# Why do some game theory textbooks explicitly require preference relations to be reflexive?

A binary relation on a set of outcomes is called a preference relation if it is complete and transitive. Completeness of course implies reflexivity. But the authors of some game theory textbooks add reflexivity as an additional condition in their definition of a preference relation. (The example below is from Maschler, Solan, & Zamir (2013).) Why do they do this?

Since completeness implies reflexivity, there can be no extremely strong reason. But here are some:

1. Students new to the language of mathematics do not always appreciate that "a pair of outcomes $$x,y$$" can consist of a single outcome, the case where $$x=y$$,

2. Sometimes, one may want to weaken a theory. For aesthetic reasons, one might prefer dropping some axiom completely instead of replacing it with a weaker alternative. There is certainly some interest in relations that are reflexive but not complete as a model of preferences. Aumann has a paper on Utility Theory without the Completeness Axiom in which he drops completeness but not reflexivity.

3. The terminology is not uniform, some authors use completeness for relations in which any two distinct elements are comparable. There are also alternative terms that might mean the same thing or not, such as total or connected.

4. Redundant definitions are not that rare and do not trouble everyone. Both Royden and Dieudonne define metric spaces in their famous analysis textbooks in a redundant way; nonnegativity follows from the other conditions.

• Thanks, Michael, this all makes sense. I speculated about 1. and 3., but didn't think of 2. and didn't even know about 4. Mar 18, 2022 at 9:05

At least for the case of the example I presented, I can now answer the question myself. I just asked Eilon Solan, one of the authors. He replied:

[...] We thought it is useful to add it for pedagogical reasons, but maybe for some people it is confusing.