# Weak preferences and negative transitivity

Let $$\succ$$ be a binary relationship on the set $$X$$ such that, given any $$x, y, z\in X$$, if $$x\succ y$$:

1. (Asymmetry): $$\neg(y\succ x)$$,
2. (Negative transitivity): $$(x\succ z) \vee (z\succ y)$$.

Let us define the abbreviations:

1. $$x\succeq y \;:=\; \neg(y\succ x)$$,

2. $$x \sim y \;:=\; x\succeq y\; \wedge \;y \succeq x$$.

As usual, the relations $$\succ, \succeq, \sim$$ denote strong preference, weak preference, and indifference.

Intuition suggests that I can conclude: $$x\succeq y \; \leftrightarrow \;(x\succ y\; \vee \;x\sim y)$$

If so, how can I derive it formally? Any useful references?

• Is the relation $\succeq$ complete? Commented Dec 19, 2021 at 3:39
• @Giskard: The definitions above are taken from Föllmer, Schied, Stochastic Finance, where it is stated that $\succeq$ completeness is implied by the asymmetry and the negative transitivity of $\succ$. Commented Dec 19, 2021 at 8:09
• Thanks for the source. Remark 2.3. claims that these properties are equivalent to $\succeq$ being transitive and complete. Can you then not take the usual $\succeq$ route to prove your intuitited statement? Commented Dec 19, 2021 at 8:44
• @Giskard: Well, usually $x\succ y\; \vee \;x\sim y$ is taken as the definition of $x\succeq y$. Commented Dec 19, 2021 at 9:10

Probably it can be done easier if you do both steps separately ($$\implies$$ and $$\impliedby$$), but here is a proof that does both at the same time: \begin{align*} &x\succ y \vee x\sim y\\ \iff\;& x \succ y \vee (x\succeq y \wedge y \succeq x) & \text{Definition of \sim} \\ \iff\;& (x \succ y \vee x\succeq y) \wedge (x \succ y \vee y \succeq x)& \text{Distributivity}\\ \iff\;& [x \succ y \vee \neg( y \succ x)] \wedge [x \succ y \vee \neg(x\succ y)]& \text{Definition of \succeq}\\ \iff\;& x \succ y \vee \neg( y \succ x)& \text{LEM}\\ \iff\;& [x \succ y \vee \neg( y \succ x)] \wedge[x \succ y \to \neg( y \succ x)]& \text{Asymmetry}\\ \iff\;& [x \succ y \vee \neg( y \succ x)] \wedge[\neg(x \succ y) \vee \neg( y \succ x)]& \text{Definition of \to}\\ \iff\;& [x \succ y \wedge \neg(x \succ y)] \vee \neg( y \succ x)& \text{Distributivity}\\ \iff\;& \neg( y \succ x)& \text{Contradiction}\\ \iff\;& x \succeq y& \text{Definition of \succeq}\\ \end{align*}