1. Local Computation Mechanism Design Shai Vardi, Tel Aviv University Joint work with Avinatan Hassidim & Yishay Mansour Men’s preferences first second Women’s preferences first second third fourth

2. Local Computation Mechanism Design Shai Vardi, Tel Aviv University Joint work with Avinatan Hassidim & Yishay Mansour Men’s preferences first second Women’s preferences first second third fourth

3. Local Computation Mechanism Design? Mechanism Design using Local Computation Algorithms What is

4. Which items does each bidder get? How much does each bidder pay? Meet some requirements, e.g., High social welfare, Truthfulness, Pareto efficiency, …. A mechanism for a combinatorial auction

5. Which items does Bob get? How much does Bob pay? Bob Meet some requirements, e.g., High social welfare, Truthfulness, Pareto efficiency, ….

6. When do we need LCAs?  Huge input.  Not enough time or space to compute the entire solution.  Only need small parts of the solution at any one time. Local Computation Algorithms (LCAs)

7. Local Computation Algorithms LCAs implement query access to a global solution. We require LCAs to be:  Fast – at most polylogarithmic in the input size per query.  Space-efficient – at most polylogarithmic overall.  Replies to all queries are consistent with the same solution.

8. Some motivation  Imagine a huge auction, with millions of items and hundreds of thousands of buyers. An item arrives to be shipped. We don’t want to have to compute the result of the entire auction, just to know to which buyer to ship the item….  There is a cloud with millions of computers on which we would like to schedule millions of jobs. We are queried on a job and would like to reply on which machine it should run. But we don’t want to compute the entire schedule….

9. stable matching The problem: There is a group of men and women. Each man has a preference list over the women, and each woman has a preference list over the men. A stable matching is one in which there is no man and woman who both prefer each other over their matched partner. We would like to query a man/woman to find their partner in some stable matching. Local

10. The Gale-Shapley algorithm (1962) Men’s preferences first second Women’s preferences first second third fourth (male courtship) phillip phyllis flynn jen jane bob mike

11. Restrictions in previous work  Truncation strategies in matching markets. Roth and Rothblum, 1999.  Marriage, honesty, and stability. Immorlica and Mahdian, 2005. We use the model of Immorlica and Mahdian – the men’s lists are truncated and selected independently and uniformly at random.

12. Local mechanism for stable matching

13. Stable matching LCA

16. 1) Why is it an LCA?

17. 3 3 3 3 3 3 2 21 1 Simulating the Gale-Shapley algorithm Men’s preferences first second Women’s preferences first second third fourth jen jane round queries reply 1 3 jen 2 5 none 3 11 jane bob phillip phyllis flynn k=2 mike

18. 2) How many men are left unmatched? k=1

19. 3) How many men are disqualified?

20. Local mechanism for stable matching

21. Some more local computation mechanisms

22. A general result on the Gale Shapley algorithm

23. Thank you for your attention!