# What would you call a preference relation that is intransitive yet complete?

I am trying to make sense of the following terminology and putting it into a table helps me keep concepts straight:

If I am understanding correctly, irrational behavior is describe as a preference relation that is both irrational and incomplete. Is that correct? If so, what would you call a preference relation that is complete, yet intransitive?

Thank you,

Complete but intransitive preferences are Cyclical Preferences.

An example of complete but intransitive preferences is in the game of "rock, paper, scissors"

In this case we can have $$\textrm{Rock}\succ \textrm{Scissors} \textrm{ and Scissors}\succ \textrm{Paper}$$ but not: $$\textrm{Rock}\succ \textrm{Paper}$$

A classic example of Cyclical preferences in economic theory is the Condorcet paradox .

I hope this helps

• Thank you, John! Mar 22 at 19:42

A preference is rational if it is complete and transitive. You need both properties satisfied for it to be rational. So an intratransitive and complete preference will still be irrational.

Indifference and strong preference on the other hand are relations derived from the preference relation itself.

The preference relation $$\succcurlyeq$$ itself says $$x$$ is at least as good as $$y$$, where $$x$$ and $$y$$ are two alternatives available to a decision maker. Formally $$x \succcurlyeq y$$ .

Strong preference relation $$x \succ y$$ (read as $$x$$ is strictly preferred to $$y$$), $$\iff$$ $$x \succcurlyeq y$$ and NOT $$y \succcurlyeq x$$.

Indifference relation $$x \sim y$$ (read as x is indifferent to y) $$\iff$$ $$x \succcurlyeq y$$ and $$y \succcurlyeq x$$

• Thank you, Rumi! Mar 22 at 19:41