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Computational Mechanism Design

Computational Mechanism Design. Nathanaël Hyafil Depth Oral Examination. Outline. Background Mechanism Design Vickrey Clarke Groves I. The three issues and direct ‘solutions’ Valuation Complexity Computation Complexity Communication Complexity II. Sequential Mechanisms Price-based

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Computational Mechanism Design

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  1. Computational Mechanism Design Nathanaël Hyafil Depth Oral Examination

  2. Outline • Background • Mechanism Design • Vickrey Clarke Groves • I. The three issuesand direct ‘solutions’ • Valuation Complexity • Computation Complexity • Communication Complexity • II. Sequential Mechanisms • Price-based • General query-based • III. Automated Mechanism Design • IV. Future Research

  3. Mechanism Design • Choose global outcome according to some objective (Social Welfare, Revenue, ...) • Objective value depends on private information held by self-interested agents  Elicitation + Incentives • Applications: • Economics: Auctions, Public Goods, ... • E-Commerce: multi-attribute Negotiation (Supply chain), ... • Comp.Sc.: Networks, Autonomic computing, ...

  4. Vickrey Clarke Groves (VCG) • Objective: Social Welfare • Agents report full valuation function • Outcome: that maximizes SW given reports • Payment: cost of your presence to the others • find outcome that maximizes SW without i, i • truth-telling: dominant strategy • wide application: quasi-linear environments

  5. 1. Valuation complexity • Valuation: a value for every outcome • Large outcome spaces (e.g. CA): • exponential number of values to compute • 1-item: hard to evaluate valuation exactly • e.g., 1 hour of CPU time

  6. 1. Valuation complexity • Work focused on analyzing valuation complexity in standard auctions (Parkes-04, Larson&Sandholm-01,03,04) • No direct solution to valuation problem but indirectly through work on communication complexity

  7. 2. Computational complexity • Problem: Large Outcome Spaces • e.g., Combinatorial Auctions: number of allocations is K^N • Finding efficient allocation (WD): • NP-complete (RPH-98) • VCG: must do this (n+1) times, for n agents

  8. Solutions for Combinatorial Auctions • Faster optimal allocation: (SSGL-01) • faster on ‘common’ instances • Branch & Bound • Heuristic search • Special cases: (RPH-98,Nisan-00,Tenn.-00) • Structure in preferences that make it ‘easy’ • Approximate allocation: …

  9. Approximate Allocation • Also intractable in worst case, if guarantees on quality of approximation • (Hastad 99) • Lose Incentive Compatibility in VCG • (LOCS 2002; NR 2000) • Settle for ‘hard’ to manipulate • -improvement NP-hard (Sanghvi & Parkes 04) • feasibly-dominant (Nisan&Ronen 00)

  10. 3. Communication complexity • Valuation: one real number for every outcome • communication costs: • problem for large outcome spaces, e.g. CA: • worst case exponential (Nisan & Segal 2003) • even 1-item auctions for low value resources (bandwidth, CPU time...) • cost of sending 1 real number is significant • privacy / secrecy issues

  11. Communication in 1-Item Auctions • Priority Games: (Blumrosen & Nisan 02 ; BN & Segal 03) • hard limit on communication: k bits/agent • 2k bids possible per agent

  12. Priority Games • ex: 2 agents, 1 bit per agent B A

  13. Priority Games: allocation • if one sends strictly higher bid: he wins B A

  14. Priority Games: allocation • if tie: fixed order; here B > A B A

  15. Priority Games • payment rule: • 2k thresholds per agent; Here (0,tA) ; (0,tB) B A

  16. Priority Games • Dominant strategy: A bids ‘1’ if and only if vA tA(Threshold strategy) B A

  17. Priority Games • allocation is not (always) optimal • Any set of thresholds induces a dominant (threshold) strategy • Given a prior, they show how to optimize thresholds to limit loss in Social Welfare

  18. Communication in CAs • Complete Languages: • fully expressive • concise on ‘important’ classes of valuations • easy to interpret • Restricted Languages: • restrict bidding (e.g., to some bundles) • report ‘closest’ allowed valuation

  19. Complete Language:LGB(BH 2001) • Ex: Machine m ; Resources r1,r2,r3, … • [ (m  r1;p1)  (m  r2;p2)  … ; p0 ] • Logical combinations of bids and goods •  ,  ,  • prices in any sub-formula

  20. LGB (BH 2001) • more compact (sometimes exponentially) • easily interpretable • easily expressed • IP for Winner Determination with LGBbids: • faster than standard (flat) IP and heuristic search • much larger problems

  21. Restricted Languages • VCG with restricted revelation not always IC • truth-telling: reporting ‘closest’ valuation • if true valuation not allowed, possibly lie • restrict to classes of valuations (Ronen 01) • feasible-dominance • less successful: unrealistic assumption(s) • restrict to subset of bundles (HKMT-03) • IC: dominant for some choices of subset • analysis of loss in SW ; (not very informative)

  22. II. Sequential Mechanisms • Addressing the three issuesseparately in VCG-based mechanisms: • not successful; everything ‘connected’ • e.g., NR-00: less computation, more communication • Move away from VCG to get: • Approximate outcome without losing IC • Same Vickrey outcome without full revelation

  23. Price-based: One-Item auctions • ASIA: (Kress&Boutilier 04) • ‘sequential partitioning’: 1 price / round • report ‘in’ or ‘out’ ( 1 bit/round/agent) • exact allocation OR no allocation • DS: stay ‘in’ if and only if value  price! • ‘Optimal’ price increase rule (MDP solution) • communication vs. efficiency

  24. Price-based: CAs • iBundle and extensions (Parkes01,Parkes&Ungar-00): • announce prices for each bundle at each round • (truthful) myopic best-response: bid on bundle that maximizes (true) utility given prices • at least ex-post equilibrium • with price adjustments: non-dynamic rev. dominant

  25. Price-based: CAs • Computation: • more WD to solve but smaller instances • empirically faster • Communication: • more bids to send but of smaller size • price posting

  26. Query-based:Heuristic Search(Conen&Sandholm 01,02) , (Hudson&Sandholm 03) • Centralized elicitation with: value, order, rank queries • Exponential worst case, but in practice... • Information structures: rank lattice, order graph • Incentives only through Vickrey payments: • in some cases ‘for free’ • Communication: empirically (on ‘important’ valuations), small fraction of valuation revealed • Computation: exponential complexity to maintain information structures

  27. Query-based:Learning Theory(SCS04,ZBS-03,BJSZ04, LP04) • Learn classes of functions while minimizing number of queries to an oracle • Queries: membership=value, equiv.=demand • Important: learn allocation function, not valuations! (Lahaie & Parkes 04) • IC: through Vickrey payments; not free • But: exact learning, even if cost > benefits

  28. Query-based: Decision Theory • Before: minimize number of queries to get optimal decision • DT approach: elicit information to improve decision, if it is worth the cost • Value Of Information starting to be used: • Chajewska, Koller, Parr 2000 • Boutilier 2002

  29. POMDP model of PE(Boutilier-02) • Sequentially optimal elicitation • trades-off quality of decision with cost of elicitation • arbitrary cost models • computationally very complex: approximation needed • no IC; Vickrey payments don’t apply

  30. III. Automated Mechanism Design • VCG: one mechanism for all priors (SW) • Myerson auction: one algorithm, outputs an auction for each prior (Revenue) • AMD = Myerson for general MD • search for mechanism as an LP • IC and IR imposed as constraints • LP objective = designer ’s expected objective

  31. Automated Mechanism Design • Problems: • Continuous outcome space • Continuous type / action space • Partial revelation • Priors hard to quantify

  32. Past Research: Strict Uncertainty • Hyafil & Boutilier, UAI 2004: • Games with strict (qualitative) uncertainty • cannot Maximize Expected Utility • should Minimize Max Regret • Minimax-Regret Equilibrium • Minimax-Regret (Partial Revelation) AMD • sequence LPs and MIPs

  33. IV. Current / Future Research • Partial Revelation: • partition agent type space • report subset containing type • pick outcome despite remaining uncertainty (approximately optimal) • Partial Revelation objectives: • ex-post expected full objective • Min ex-post MaxRegret wrt full objective • IC: dominant / BayesNash / MinimaxRegret

  34. Current / Future Research • General case: VCG-like algorithm • need ‘ tie-breaking ’ and payment rule • AMD: optimize mechanism + partition • non-linear LP objective • quadratic constraints • for direct mechanisms with dominant strategies, in general setting: ‘impossible’? • BayesNash? Minimax Regret? • Sequential mechanisms

  35. Current / Future Research • specific application: multi-attribute negotiation • incentives issues • computational issues • MinimaxRegret-based Mechanism Design • use MMR as guide for sequential elicitation • incentives through Vickrey payments • more useful incentives properties?? • (MinimaxRegret-Equilibrium applications)

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