Using equation (7.8), verify that the strict upper contour...

Using equation (7.8), verify that the strict upper contour sets in Figure 7.8 are correct in depicting that $(2 / 3,1 / 3) \notin P_i(1 / 3,2 / 3)$ and $(1 / 3,2 / 3) \notin P_i(2 / 3,1 / 3)$. Demonstrate by example that these preferences are in fact intransitive.

Key Concepts

Strict Upper Contour Sets

A strict upper contour set of a particular alternative is the collection of all other options that a decision maker strictly prefers over it. In utility theory, these sets help in visualizing and verifying the structure of preferences, and they are depicted often in diagrams to capture the set of alternatives that lie 'above' a given threshold defined by an alternative’s utility level.

Preference Relations

A preference relation is an ordering that reflects how a decision maker compares different alternatives based on the satisfaction or utility they derive from each. In the context of economic choice and decision theory, these relations are used to formalize the notion of one option being preferred to another, and they form the basis for constructing utility functions and contour sets.

Transitivity of Preferences

Transitivity is a key property in rational choice theory which stipulates that if an individual prefers option A over option B and option B over option C, then they should also prefer option A over option C. This consistency condition is essential for the existence of a utility function that represents the decision maker's preferences.

Intransitive Preferences

Intransitive preferences occur when the transitivity condition is violated, meaning that even if a decision maker prefers option A over option B and option B over option C, they might not prefer option A over option C. Intransitivity can lead to cyclical choices and may be illustrated using examples and diagrammatic contour sets, indicating that preference ordering is not consistent across all choices.

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