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I've been doing my own reading on non-rational preference relations.
I'm currently under the impression that transitivity follows as a direct result of completeness of preferences. However my (much more advanced colleagues) have told me this is not necessarily the case.
I'm still to understand how this can be. What would such preferences look like?
I'm quite surprised nobody has picked the obvious one:
Complete, definitely not transitive.
Completeness: given any pair, I have a preference, I can make a choice.
If had to choose between marrying Rachel and Monica, I would go for Rachel. Good looks, fun, etc.
If had to choose between marrying Chandler and Rachel, I would go for Chandler. Corny sense of humor but aware of it, even temperament, etc.
If had to choose between marrying Monica and Chandler, I would go for Monica. Loyal, very diligent etc.
Wait, if I had to choose between Rachel, Monica and Chandler, who would I marry? (what is my ordering, who is my 1st choice)
Here's an example of an incomplete but transitive preference.
Consider three fruits, an apple ($A$), a banana ($B$), and a coconut ($C$). I cannot choose between individual fruits, i.e. I don't have a preference over $A$, $B$, or $C$ --- not that I'm indifferent between them, I just can't compare them. However, I do prefer more variety to less, namely, I'd choose a bundle with two fruits, e.g. $\{A,B\}$ over a bundle with only one fruit, say $\{C\}$.
It's easy to see that my preference over bundles of fruits (the powerset of $\{A,B,C\}$) is incomplete, since it is not defined over singletons. However, my preference is still transitive, in that whenever bundle $X$ is preferred to $Y$ and $Y$ preferred to $Z$, $X$ is preferred to $Z$, for example, $\{A,B,C\}\succ\{B,C\}\succ\{C\}\;\Rightarrow\;\{A,B,C\}\succ\{C\}$.
It may be helpful to consider a phenomenon from politics called the Condorcet paradox. This is a situation that can happen in votes, in which the overall population would vote for A over B, would vote for B over C, and would vote for C over A. It is not a purely theoretical possiblity: it is the present reality in the UK over the best resolution to the Brexit crisis. Either link gives a good explanation of how the situation can arise, but for completeness the simplest example has three voters and three candidates with preferences as follows:
1: A > B > C
2: B > C > A
3: C > A > B
When taken to a vote, 2/3 would prefer A to B, 2/3 would prefer B to C, and 2/3 would prefer C to A. Transitivity would require that A be prefered over C. Therefore the preference of the group (although not the individuals) is complete and non-transitive.
From what I understand from @Giskard's answer is that the fact that we have comparablity of items over sets of 2, does not necessarily imply the fact that we have comparablity over sets of 3.
Consider the case where we are considering bundles $X=\{x_1,x_2\}$,$Y=\{y_1,y_2\}$ and $Z=\{z_1,z_2\}$. Suppose now we have for the pairwise comparison between bundles: $$(1)\ \{X,Y\}\in A,\ \ \ Y\in C(A)$$ $$(2) \ \{Y,Z\}\in B,\ \ \ Z\in C(B)$$ $$(3) \ \{Z,X\}\in D,\ \ \ X\in C(D)$$
Where $C(\cdot)$ is the choice in environment $A,B,D$ ect.
We have shown that we in fact do have completeness of preferences since we can compare any two bundles. However, transitivity is not established because given the above results we cannot say for $ \{X,Y,Z\}\in Q$ there is a choice that follows since we have (3) as our source of irrationality fudging the transitivity of preferences.
Graphically we have the following situation:
