Tight Analysis of Randomized Truthful Scheduling: An $(m+1)/2$ Lower Bound

The Nisan-Ronen conjecture, a central problem in algorithmic game theory, asserts that no deterministic truthful mechanism for makespan minimization on $m$ unrelated machines can achieve an approximation ratio better than $m$. Christodoulou, Koutsoupias, and Kovács resolved it in 2023, but the optimal approximation ratio achievable by randomized truthful mechanisms remained open. Recently, Anastasiadis, Christodoulou, Koutsoupias, Kovács, and Schecker extended techniques from the deterministic proof to establish a $(3/2-\sqrt{2})(m-1)$ lower bound for universally truthful mechanisms with discrete support and gave a universally truthful mechanism with approximation ratio $m/2+o(m)$. For the broader class of truthful-in-expectation (TIE) mechanisms, however, the gap remained large: the best known upper bound was $(m+5)/2$, while the lower bound was only $2-1/m$. We prove a lower bound of $(m+1)/2$, determining the optimal TIE approximation ratio up to an additive constant as $m/2+Θ(1)$. Our result also improves the leading constant in the lower bound for universally truthful mechanisms from $3/2-\sqrt{2} \approx 0.0858$ to $1/2$. Together with their $m/2+o(m)$ upper bound, our result determines the optimal approximation ratio for universally truthful mechanisms up to lower-order terms.