Random priority item allocation
Random priority (RP), also called Random serial dictatorship (RSD), is a procedure for dividing indivisible items fairly among people. Suppose formula_1 partners have to divide formula_1 (or fewer) different items among them. Since the items are indivisible, some partners will necessarily get the less-preferred items (or no items at all). RSD attempts to insert fairness into this situation in the following way. Draw a random permutation of the agents from the uniform distribution. Then, let them successively choose an object in that order (so the first agent in the ordering gets first pick and so  on). Properties. RSD is a truthful mechanism when the number of items is at most the number of agents, since you only have one opportunity to pick an item, and the obviously dominant strategy in this opportunity is to pick the best available item. RSD is ex-ante envy-free (EF), since each agent has the same chance to appear in each position in the ordering. Obviously, it is not ex-post envy-free, since an agent who happens to be last in the random permutation might envy agents who appear earlier. RSD always yields an ex-post Pareto efficient  (PE) outcome. However, it is not ex-ante PE when the agents have Von Neumann-Morgenstern utilities over random allocations, i.e., lotteries over objects (Note that ex-ante envy-freeness is "weaker" than ex-post envy-freeness, but ex-ante Pareto-efficiency is "stronger" than ex-post Pareto-efficiency). As an example, suppose there are three agents, three items and the VNM utilities are: RSD gives a 1/3 chance of every object to each agent (because their preferences over sure objects coincide), and a profile of expected utility vector (0.6, 0.4, 0.4). But assigning item y to Alice for sure  and items x,z randomly between Bob and Carl yields the expected utility vector (0.8, 0.5, 0.5). So the original utility vector is not Pareto efficient. When the rankings of the agents over the objects are drawn uniformly at random, the probability that the allocation given by RSD is ex-ante PE approaches zero as the number of agents grows. An alternative rule, the probabilistic-serial rule, is ex-ante PE (which implies ex-post PE) and ex-ante EF, but not truthful. There is provably "no" mechanism that is symmetric, truthful and ex-ante PE. Generalizations. More objects than  agents. When there are more than formula_1 objects, some agents may get more than one object. There are several ways to extend RSD to this case. General decision-making. RSD can be defined for the more general setting in which the group has to select a single alternative from a set of alternatives. In this setting, RSD works as follows: First, randomly permute the agents. Starting with the set of all alternatives, ask each agent in the order of the permutation to choose his favorite alternative(s) among the remaining alternatives. If more than one alternative remains after taking the preferences  of all agents into account, RSD uniformly randomizes over those alternatives. In the item division setting mentioned earlier, the alternatives correspond to the allocations of items to agents. Each agent has large equivalence classes in his preference, since he is indifferent between all the allocations in which he gets the same item. In this general setting, if all agents have strict preferences over the alternatives, then RSD reduces to drawing a random agent and choosing the alternative that the agent likes best. This procedure is known as random dictatorship (RD), and is the unique procedure that is efficient and strategyproof  when preferences are strict. When agents can have weak preferences, however, no procedure that extends RD (which includes RSD) satisfies both efficiency and strategyproofness.