How to change this definition of a voting rule?

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?