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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?

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    $\begingroup$ Being decisive (single-valued) and strategy-proof works. $\endgroup$
    – Michael Greinecker
    Commented Jul 8 at 8:03
  • $\begingroup$ 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? $\endgroup$
    – EoDmnFOr3q
    Commented Jul 8 at 8:06
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    $\begingroup$ 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. $\endgroup$
    – Michael Greinecker
    Commented Jul 8 at 8:12
  • $\begingroup$ 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. $\endgroup$
    – EoDmnFOr3q
    Commented Jul 8 at 8:43

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