# Collections of axioms that imply monotonicity?

I am currently searching for collections of axioms that imply monotonicity.

Let $$N$$ be a finite set of players, let $$A$$ be a finite set of alternatives, let $$\mathcal{P}$$ be the set of all linear order profiles on $$A$$, and let $$F:\mathcal{P}\to 2^A\backslash\{\emptyset\}$$ be a social choice correspondence.

Given any player $$i\in N$$, any alternative $$x\in A$$ and any linear order profile $$P\in\mathcal{P}$$, let $$L_i(x,P)=\{y\in A\mid xP_iy\}$$ be player $$i$$'s lower contour set at $$x$$.

A social choice rule $$F:\mathcal{P}^N\to 2^A\backslash\{\emptyset\}$$ is monotonic if and only if for all alternatives $$x\in A$$ and all linear order profiles $$P,P'\in\mathcal{P}$$, the following is true: if $$x\in F(P)$$ and $$L_i(x,P)\subseteq L_i(x,P')$$ for all $$i\in N$$, then $$x\in F(P')$$.

I am looking for collections of axioms that, once combined, imply monotonicity. Do you know any?

• Being decisive (single-valued) and strategy-proof works. Commented Jul 8 at 8:03
• Thank you for your comment. I suspect this comes from the fact that decisivenes and strategy-proofness, together with the unrestricted preference domain; imply dictatorship, which is obviously monotonic. Any other one that comes to mind, involving perhaps some form of IIA? Commented Jul 8 at 8:06
• To get to dictatorship, you need to also assume that there are at least three alternatives and that every alternative can be elected. It is generally true that monotonicity and strategy-proofness are equivalent on a full domain for social choice-functions. Commented Jul 8 at 8:12
• Thank you again for your comment—I didn’t know the equivalence holds so generally. Given the relationship between IIA and strategy-proofness found by Satterthwaite (1975), I’m kinda surprised there’s no known collection of axioms containing some form of IIA that together imply monotonicity. Commented Jul 8 at 8:43