Random priority item allocation
Random serial dictatorship (RSD),[1] also called: random priority (RP),[2] is a procedure for dividing indivisible items fairly among people.
Suppose partners have to divide (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 fair, at least in the sense of equal treatment of equals, since each agent has the same chance to appear in each position in the ordering.
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 just to pick the best available item.
However, RSD is not Pareto efficient when the agents have Von Neumann-Morgenstern utilities over random allocations (lotteries over objects). In fact, there exits no mechanism that satisfies symmetry, truthfulness and Pareto efficiency.[3]
As an example, suppose there are three agents, three items and the VNM utilities are:
| Item x | Item y | Item z | |
|---|---|---|---|
| Alice | 1 | 0.8 | 0 |
| Bob | 1 | 0.2 | 0 |
| Carl | 1 | 0.2 | 0 |
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.[2]
In fact, when the rankings of the agents over the objects are drawn uniformly at random, the probability that the allocation given by RSD is Pareto efficient approaches zero as the number of agents grows.[4]
Generalization
RSD can also 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.[5] When agents can have weak preferences, however, no procedure that extends RD (which includes RSD) satisfies both efficiency and strategyproofness.[6]
See also
References
- ^ Abdulkadiroglu, Atila; Sonmez, Tayfun (1998). "Random Serial Dictatorship and the Core from Random Endowments in House Allocation Problems". Econometrica. 66 (3): 689. doi:10.2307/2998580. JSTOR 2998580.
- ^ a b Bogomolnaia, Anna; Moulin, Hervé (2001). "A New Solution to the Random Assignment Problem". Journal of Economic Theory. 100 (2): 295. doi:10.1006/jeth.2000.2710.
- ^ Zhou, Lin (1990). "On a conjecture by gale about one-sided matching problems". Journal of Economic Theory. 52: 123. doi:10.1016/0022-0531(90)90070-Z.
- ^ Manea, Mihai (2009). "Asymptotic ordinal inefficiency of random serial dictatorship". Theoretical Economics. 4 (2): 165–197.
- ^ Gibbard, Allan (1977). "Manipulation of schemes that mix voting with chance". Econometrica. 45 (3): 665–681.
- ^ Brandl, Florian; Brandt, Felix; Suksompong, Warut (2016). "The impossibility of extending random dictatorship to weak preferences". Economics Letters. 141: 44–47. arXiv:1510.07424. doi:10.1016/j.econlet.2016.01.028.