<p>Note that next he defines <span class="math-container">$x R y$</span> as <span class="math-container">$\neg y P x$</span>. If we take the contrapositive of (1b), we get:
<span class="math-container">$$
(\neg x P y \wedge \neg y P z ) \to \neg x P z,
$$</span>
which then translates to the usual transitivity condition on <span class="math-container">$R$</span>,
<span class="math-container">$$
(y R x \wedge z R y) \to z R x.
$$</span>
Also note that condition (1.a) gives in this notation,
<span class="math-container">$$
\neg (x P y \wedge y P x) \equiv (y R x \vee x R y),
$$</span>
which states that <span class="math-container">$R$</span> is complete.</p>
<p>So condition (1.a) and (1.b) simply require <span class="math-container">$R$</span> to be a complete and transitive relation.</p>

CC BY-SA 4.0
2024-11-25T17:33:16.473Z
I had this implication, but I don't understand why R total reflexive transitive => (1.b). Isn't something more assumed?
Zlotz
CC BY-SA 4.0
2024-11-25T11:46:56.817Z
2024-11-25T11:46:56.817Z
tdm
<p>OK, I've formalized in Coq the correspondence between <a href="https://en.wikipedia.org/wiki/Weak_ordering#Strict_weak_orderings" rel="nofollow noreferrer">the &quot;math standard&quot; preference relations</a> and Gibbard's ordering: see <a href="https://github.com/mcoulont/WeakOrdering" rel="nofollow noreferrer">the Github repo</a></p>

CC BY-SA 4.0
2024-12-14T00:28:00.467Z
2024-12-14T00:28:00.467Z
Zlotz
<p>I'm reading Alan Gibbard's <a href="https://www.eecs.harvard.edu/cs286r/courses/fall11/papers/Gibbard73.pdf" rel="nofollow noreferrer">Manipulation of Voting Schemes: A General Result</a> and I'm a bit astonished by his definition of the concept of ordering.</p>
<p>I'm talking about the following condition parge 594: <span class="math-container">$$(1b): \forall x, y, z,  (xPz \implies (xPy \lor yPz))$$</span></p>
<p>Why would <span class="math-container">$y$</span> be forcefully comparable with either <span class="math-container">$x$</span> or <span class="math-container">$z$</span>?</p>

<game-theory><social-choice><voting>
Curious definition of ordering in Gibbard's "Manipulation of Voting Schemes: A General Result"
Zlotz
# Curious definition of ordering in Gibbard's "Manipulation of Voting Schemes: A General Result"

(Asked by: username_0 on 2024-11-25)

I'm reading Alan Gibbard's [Manipulation of Voting Schemes: A General Result](https://www.eecs.harvard.edu/cs286r/courses/fall11/papers/Gibbard73.pdf) and I'm a bit astonished by his definition of the concept of ordering.

I'm talking about the following condition parge 594: $$(1b): \forall x, y, z, (xPz \implies (xPy \lor yPz))$$

Why would $y$ be forcefully comparable with either $x$ or $z$?

**Tags:** game-theory, social-choice, voting

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## 2 answers

### Answer by username_1 on 2024-11-25. Score: 2

Note that next he defines $x R y$ as $\neg y P x$. If we take the contrapositive of (1b), we get:
$$
(\neg x P y \wedge \neg y P z ) \to \neg x P z,
$$
which then translates to the usual transitivity condition on $R$,
$$
(y R x \wedge z R y) \to z R x.
$$
Also note that condition (1.a) gives in this notation,
$$
\neg (x P y \wedge y P x) \equiv (y R x \vee x R y),
$$
which states that $R$ is complete.

So condition (1.a) and (1.b) simply require $R$ to be a complete and transitive relation.

> **Comments:**
> **username_0**: I had this implication, but I don't understand why R total reflexive transitive => (1.b). Isn't something more assumed?  

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### Answer by username_0 on 2024-12-14. Score: 1

OK, I've formalized in Coq the correspondence between [the "math standard" preference relations](https://en.wikipedia.org/wiki/Weak_ordering#Strict_weak_orderings) and Gibbard's ordering: see [the Github repo](https://github.com/mcoulont/WeakOrdering)


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