Aggregating the single crossing property

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Optimal information disclosure: A linear programming approach (2018, background): “This standard single-crossing assumption holds if and only if, for all s1 s2 ∈ S, functions ũ(r s1) and ũ(r s2) of r ∈ R satisfy signed-ratio monotonicity (Theorem 1 of Quah and Strulovici 2012).”
Certifiable Pre‐Play Communication: Full Disclosure (2014, unknown): “…revealing equilibrium with extremal beliefs, after any unilateral deviation of some player j in the communication stage, the oﬀ-path beliefs of all players i (cid:54) = j assign probability 1 to the same type t j ∈ T j of player j independently of the message vector m − j and their own type t i .”
Persuasion of a Privately Informed Receiver (2017, unknown): “Function I(r) = g′ (r) + ρg (r) crosses the horizontal axis at most once and from above (from below) on Ω for all ρ ∈ R if and only if g′ (r) /g (r) is nonincreasing (nondecreasing) on Ω by Proposition 1 of Quah and Strulovici (2012). □”