ML Wiki

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

Definition

• 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 $A$

Example 1

• 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

Example 2

Suppose there exist two ways to subscribe to some newspaper:

• paper subscription $P$: 100 USD
• web version $W$: 60 USD

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:

• $P$ for 100 USD: 30%
• $W$ for 60 USD: 70%

After:

• $P$ for 100 USD: 0%
• $W$ for 60 USD: 30%
• $P+W$ for 100 USD: 70%

The part of readers switched $\Rightarrow$ More money

---

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

• Risk of Manipulation

Risk of Manipulation

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

• adding a new candidate or
• deleting a candidate

This manipulation is also sometimes called control.

Manipulation:

• suppose somebody knows the individual rankings
• they may propose a new candidate that will take some votes
• this way influencing the final result

Methods that suffer from the manipulation:

• Plurality Voting
• Two-Round Voting
• Condorcet's Rule
• Borda's Rule

Sources

• Decision Engineering (ULB)
• EC228 Voting Theory Lecture Notes [1]