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.
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.