On Strategy-Proof Allocation without Payments or Priors

In this paper we study the problem of allocating divisible items to agents without payments. We assume no prior knowledge about the agents. The utility of an agent is additive. The social welfare of a mechanism is defined as the overall utility of all agents. This model is first defined by Guo and Conitzer [7]. Here we are interested in strategy-proof mechanisms that have a good competitive ratio, that is, those that are able to achieve social welfare close to the maximal social welfare in all cases. First, for the setting of n agents and m items, we prove that there is no (1/m+e)-competitive strategy-proof mechanism, for any e>0. And, no mechanism can achieve a competitive ratio better than $4/\sqrt{n}$ , when $m \ge \sqrt{n}$ . Next we study the setting of two agents and m items, which is also the focus of [7]. We prove that the competitive ratio of any swap-dictatorial mechanism is no greater than $1/2 + 1/\sqrt{\left\lbrack\log{m}\right\rbrack }$ . Then we give a characterization result: for the case of 2 items, if the mechanism is strategy-proof, symmetric and second order continuously differentiable, then it is always swap-dictatorial. In the end we consider a setting where an agent's valuation of each item is bounded by C/m, where C is an arbitrary constant. We show a mechanism that is (1/2+e(C))-competitive, where e(C)>0.