I am working on a social choice problem that must allow for alternative sets of different sizes. In the paper I'm struggling to write, the environment is a $5$-tuple $(N,A_\tau,\mathcal{P}(A_\tau),\Delta(A_\tau),\sigma)$. First, $N=\{1,\dots,n\}$ is a finite voter set, where $n\geqslant2$. Second, $A_\tau=\{a_1,\dots,a_\tau\}$ is a finite alternative set, where $\tau\in\mathbb{N}=\{1,2,\dots\}$. Third, \begin{gather} \mathcal{P}(A_\tau)=\bigl\{P\mid P=(P_i)_{i\in N}:P_i\text{ is voter $i$'s strict order over }A_\tau\bigr\} \end{gather} is the set of all strict preference profiles over some alternative set: namely, for all alternatives $x,y\in A_\tau$ and all voters $i\in N$, $x P_iy$ if and only if voter $i$ strictly prefers $x$ over $y$. Fourth, \begin{gather} \Delta(A_\tau)=\left\{\delta\in[0,1]^{A_\tau}\mid\sum_{x\in A_\tau}\delta(x)=1\right\} \end{gather} is the set of all lotteries over some alternative set. And fifth, \begin{gather} \sigma:\bigcup_{\tau\in\mathbb{N}}\mathcal{P}(A_\tau)\to\bigcup_{\tau\in\mathbb{N}}\Delta(A_\tau) \end{gather} is a voting rule that satisfies $\sigma(P)\in\Delta(A_\tau)$ for all numbers $\tau\in\mathbb{N}$ and all strict preference profiles $P\in\mathcal{P}(A_\tau)$.
The problem that I face is that this definition of a voting rule allows for some funny objects which I would like to rule out. These funny objects are voting rules that change their logic depending on the size of the alternative set. To see so, construct a voting rule as follows:
- If $\tau\leqslant3$ (i.e., there are at most three alternatives), our voting rule assigns identical weight to all alternatives that are top-ranked by the largest number of voters and null weight to all other alternatives (i.e., plurality rule);
- If $\tau\geqslant4$ (i.e., there are four or more alternatives), our voting rule assigns weight one to the alternative that is top-ranked by voter $1$ (i.e., dictatorship of voter $1$).
Can anybody help me figure out how to change my definition of a voting rule to ensure that a voting rule does not change its logic depending on the size of the alternative set?
