Games with Comparative Utility Functions

The central objects of investigation of this paper are games with comparative utility functions. This concept can be considered as a relativistic approach to expected utility theory. The approach covers also the classical idea of utility functions. In some recent research papers the philosophic justification for games of this type has been discussed in more detail. The notion of a Nash Equilibrium also applies to the context of noncooperative games with comparative utility functions for the players. We will show that equilibria in the sense of Nash can be characterized by the solutions of complementarity problems in a very similar way as for classical non-cooperative games. Moreover, the question of existence of pure strategy Nash Equilibrium will be discussed. If the utility functions are realizations of random variables, the frequency of pure strategy equilibrium appearance for games with a large number of options follows the (...)-rule. This behavior has already been derived for classical noncooperative games, but the stochastic properties are somewhat different for comparative games.