Regular preorders and behavioral indifference

In this article, we discuss some aspects of Eliaz and Ok ’s (Games Econ Behav 56:61–86, 2006) choice theoretical foundations of incomplete preferences. Our aim is to clarify some aspects of their regularity condition for preorders, showing that, as far as rationalization of a choice correspondence alone is concerned, no further restriction is imposed by requiring the preorder to be regular. However, if one is also interested in capturing other observable aspects of the individual’s choice procedure, such as Eliaz and Ok’s notion of c-incomparability or the notion of behavioral indifference introduced in this paper, then the only option is to use a regular preorder. We also argue that their weak axiom of revealed non-inferiority (WARNI) is too strong a property if our aim is to characterize the rationalizability of a choice correspondence by a (possibly incomplete) preorder. Finally, as we have mentioned above, we introduce the notion of behavioral indifference and argue, not only that Eliaz and Ok’s notion of c-incomparability (observable incomparability) can be derived from it, but also that it has a wider range of applicability than their notion of observable incomparability.