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### Abstract

Utility functions form an essential part of game theory and economics.
In order to guarantee the existence of utility functions most of the
time sufficient properties are assumed in an axiomatic manner. One
famous and very common set of such assumptions is that of expected
utility theory. Here, the rationality, continuity, and independence of
preferences is assumed. The von-Neumann-Morgenstern Utility theorem
shows that these assumptions are necessary and sufficient for an
expected utility function to exists. This theorem was proven by
Neumann and Morgenstern in ``Theory of Games and Economic
Behavior'' which is regarded as one of the most influential
works in game theory. The formalization includes formal definitions of
the underlying concepts including continuity and independence of
preferences.