1. Qing Cui 2014/09/30

2.  Introduction of matching theory  Stable Marriage, extensions and maximum-weighted stable matching problem. By Prof. Chen. 2014.3.3  Matching markets and market clearing price. By Prof. Wang. 2014.3.10  Deep understanding on matching  Two fundamental algorithms (stability) and several perspectives on dimensionality. By Qing. 2014.4.14  Stable roommates and the connection with two fundamental algorithms. By Zisang. 2014.5.5  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. 2014.5.5.  Solving scheduling problem by market clearing price. By Qing. 2014.5.26 and 2014.6.3.

3.  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)

4.  Two papers  Abraham Othman, Tuomas Sandholm, and Eric Budish. 2010. 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): 913-935.  One professor  http://se.shufe.edu.cn/structure/zh/shizhidw/xfjjx_con_28484_1.htm http://se.shufe.edu.cn/structure/zh/shizhidw/xfjjx_con_28484_1.htm  http://shufemd.weebly.com/2014-spring.html http://shufemd.weebly.com/2014-spring.html  Part of one book (in our small seminar)  Two-Sided Matching A Study in Game-Theoretic Modeling and Analysis

5.  Abraham Othman, Tuomas Sandholm, and Eric Budish. 2010. 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  https://spike.wharton.upenn.edu/mbaprogram/course_match/ https://spike.wharton.upenn.edu/mbaprogram/course_match/

6.  Definition of course allocation problem  Budish’s approximate CEEI mechanism (A-CEEI)  Why traditional solutions fail  The algorithm: agent level and master level  Experiments

7.  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.

10. 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)

11.  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

12.  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.

13.  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.

14.  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

16.  Cost most time of the algorithm  Can be parallized

17.  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

22.  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