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Independence to Third Alternatives

Independence to Third Alternatives, or Independence to Irrelevant alternatives is a principle of Voting Theory.

it says that if another alternative is added or removed, the position of a candidate should remain at least as good as it was
this is an important principle in Arrow's Impossibility Theorem
also in MCDA methods violation of this principle leads to Rank Reversal

Suppose we have a set of alternatives (candidates) $A^*$
Let us consider 4 different individual rankings over the sets $A^*$ and $A$ s.t. $A \subset A^*$

(note that $A$ is a strict subset of $A^*$)

The rankings are $R_1, R_2$ and $R'_1, R'_2$
$S_1, S_2, S'_1, S'_2$ are indifference relations defined by these orderings (respectively)
we assume that $R_1 \equiv R'_1$ and $R_2 \equiv R'_2$ for the set $A$
that is, $\forall x,y \in A$:
1. $x \ S_1 \ y \iff x \ S'_1 \ y$
2. $x \ S_2 \ y \iff x \ S'_2 \ y$

Example:

$A^* = \{x, y, z\}$
- $R_1: x > y > z, R'_1: z > y > z$
- $R_2: z > y > x, R'_2: y > x > z$
now restrict ourselves to $A = \{x, y\} \subset A^*$
- $R_1 \equiv R'_1$ and $R_2 \equiv R'_2$
- the only thing that changes is the relative position of $z$ within the pairs of rankings

A voting method $H$ is independent to third alternatives if

the global ordering produced by $H$ under the set $A$ is the same for both rankings such rankings:
$H(R_1, R_2) |_A \equiv H(R'_1, R'_2) |_A$

In other words, ordering of the set $A^* - A$ is irrelevant to the choice over $A$

Consider two dishes: beef and lamb
The choice between these two alternatives should not change when pork is also available
pork is irrelevant alternative to the preference ordering of beef and lamb

Suppose there exist two ways to subscribe to some newspaper:

Note that for the publisher the web version costs nearly nothing

The publisher proposes the following:

$P$: 100 USD, $W$: 60 USD, $P+W$ also 100 USD.
in this case we see that no rational decision taker will ever take just $P$, but always $P+W$

Before the preposition the distribution of readers could be this:

After:

The part of readers switched $\Rightarrow$ More money

There are several ways in which a method may suffer from dependence to 3rd alternatives:

A method suffers from the Risk of Manipulation if the outcome of an election can be changed by

This manipulation is also sometimes called control.

Manipulation:

Methods that suffer from the manipulation: