In economics, an ordinal utility function is a function representing the preferences of an agent on an ordinal scale. Ordinal utility theory claims that it is only meaningful to ask which option is better than the other, but it is meaningless to ask how much better it is or how good it is. All of the theory of consumer decision-making under conditions of certainty can be, and typically is, expressed in terms of ordinal utility. For example, suppose George tells us that "I prefer A to B and B to C". George's preferences can be represented by a function u such that: But critics of cardinal utility claim the only meaningful message of this function is the order ; the actual numbers are meaningless. Hence, George's preferences can also be represented by the following function v: The functions u and v are ordinally equivalent – they represent George's preferences equally well. Ordinal utility contrasts with cardinal utility theory: the latter assumes that the differences between preferences are also important. In u the difference between A and B is much smaller than between B and C, while in v the opposite is true. Hence, u and v are not cardinally equivalent. The ordinal utility concept was first introduced by Pareto in 1906. Notation Suppose the set of all states of the world is and an agent has a preference relation on . It is common to mark the weak preference relation by , so that reads "the agent wants B at least as much as A". The symbol is used as a shorthand to the indifference relation: , which reads "The agent is indifferent between B and A". The symbol is used as a shorthand to the strong preference relation: , which reads "The agent strictly prefers B to A". A function is said to represent the relation if: Related concepts Indifference curve mappings Instead of defining a numeric function, an agent's preference relation can be represented graphically by indifference curves. This is especially useful when there are two kinds of goods, x and y. Then, each indifference curve shows a set of points such that, if and are on the same curve, then . An example indifference curve is shown below: Each indifference curve is a set of points, each representing a combination of quantities of two goods or services, all of which combinations the consumer is equally satisfied with.