Savage showed us how to infer an agent's subjective probabilities and utilities from the bets which the agent accepts or rejects. But in a game theoretic situation an agent's beliefs are not just about the world, but also about the probable actions of other agents, which will depend on their beliefs and utilities. Moreover, it is unlikely that agents know the precise subjective probabilities or cardinal utilities of other agents. An agent is more likely to know something about the preferences of other agents and something about their beliefs. In view of this, the agent is unlikely to to have a precise best action which we can predict, but is more likely to have a set of “not so good” actions which the agent will not perform.

Ann may know that Bob prefers chocolate to vanilla to strawberry. She is unlikely to know whether Bob will prefer vanilla ice cream or a 50-50 chance of chocolate and strawberry. So Ann's actions  and her beliefs need to be understood in the presence of such partial ignorance. We propose a theory which will let us decide when Ann is being irrational, based on our partial knowledge of her beliefs and preferences, and assuming that Ann is rational, how to infer her beliefs and preferences from her actions.

Our principal tool is a generalization of rational behavior in the context of ordinal utilities and partial knowledge of the game which the agents are playing.