  Human computation or crowdsourcing involves joint inference of the
ground-truth-answers and the worker-abilities by optimizing an objective
function, for instance, by maximizing the data likelihood based on an assumed
underlying model. A variety of methods have been proposed in the literature to
address this inference problem. As far as we know, none of the objective
functions in existing methods is convex. In machine learning and applied
statistics, a convex function such as the objective function of support vector
machines (SVMs) is generally preferred, since it can leverage the
high-performance algorithms and rigorous guarantees established in the
extensive literature on convex optimization. One may thus wonder if there
exists a meaningful convex objective function for the inference problem in
human computation. In this paper, we investigate this convexity issue for human
computation. We take an axiomatic approach by formulating a set of axioms that
impose two mild and natural assumptions on the objective function for the
inference. Under these axioms, we show that it is unfortunately impossible to
ensure convexity of the inference problem. On the other hand, we show that
interestingly, in the absence of a requirement to model "spammers", one can
construct reasonable objective functions for crowdsourcing that guarantee
convex inference.
