Formal proof of the linear-cost monotonicity assumption for success probability

Establish whether the monotonicity property stated as Assumption (monotonicity) holds: for any linear cost function c_lin(x) = a·x with a > 0 and fixed budget B, prove or refute that the success probability of electing the correct outcome under symmetric committees is monotone decreasing in the number of delegation representatives (DReps) when each DRep uses the corresponding budget-limited maximum effort. Specifically, show that succ(x, k) ≤ succ(x′, k′) whenever x ≤ x′ and k = ⌊B / c_lin(x)⌋, k′ = ⌊B / c_lin(x′)⌋, where succ(x, k) is the probability that the weighted majority of k DReps—each exerting effort x and voting correctly with probability 1/2 + x under proportional-weighting w_i = x / (kx)—selects the correct outcome with random tie-breaking.

Background

The paper analyzes optimal committee sizes and reward mechanisms for delegated voting where each DRep’s effort x simultaneously increases their probability of voting correctly (to 1/2 + x) and their attracted delegation, with symmetric committees yielding equal weights. In the concave–convex cost regime, the authors propose bounds on optimal efforts that depend on a monotonicity condition for linear costs.

Assumption (monotonicity) posits that, for linear costs c_lin(x) = a·x under a fixed budget B, increasing the number of DReps (and thus lowering the per-DRep maximum permissible effort) should not increase the success probability. The authors rely on this assumption to strengthen bounds (Theorem cocave-convex-tangent), have verified it empirically for several parameter choices, and provide a weaker formal result in the appendix, but they lack a complete proof for the general case.