Regret-based Incremental Partial Revelation Mechanism Design

1. Regret-based Incremental Partial Revelation Mechanism Design Nathanaël Hyafil, Craig Boutilier AAAI 2006 Department of Computer Science University of Toronto

2. $$ $$ $$ $$ $$ $$ $$ $$ Bargaining for a Car Luggage Capacity? Two Door? Cost? Engine Size? Color? Options?

3. Mechanism Design • Mechanism design tackles this: • Design rules of game to induce behavior that leads to maximization of some objective(e.g., social welfare, revenue, ...) • Objective value depends on private information held by self-interested agents  Elicitation + Incentives

4. “Computational” Mechanism Design • The interesting questions: • what preference info is relevant to the task at hand? • when is the elicitation effort worth the improvement it offers in terms of decision quality? • how to deal with incentives ?

5. Overview • Mechanism Design Background • Incremental Partial Revelation Mechanism • Regret-based iPRMs • Experimental results • Conclusion / Future Work

6. Basic Social Choice Setup • Choice of x from outcomes X • Agents 1..n: typetiTi and valuationvi(x, ti) • Type vectors: tT • Goal: implement social choice functionf: T  X • e.g., social welfare SW(x,t) =  vi(x, ti) • Quasi-linear utility: • ui(x, i ,ti ) = vi(x, ti ) - i • Our focus: social welfare maximization

7. Basic Mechanism Design • A mechanismm consists of three components: • actions Ai • allocation function O: A X • payment functions pi : A R • Mechanism is incentive compatible: • In equilibrium, agents reveal truthfully • Ex-post IC • Assume others tell the truth and agent i knows the others’ types • Then agent i should tell the truth

8. Properties • Mechanism is efficient: • maximizes social welfare given reported types: •  -efficient: within  of optimal social welfare • Ex post individually rational: • no agent can lose by participating • -IR: can lose at most 

9. Direct Mechanisms • Revelation principle: focus on direct mechanisms where agents directly and (in eq.) truthfully reveal their full types • For example, Groves scheme (e.g., VCG): • choose efficient allocation and use payment function: • incentive compatible in dominant strategies • efficient, individually rational

10. Cost of Full Revelation • Communication costs • Computation costs • Cognitive costs • Privacy costs INTRACTABLE! Partial revelation?

11. Partial Revelation • Full revelation: • Not always necessary for optimal decision • When necessary, not always worth the costs • Partial revelation: • Elicit just enough to make optimal decision • Trade-off elicitation costs with decision quality • Can we maintain incentives?

12. Existing Work on Partial Revelation [Conen,Hudson,Sandholm, Parkes, Nisan&Segal, Blumrosen&Nisan] • Most Work: • require enough revelation to determine optimal allocation and VCG payments • hence can’t offer savings in general [Nisan&Segal05] • Exception:Priority games [Blumrosen&Nisan 02] • specific settings (1-item, combinatorial auctions)

13. Overview • Mechanism Design Background • Incremental Partial Revelation Mechanism (iPRM) • Regret-based iPRMs • Experimental results • Conclusion / Future Work

14. Incremental Partial Revelation Mechanisms (iPRMs) • iPRM interacts with agents: • set of queriesQi (e.g. standard gamble:“v( car ) >5?”) • response rRi(qi) interpreted as partial type i (r)Ti(e.g. bounds on each parameter) • Formal Model (see paper)

15. iPRMs • Goal: • Trade-off quality of alloc. with revelation costs • Maintain acceptable incentives properties • At each step, given , choose between: • Terminating (which allocation?) • Eliciting (which query?)

16. Minimax Regret: Utility Uncertainty • Regret : • Max regret of x given  : • MMR-optimal allocation: x* = arg minx MR(x, )

17. Overview • Mechanism Design Background • Incremental Partial Revelation Mechanism • Regret-based iPRMs • Experimental results • Conclusion / Future Work

18. Regret-based Elicitation • Find query to reduce MMR level? • Several heuristics proposed for preference elicitation. • We adapt Current Solution Strategy (CSS) • Focus elicitation on allocations involved in regret

19. Allocation Elicitation • Proposed allocation elicitation algorithm • Using SW-regret computation and elicitation • See paper for details • Allocation elicitation phase terminates with •  -efficient allocation • Partial type 

20. Incentive Properties • Let mechanism M = (x* , piT), with • -efficient allocation function x* • payments: piT(x* ; ) = maxt  piVCG(x* ; t) • Theorem 1: • M is -efficient, - ex post IR, - ex post IC •  = +() • (): bound on payment uncertainty

21. Approximate Incentives •  : bound on utility gain • But gain from manipulation outweighed by costs of manipulation • don’t know types of others • must simulate mechanism • Formal, approximate IC  practical, exact IC

22. 2 Phase Approach • Bound on manipulability: •  + () : not a priori • If () too large: • Elicit to reduce payment uncertainty • Payment elicitation strategy: based on CSS (P-CSS) • Terminates with a priori bounds • ( + ) -IC • -IR , -efficiency

23. Direct Optimization • Causes of manipulability: • efficiency loss + payment uncertainty • MMR w.r.t. SW only accounts for efficiency loss • Should minimize global worst-case manipulability: • u(best lie) - u(truth) • efficiency loss bounded by worst-case manipulability • Formulate as regret optimization and elicitation • ask queries that directly reduce global manipulability

24. Single Phase Approach • Theorem 2:For M = (x* , piT), • If  =max worst case manipulability • Then M is • -efficient • - ex post IC • - ex post IR

25. Overview • Mechanism Design Background • Incremental Partial Revelation Mechanism • Regret-based iPRMs • Experimental results • Conclusion / Future Work

26. Elicitation Strategies • Two Phase (2P): • SW loss and payment uncertainty for elicitation and decisions •  Two Phase ( 2P): • SW loss and payment uncertainty for elicitation • Manipulability for decisions • Common-Hybrid (CH): • Manipulability for elicitation and decisions • Myopically Optimal (MY): • Simulate all queries, ask best

27. Test Domains • Car Rental Problem: • 1 client , 2 dealers • Car: 8 attributes, 2-9 values, ~12000 cars • factored valuation/costs: 13 factors, size 1-4 • Total 825 parameters • Small Random Problems: • supplier-selection, 1 buyer, 2 sellers • 81 parameters

28. Results: Car Rental Initial regret: 99% of opt SW Zero-regret: 71/77 queries Avg remaining uncertainty:92% vs 64% at zero-manipulabilityAvg nb params queried: 8% • relevant parameters • reduces revelation • improves decision quality

29. Results: Random Problems

30. Conclusion • Theoretical model for iPRMs • Class of iPRMs with approximate incentives • Key point: • Approximation trade off cost vs. quality • Formal, approximate IC  practical, exact IC • Applicable to general mechanism design • Empirically very effective

31. Current + Future Work • More heuristics + test domains • Formal model manipulation and revelation costs  formal, exact IC  explicit revelation/quality trade-off • Sequentially optimal elicitation • One-shot partial revelation mechanisms“Mechanism Design with Partial Revelation” draft 2006

32. Questions?