House Allocation with Existing Tenants and the Stable Roommate Problem

Transcription

1 House Allocation with Existing Tenants and the Stable Roommate Problem Christopher Ziegler Technische Universität München May 8, 2014 Christopher Ziegler (TUM) House Allocation and Roommate Matching May 8, / 14

2 Overview 1 House Allocation with Existing Tenants The problem itself Top trading Cycles mechanism Desirable properties of the outcome of the algorithm The algorithm itself Applications 2 Stable Roommate Problem The problem itself Algorithm for the stable roommate problem Phase 1 Preparation for Phase 2 Phase 2 Applications 3 Conclusions Christopher Ziegler (TUM) House Allocation and Roommate Matching May 8, / 14

3 House Allocation with Existing Tenants There are two sets, namely H(ouses) and A(gents). A has a preference ranking, which is a total order over H, but H has no preference ranking over A. Goal: find pairs of one element in A and one in H such that the people in A are as happy as possible. Some elements of H and A are yet paired and some not. You can not force an existing tenant out of his house. Christopher Ziegler (TUM) House Allocation and Roommate Matching May 8, / 14

4 Desirable Properties of the outcome of the algorithm Pareto-Efficiency You can not make the house of an agent a more desirable for a without making the house of another agent a worse for a. Individual Rationality When you participate in the algorithm as an agent a A, it does not worsen your house if you had one before the algorithm starts. This is important to increase the participation of the tenants. Strategy-Proofness There is no preference ranking such that an agent a A gets a better outcome by misrepresenting his/her individual preference ranking. Christopher Ziegler (TUM) House Allocation and Roommate Matching May 8, / 14

5 Algorithm Fix an ordering over the agents. Build a directed graph G = (V, E) with V=A H and the following edges. Let each agent point to his/her favourite house among the current houses. Let each occupied house point to its occupant, each available house to the current agent with the highest priority. Find a cycle, for example with depth-first search (it exists, why?). Assign each agent in the cycle the house he/she pointed to. Remove agents and houses in the cycle and go to bullet point number 2. Christopher Ziegler (TUM) House Allocation and Roommate Matching May 8, / 14

6 Proof of some properties Theorem (This algorithm is Pareto-efficient) Any agent who gets his house in the first step gets his/her favourite choice, which clearly can not be improved. Any agent a who gets his house in the i-th step gets his/her favourite house among the remaining houses. Hence, a s house can only be improved by taking the house from an agent b who obtained his/her house in prior steps, which would worsen b s house. Theorem (This algorithm is individually rational) Consider a tenant a of the house h. h points to a until a gets removed, because h has only one outgoing edge. Therefore a gets his/her favourite house among a set of houses which is a superset of the set of his/her house. Hence, a can not get a worse house than his/her old one. Christopher Ziegler (TUM) House Allocation and Roommate Matching May 8, / 14

7 Applications Assign elementary schools/high schools. Assign workplaces/parking places. Sharing out duties/calculations in distributed systems (with different compters). College housing. Welfare housing. Not(!) normal housing, because landlords/landladies have preferences, too. Christopher Ziegler (TUM) House Allocation and Roommate Matching May 8, / 14

8 Stable Roommate Problem R is a set of an even number of roommates. Every r R has a preference ranking over R \ {r}. The aim is to partition the elements of R in pairs stable or discovering it is impossible to do so. Stability Two elements of R, who are not paired in the partition do not prefer each other over their own partners. Christopher Ziegler (TUM) House Allocation and Roommate Matching May 8, / 14

9 Phase 1 Every agent a in the set of R proposes to his/her favourite agents in a s preference list until a gets maybe as an answer. An agent b who gets a proposal from a returns immediately no, if b has received a proposal from someone, who b prefers over a. Else, b returns maybe to a and eventually no to someone, namely c who obtained a maybe from b before, because now b has a better proposal. Then, of course c has to propose to someone else. The phase ends, when every agent holds a proposal or when an agent is rejected by every other agent. Christopher Ziegler (TUM) House Allocation and Roommate Matching May 8, / 14

10 Preparation for Phase 2 Now, we consider agent y holding a proposal from agent x. Consider the list of possible roommates for y, initially R \ {y}. z R : if y prefers x over z: delete z from y s list and delete y from z s list. Christopher Ziegler (TUM) House Allocation and Roommate Matching May 8, / 14

11 Phase 2 If every agent has one agent on his/her current list, then match them and return the matching. Else if an agent has an empty list then there is no matching. Else: repeat the following steps until one of these conditions are satisfied. find cyclic sequence a 0,..., a r 1 such that for i = 0,..., r 1 the second agent in a i s current preference list is the first one in a i+1 mod r s. This person is denoted by b i+1 mod r This can be done very easily, set a 0 an arbitrary agent, whose preference list contains two or more elements. b i+1 : second person in a i s current list a i+1 :last person in b i+1 s list (such that b i+1 is the first one on a i+1 ) We stop, until an a i = a i j, for a fitting j because then we have a cycle. Now we force each b i to reject the proposal of a i and therefore we force each a i to propose to b i+1 mod r. Christopher Ziegler (TUM) House Allocation and Roommate Matching May 8, / 14

12 Applications Finding roommates (e.g. in college housing). Dating agency/marriage bureau for homosexual people. Assigning bureaus, that are built for two workers. Ride sharing (in cities, who forbid to drive alone in a car, because of air pollution etc.). Christopher Ziegler (TUM) House Allocation and Roommate Matching May 8, / 14

13 Conclusions The Top trading cycles mecanism allocates houses with existing tenants. satisfies Pareto-efficiency, individual rationality and strategy-proofness. The stable roommates algorithm checks whether a stable matching exists. if so, the algorithm computes it. has complexity O(n 2 ). Christopher Ziegler (TUM) House Allocation and Roommate Matching May 8, / 14

14 References Atila Abdulkadiroglu, Tayfun Sönmez: House Allocation with Existing Tenants, in the Journal of Economic Theory, 1999 Robert W. Irving: An Efficient Algorithm for the Stable Roommates Problem, in the Journal of Algorithms, 1985 Szilvia Pápai: Strategyproof Assignment by Hierarchical Exchange, in the Econometrica, 2000 Christopher Ziegler (TUM) House Allocation and Roommate Matching May 8, / 14