Debreu's representation theorems In economics, the Debreu's theorems are preference representation theorems—statements about the representation of a preference ordering by a real-valued utility function. The theorems were proved by Gerard Debreu during the 1950s. ## Background Suppose a person is asked questions of the form "Do you prefer A or B?" (when A and B can be options, actions to take, states of the world, consumption bundles, etc.). All the responses are recorded and form the person's preference relation. Instead of recording the person's preferences between every pair of options, it would be much more convenient to have a single utility function - a function that maps a real number to each option, such that the utility of option A is larger than that of option B if and only if the agent prefers A to B. Debreu's theorems address the following question: what conditions on the preference relation guarantee the existence of a representing utility function? ## Existence of ordinal utility function The 1954 Theorems say, roughly, that every preference relation which is complete, transitive and continuous, can be represented by a continuous ordinal utility function. ### S