Our work deals with the important problem of globally characterizing truthful mechanisms where players have multi-parameter, additive valuations, like scheduling unrelated machines or additive combinatorial auctions. Very few mechanisms are known for these settings and the question is: Can we prove that no other truthful mechanisms exist? We characterize truthful mechanisms for n players and 2 tasks or items, as either task-independent, or a player-grouping minimizer, a new class of mechanisms we discover, which generalizes affine minimizers. We assume decisiveness, strong monotonicity and that the truthful payments (The (normalized) payments are uniquely determined by the allocation function of the mechanism; thus the assumptions concern properties of the allocation.) are continuous functions of players' bids.

We study the problem where a task (or multiple unrelated tasks) must be executed, there are multiple machines/agents that can potentially perform the task, and our objective is to minimize the expected sum of the agents' processing times. Each agent does not know exactly how long it will take him to finish the task; he only knows the distribution from which this time is drawn. These times are independent across agents and the distributions fulfill the monotone hazard rate condition. Agents are selfish and will lie about their distributions if this increases their expected utility. We study different variations of the Vickrey mechanism that take as input the agents' reported distributions and the players' realized running times and that output a schedule that minimizes the expected sum of processing times, as well as payments that make it an ex-post equilibrium for the agents to both truthfully report their distributions and exert full effort to complete the task. We devise the ChPE mechanism, which is uniquely tailored to our problem, and has many desirable properties including: not rewarding agents that fail to finish the task and having non-negative payments.