Voting Theory Relations

voting-theory

The relations that are defined for Voting Theory principles and methods. Also used in theorems such as May’s Theorem or Arrow’s Impossibility Theorem

let $A = { a_1, …, a_k }$ be the set of candidates
let $V = { 1, 2, …, N }$ be the set of voters; there are $N$ voters
each voter can express his preference on the basis of a ‘‘total order’’
- i.e. he has to rank all the candidates
- $R_j(a_i)$ is the position of candidate $a_i$ in the ranking of voter $j$

Based on this notation let us define the following relations:

Preference (or Strong Preference)
Indifference
“At least as good as” Relation (or Weak Preference) - the combination of preference and indifference

Preference Relation

$>$ (or $P$) is the (strong) preference relation, voters use it to express the preference

Example:

It satisfies three axioms:

'’completeness’’
- for any $x$ and $y$ either $x < y$ or $y < x$
'’transitivity’’ (or ‘‘consistency’’)
- $\forall x, y, z \in A: x > y \land y > z \Rightarrow x > z$
'’asymmetric’’
- $\forall x, y: (x > y) \Rightarrow \overline{ y > x }$

Notation

$P_i$ shows the individual preference of voter $i$
- $x \ P_1 \ y$ means that voter 1 prefers $x$ to $y$
$P$ shows the global aggregated preference

$\sim$ or $I$ is the indifference relation, voters use it to express that both candidates are equally good

Properties:

indifference is not always transitive

cups of coffee

- but in some cases is: for instance, in the Arrow’s Impossibility Theorem it’s considered transitive

Notation

$\geqslant$ or $S$ - means “at least as good as” - indifferent or better, sometimes referred as ‘‘weak preference’’

$S \equiv (P \lor I)$ or $\geqslant \equiv [< \lor \sim]$
the opposite of $a \ S \ b$ is $b \ P \ A$:
- $\overline{a \ S \ b} \equiv (a \ \overline{S} \ b) \equiv (a \ \overline{P \lor I} \ b) \equiv (a \ (\overline{P} \land \overline{I}) \ b) \equiv a \ \overline{P} \ b \equiv b \ P \ a$
- not preferred and not indifferent

Properties

for any pair $(x, y), x,y \in A, x \ne y$

$x \ S \ y \not \Rightarrow y \ P \ x$

- ‘‘completeness’’

- $\forall x, y \in A:$ either $x \ S \ y$ or $y \ S \ x$

'’transitivity’’ (or ‘‘consistency’’)
- $\forall x, y, z \in A: x \ S y \land y \ S \ z \Rightarrow x \ S \ z$

We can use express Preference and Indifference via this relation:

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