Comparative optimality of larger committees under convex costs beyond the 3-vs-1 case

Determine whether, under convex cost functions c(x), committees of size k″ are better than committees of size k′ for k″ > k′ > 3 in terms of the success probability succ achieved by symmetric efforts subject to the budget constraint. Specifically, characterize for general convex c(x) and budget B whether succ(x*(k″), k″) ≥ succ(x*(k′), k′), where x*(k) is the optimal symmetric effort satisfying k·c(x*(k)) = B.

Background

In the convex cost regime, the authors show that whether three DReps outperform one DRep depends on both the budget and the exponent in power-law costs c(x) = x^β (Theorem th_three-vs-one), illustrating that larger committees are not uniformly better. Extending this comparison to larger committee sizes requires handling many combinatorial outcome events.

The authors explicitly note that generalizing the 3-vs-1 comparison to k″ vs. k′ > 3 is complex and leave it as an open problem, emphasizing the need for a broader characterization of when larger committees yield higher success probability under convex costs.