# What does non degeneracy mean for a preference?

I saw a non-degeneracy assumption in Gilboa and Schmeidler's paper (maxmin expected utility with non-unique prior).

The statement is "Not for all $f$ and $g$ in $L$, $f \geq g$".

So can you explain the intuition for this assumption and give some example so that I can understand the "non-degeneracy" property better?

• My first guess would be something related to a degenerate probability distribution. But please provide a bit more details: a link to the paper or the full statement of the assumption. May 13 '17 at 5:56
• Hi, here is a link: researchgate.net/publication/…. The non-degeneracy assumption is on page 144 of the original paper, under the name of A.6.
• Perhaps it is easier to understand non-degeneracy if we look at the contrapositive of the statement: There exist (horse lotteries) $f,g\in L$ such that $f>g$. In other words, $\geq$ is not a preference relation that treats all horse lotteries in $L$ equally; it strictly prefers some $f$ to some $g$. Unfortunately I can't produce examples to help elucidate this, as I'm not that familiar with this field. Maybe looking up Anscombe-Aumann's definition of subjective probability will help. May 13 '17 at 8:11
It means what Herr K. has guessed: there are acts $f$ and $g$ that the decision maker does not deem equivalent from the preference standpoint. In other words, the DM expresses a strong preference in at least one comparison he makes. a preference that does not satisfy it is a AA subjective expected utility with a constant index $u$ (i.e. such that the representation is $\int u dp$ for an arbitrary belief $p$). On the other hand if at least for one comparison the DM says $f$ is strictly better than $g$ then you satisfy it.
• I might add: the reason for this assumption is so that $u(x) = c$ for some constant $c$ does not rationalize the preference. This state of affairs is both boring and destroys any uniqueness results (since beliefs no longer play any role they can be arbitrary).