Computational Mechanism Design Nathanaël Hyafil Depth Oral Examination
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
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, ...
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
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
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
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
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: …
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)
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
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
Priority Games • ex: 2 agents, 1 bit per agent B A
Priority Games: allocation • if one sends strictly higher bid: he wins B A
Priority Games: allocation • if tie: fixed order; here B > A B A
Priority Games • payment rule: • 2k thresholds per agent; Here (0,tA) ; (0,tB) B A
Priority Games • Dominant strategy: A bids ‘1’ if and only if vA tA(Threshold strategy) B A
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
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
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
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
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)
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
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
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
Price-based: CAs • Computation: • more WD to solve but smaller instances • empirically faster • Communication: • more bids to send but of smaller size • price posting
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
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
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
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
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
Automated Mechanism Design • Problems: • Continuous outcome space • Continuous type / action space • Partial revelation • Priors hard to quantify
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
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
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
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)
