Ordinal aggregation results via Karlin's variation diminishing property

When is the weighted sum of quasi-concave functions quasi-concave? We answer this, extending an analogous preservation of the single-crossing property in QS: Quah and Strulovici (2012). Our approach develops a general preservation of n-crossing properties, applying the variation diminishing property in Karlin (1956). The QS premise is equivalent to Karlin's total positivity of order two, while our premise uses total positivity of order three: The weighted sum of quasi-concave functions is quasi-concave if each has an increasing portion more risk averse than any decreasing portion.