Private Matchings and Allocations
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Presentation at STOC 2014
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Private Matchings and Allocations
- 1. Private Matchings and Allocations Joint work with Justin Hsu (Penn) Zhiyi Huang (HKU) Aaron Roth (Penn) Tim Roughgarden (Stanford) Speaker: Steven Wu University of Pennsylvania STOC 2014
- 3. An important special case: 𝑚 goods 𝑗 ∈ [𝑚] 𝑛 agents i ∈ [𝑛] 1 2 3 A B C • Unit demand valuations • Equivalent to max-weight matching
- 4. Our Goal High social welfare allocation Privacy (without revealing individual private valuations)
- 5. D Differential Privacy Algorithm ratio bounded Alice Bob Chris Donna ErnieXavier
- 6. Differential Privacy • An algorithm A with domain X and range R satisfies ε-differential privacy if for every outcome r and every pair of databases D, D’ differing in one record: Pr[ A(D) = r ] ≤ (1 + ε)Pr[ A(D’) = r] • Domain: Reported valuation functions • Range: Matchings
- 7. Problem: Assignment Reveals Preference • Problem: High welfare matching will give people what they want.
- 10. Joint Differential Privacy (KPRU’14) (MM09, GLMRT10)
- 11. Supply Assumption • We need multiple copies for each type of good even under JDP. • How many? Impossible Trivial
- 12. Main Result Theorem: There is a JDP algorithm in the that solves the max-weight matching problem with n people and k types of goods with supply at least s each, and outputs a matching of weight OPT – αn whenever:
- 13. A Framework for JDP The “Billboard Model”
- 14. “Low information” Signal o From the signal, every bidder can figure out what item they are matched to in a matching o Does not reveal each individual’s private data • Think: Prices
- 15. Max Matchings (A Sketch) A remarkable algorithm for Max-Matchings: [Kelso and Crawford ’82]
- 17. Welfare
- 18. Prices as information Claim: Bidders just need to see the prices 1. Prices are sufficient to identify the favorite good 2. When price raises again, a bidder is unmatched 3. Bidders are matched to the last thing they bid on • Just need to count how many bids each good received!
- 19. Privately Maintaining Counts 1 0 0 1 1 1 0 1 032 • Private (noisy) counters under continual observation [DNPR10, CSS10] • Given a stream of T bits, maintain an estimate of the running count with accuracy o Single Stream of sensitivity 1
- 20. Privately Maintaining Counts 1 1 1 1 1 1 0 0 018 1 0 0 1 1 1 0 1 032 0 0 0 1 1 1 1 1 0192 • A straightforward generalization: K counters on K streams that collectively have sensitivity Δ gives accuracy
- 21. Lower Sensitivity Stopping the auction early with a new condition • Sensitivity
- 22. Counter Error • Error per bid counter
- 23. Supply • Goods might also be under/over-allocated by E. o Doesn’t reduce the welfare by more than (1-α) factor if
- 24. Main Theorem Theorem: There is a private algorithm in the billboard model that solves the max-weight matching problem with n people and k types of goods with supply at least s each, and outputs a matching of weight OPT – αn whenever:
- 25. Extensions • Results extend to the allocation problem when buyers have gross substitute preferences.
- 26. Conclusions • Some problems that can’t be solved under DP can be solved under joint-DP. o If the output is partitioned among the agents o The agent’s output is allowed to be sensitive in his input. • Billboard model: interesting framework to design a joint-DP algorithm?
- 27. Private Matchings and Allocations Joint work with Justin Hsu (Penn) Zhiyi Huang (HKU) Aaron Roth (Penn) Tim Roughgarden (Stanford) Speaker: Steven Wu University of Pennsylvania
Hinweis der Redaktion
- Of course, since our talk is the last talk of this session. The first notion of privacy we would look at is differential privacy. Some of you might have seen this pictures many times. It says that if we change the input database by one record, the output distribution induced by our randomized algorithm is not affected by much.
- If there is no contention for goods, a high welfare matching has to give people what they want. A high welfare allocation will give people what they want. Consider a scenario where people either like alcohol or cigarettes. The tension between social welfare and privacy makes it impossible to work with standard differential privacy.
- Not the songs you heard on radio. A nice decentralized model
- The idea is to have our mechanism broadcast some low information signal to all of the agents. The signal itself should satisfy standard DP. And a natural candidate for such a signal in allocation problem is prices.