Quick review of binary relations generally and preference relations specifically. The minimum requirement of binary relations to make them preference relations is asymmetry and negative transitivity. These two things imply the ‘normal’ qualities of preference relations (irreflexivity, transitivity and acyclicity). Phew, I was worried they might not.

It would be tedious to display the whole set of options to agents and have them state their preferences, e.g. “I prefer x to y, y to z and z to x… oops, I mean x to z.” Another tact is choice theory or revealed preference theory. A choice function is a function that takes a subset of the total number of items to choose from and picks a subset, i.e. the agents choice(s). The interesting question is when are there correspondences between preference relations and choice functions. The weak axiom is explored via Houthakker’s axiom and Sen’s alpha and beta properties (pg 13-15).

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