Is the sum of quasi concave functions quasi concave

Question

Is the sum of quasi-concave functions a quasi concave function? I presume that's not in the case in general, but under which conditions is this true?

Answer 1

I will provide a criterion under which certain sums preserve quasiconcavity. Since it is not covered in traditional convex optimization textbooks ala Boyd & Vandenberghe (2004), since it is quite recent it might help others.

The reference which helped me out is: Quah, John K. H., and Bruno Strulovici. “AGGREGATING THE SINGLE CROSSING PROPERTY.” Econometrica, vol. 80, no. 5, 2012, pp. 2333–48. JSTOR, http://www.jstor.org/stable/23271450.

The idea is the following. A function $h(x)=f(x)+g(x), x\in X$ is quasiconcave when $h'(x)$ has the single-crossing (SC) property, i.e., it crosses the $x$-intercept only once. Theorem 1 of the above reference, is:

Let $f$ and $g$ be two SC functions. Then $\alpha f + \beta g$ is an (SC) function for any nonnegative scalars $\alpha$ and $\beta$ if and only if $f$ and $g$ obey signed-ratio monotonicity of the form:

(a) at any $x′\in X$ , such that $g(x′) < 0$ and $f(x′)>0$, we have $$\frac{−g(x′)}{f(x′)}\geq \frac{−g(x′′)}{f(x′′)}\text{ when } x′′ > x′$$ and (b) at any $x′\in X$ , such that $f(x′) < 0$ and $g(x′)>0$, we have $$\frac{−f(x′)}{g(x′)}\geq \frac{−f(x′′)}{f(x′′)}\text{ when } x′′ > x′$$

Answer 2

$f(x) = x^3$ and $g(x) = 1-x^2$ are both quasiconcave but $f(x) + g(x) = x^3 - x^2 +1$ is not.

A trivial case where the sum is still quasiconcave is when both functions are concave