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For (finite) normal form games we can develop everything for pure strategies by considering ordinal utility functions to capture only a players preferences over the pure strategy space of the game.

However in order to define a notion of expect utility for mixed strategies we need positive linear combinations of utility values to make sense so we can extend the domain of each players utility function from the pure strategy space to the mixed strategy space.

What are the relevant properties of cardinal utility functions that make such combinations make sense?

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