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I am working on aggregation of preferences and something is unclear concerning one property of rules of preference namely the monotonicity.

A $\mathcal{W}$ rule of preference is a function $F$ from $\Pi=W(A)^{N}$ to $W(A)$ where $W(A)$ is a set of complete preorders and $N=\mathbb{N}\cap[1,N]$.

We say that $\mathcal{W}$ satisfies the monotonicity property if for all profiles (of preference) $p,p’\in\Pi$ with $p=p’$ except for a voter $i\in N$ for whom $x\in A$ is better ranked than in $p$, then the ranking of $x$ in $F(p’)$ cannot be worse than in $F(p)$.

I have some difficulty to represent this on an example. I tried to understand this on a single name two rounds ballot and to see if this property holds or not.

Consider that $A=\{x,y,z\}$ and $N=17$

  • $x\succ y\succ z$ for $6$ voters
  • $z\succ x\succ y$ for $5$ voters
  • $y\succ z\succ x$ for $4$ voters
  • $y\succ x\succ z$ for $2$ voters

This correspond to a profile $p$.

Now consider another profile $p’$ given by

  • $x\succ y\succ z$ for $8$ voters
  • $z\succ x\succ y$ for $5$ voters
  • $y\succ z\succ x$ for $4$ voters

We have that $p=p’$ for all voters except 2.

Now if we consider $F(p)$ it will be $ x\succ y\succ z$ (since after the first round we have $x$ and $y$ and then $x$ is preferred). And for $F(p’)$ we have $z\succ x\succ y$.

And we see that the rank of $y$ in $F(p’)$ is clearly less than the one he has in $F(p)$.

I would like to know if my understanding of this concept is good please.

coboy
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1 Answers1

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Your understanding is correct.

Monotonicity can be defined slightly differently depending on the exact situation, but in general it always means that ranking a candidate higher without otherwise changing your ballot cannot be harmful to the outcome for that candidate. Similarly, it often includes the stipulation that ranking a candidate lower cannot be beneficial.

Your example proves that your voting rule is not monotonic, and indeed runoff rules in general are not monotonic.

ADdV
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