Qing Cui 2014/09/30
Introduction of matching theory Stable Marriage, extensions and maximum-weighted stable matching problem. By Prof. Chen Matching markets and market clearing price. By Prof. Wang Deep understanding on matching Two fundamental algorithms (stability) and several perspectives on dimensionality. By Qing Stable roommates and the connection with two fundamental algorithms. By Zisang Research The solvability of stable roommates when the scale increases. By Zisang. Application on time scheduling problem Consistent preference and student project allocation. By Qing Solving scheduling problem by market clearing price. By Qing and
Elementary concepts and results (1 lesson) The structure and representation of all stable matchings (2 lessons) Building and exploiting the representation of all stable matchings (1 lesson) The stable roommates problem (2 lessons) The perfect matching on hypergraph (1 lesson)
Two papers Abraham Othman, Tuomas Sandholm, and Eric Budish Finding approximate competitive equilibria: efficient and fair course allocation. In Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems (AAMAS '10) (Similar version published in Journal of Political Economy in 2011) Hatfield, John William, and Paul R. Milgrom. "Matching with contracts."American Economic Review (2005): One professor Part of one book (in our small seminar) Two-Sided Matching A Study in Game-Theoretic Modeling and Analysis
Abraham Othman, Tuomas Sandholm, and Eric Budish Finding approximate competitive equilibria: efficient and fair course allocation. In Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems (AAMAS '10) He has commercialized this paper
Definition of course allocation problem Budish’s approximate CEEI mechanism (A-CEEI) Why traditional solutions fail The algorithm: agent level and master level Experiments
The course-allocation problem belongs to a broader class of problems called combinatorial assignment, in which a set of indivisible integer-supply objects is to be allocated amongst a set of agents with preferences over bundles, without the use of monetary transfers.
Two students {1,2} and four courses {a,b,c,d} each in unit supply. Students consume at most two courses each. Students’ utility functions are additive separable over objects. Preferences (normalized to sum to 100)
Unfair It will allocate both of the good courses to student 2 Create incentives to misreport If student 1 report (63, 33, 3, 1) then she will get both of the good courses The reason why it works well in auction is real money transfer. Winner determination problem from combinatorial auctions
In their problem, money is artificial and has no outside use There is no easy way to augment this problem so as to implement A-CEEI mechanism.
Eisenberg-Gale paradigm Assume the goods are divisible Solve for an optimal allocation as if the goods were perfectly divisible and then “round” the resulting fractional allocation to a “nearby” integer allocation – may be dangerous about fairness. Allocate students equal budgets of artificial currency and highest bidders get a seat. (PKU) No money transfer. Similar to the first problem.
Preference language How the agents report their valuation for courses The agent level Each agent searches through bundle space to find their most- preferred affordable bundle at the current prices. Use Mixed Integer Programming (MIP) The master level The center searches through price space to determine what prices to next propose to the agents. Use Tabu search
Cost most time of the algorithm Can be parallized
Tabu Neighborhood Selection Gradient Descent (global) Individual Price Adjustments The key difficulty is determining the minimum price increase to lower demand on an oversubscribed course by exactly one student General case Solve with MIPs
Preference need normalized? Select the most preferred class according to what? Value – price or the affordable best? Local search method like Tabu search Hybrid neighborhoods Parallezation