently less sensitive than one designed only to detect the presence of a signal, but still can approach the performance of a skilled analyst. A typical algorithm filters the input data and then generates a function characterizing the seismic time series. This function may be as simple as the absolute value of the series, or it may be quite complex. Event detection is accomplished by comparing the function or its short-term average (STA ) with a threshold value (THR), which is commonly some multiple of a long-term average (LTA) of a characteristic function. If the STA exceeds THR, a trigger is declared. If the event passes simple criteria, it is reported. Sensitivity, expected timing error, false-trigger rate, and false-report rate are interrelated measures of performance controlled by choice of the characteristic function and several operating parameters. At present, computational power limits most systems to one-pass, time-domain algorithms. Rapidly advancing semi-conductor technology, however, will make possible much more powerful multi-pass approaches incorporating frequency-domain detection and pseudo-offline timing.

Abstract: We present a simple and easily calculated expression for the Shapley value whenever the characteristic function is a “cost” function with the property that the cost of any subset of players is equal to the cost of the “largest” player in that subset. It turns out that a simple rule previously proposed for calculating airport landing charges generates precisely the Shapley value for an appropriately defined game.

Abstract: This paper gives a formulation of a theory of n-person cooperative games with side payments in terms of a partition function which is defined on the set of all partitions of the set of players. The results for all games with n ≤ 3 are presented. This development generalizes the von Neumann-Morgenstern theory of games in characteristic function form.