1 (One-shot) Mechanism Design with Partial Revelation Nathanaël Hyafil, Craig Boutilier IJCAI 2007 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 Partial Revelation Mechanism Design  Problem: Stating full utility is intractable Costs: communication, computational…  Partial Revelation: what preference info is relevant to decision? when is the elicitation cost worth the improvement in decision quality? how to deal with incentives ?

5 Overview  Mechanism Design Background  Partial Revelation Mechanisms (PRM)  Regret-based PRMs  Partition Optimization  Experimental Results

6 Basic Social Choice Setup  Choice of x from outcomes X (e.g. cars)  Agents 1..n: type t i  T i and valuation v i (x, t i )  Type vectors: t  T  Goal: implement social choice function f: T  X e.g., social welfare SW(x,t) =  v i (x, t i )  Quasi-linear utility: u i (x,  i,t i ) = v i (x, t i ) -  i  Our focus: social welfare maximization

7 Basic Mechanism Design  A direct mechanism M consists of three components: types T i allocation function m: T  X payment functions p i : T  R  Mechanism is incentive compatible: (IC) In equilibrium, agents reveal truthfully  Dominant Strategy IC Regardless of what others report, agent i should always tell the truth

8 Properties  Mechanism is efficient: maximizes social welfare given reported types:  -efficient: within  of optimal social welfare  Mechanism is Individually Rational: (IR) 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 Existing Work on Partial Revelation [Conen,Hudson,Sandholm, Parkes, Nisan&Segal, Blumrosen&Nisan, …]  Full revelation not always necessary for optimal decision (though worst-case is exponential: [Nisan&Segal05])  Most Approaches: require enough revelation for optimal VCG outcome sequential, not one-shot / specific settings (1-item,CAs)  BUT: optimal decision not always worth the costs Partial revelation:Trade-off elicitation costs with decision quality e.g. Priority games [Blumrosen&Nisan 02]  Can we maintain incentives?

12 Overview  Mechanism Design Background  Partial Revelation Mechanisms (PRM)  Regret-based PRMs  Partition Optimization  Experimental Results

13 Partial Revelation Mechanisms  A partial type is any subset  i  T i e.g. v(red,2doors)  [50,75], etc…  A one-shot (direct) partial revelation mechanism set  i of partial types,  i. (typically partition, not required) m:   X, chooses allocation m(  ) p i :   R, sets payment p i (  )  A truthful strategy: report  i s.t. t i   i  Goal: Tradeoff “quality” with revelation/communication costs maintain appropriate incentives

14 Partial Revelation MD: Negative Results  Partial revelation  can’t generally maximize SW must allocate under type uncertainty  Roberts: Dominant-IC  (affine) SW maximizer,  Partial revelation  no Dominant-IC  What are some solutions? relax solution concept to BNE / Ex-Post relax solution concept to approximate dominant-IC

15 Partial Revelation MD: Negative Results  Avoid Roberts by relaxing solution concept?  Bayes-Nash Equilibrium Theorem:  Bayes-Nash IC PRM with certain form of partitions  Trivial mechanism Consequences:  max expected SW = same as best trivial  max expected revenue = same as best trivial  “Useless”  Ex-Post Equilibrium:Same

16 Approximate Incentives   : bound on utility gain difference b/w u(best lie) and u(truth)  Considerable costs of manipulation: Uncertainty over others’ types Valuation + computational costs  If  is small enough Formal, approximate IC  practical, exact IC

17 Overview  Mechanism Design Background  Partial Revelation Mechanisms (PRM)  Regret-based PRMs  Partition Optimization  Experimental Results

18 Regret-based PRMs  In any PRM, how is allocation to be chosen?  x*(  ) is minimax-regret optimal decision for   A regret-based PRM: m(  )=x*(  ) for all   

19 Regret-based PRMs: Efficiency  Obs: If MR(x*(  ),  )   for all  , then regret- based PRM m is  -efficient for truthtelling agents.  We can tradeoff efficiency for elicitation effort More elicitation effort  more refined  ’s  smaller   Incentives?

20 Regret-based PRMs: Incentives  Can generalize Groves payments f i (  - i ): arbitrary type in  -i and h i (  - i ) an arbitrary function of  - i  Theorem: Let m be a regret-based PRM with partial types  and a partial Groves payment scheme. If MR(x*(  ),  )   for all  , then m is  -dominant incentive compatible

21 Approximate Incentives and IR  Can generalize Clark payments to get  -IR  A Clark-style regret-based PRM gives approximate Efficiency approximate Incentive Compatibility approximate Individual Rationality (Increased revenue from flexible payments)  Allows tradeoff “quality” vs revelation costs as long as we can find a good set of partial types

22 Overview  Mechanism Design Background  Partial Revelation Mechanisms (PRM)  Regret-based PRMs  Partition Optimization  Experimental Results

23 (One-shot) Partial Type Optimization  Designing PRM: must pick partial types we focus on bounds on utility parameters Use regret-based heuristics to estimate VOI  i : p1p1 p2p2

24 The Mechanism Tree (  1,…  i,…  n ) Worst-case Heuristic: Split  1

25 The Mechanism Tree (  ’ 1,…  i,…  n )(  ’’ 1,…  i,…  n ) (  1,…  i,…  n ) Worst-case Heuristic: Split  i

26 The Mechanism Tree (  ’ 1,…  i,…  n )(  ’’ 1,…  i,…  n ) (  ’ 1,…  ’ i,… )(  ’ 1,…  ’’ i,… ) (  1,…  i,…  n ) More details necessary to make it tractable

27 Empirical Results  Negotiation problem 1 buyer, 1 seller, 4 boolean attributes valuation/cost given by factored model (GAI)  16 values/costs specified by 8 parameters  Compare: uniform partitioning vs. regret-based heuristic worst-case  and expected  (uniform prior)

28 Empirical Results average  = vs 11 bits (40% savings) worst  = vs 11 bits (50% savings)

29 Empirical Results  Mechanism accounts for all types Initial regret: % of optimal (depending on actual type vector)  With 11 bits (1.4 bits/param, 0.7 bits/good): 20-56% of optimal (regret) vs 30-86% (uniform) 60% reduction of  vs 38%

30 Contributions  Negative Results Exact incentives “useless”  Regret-based PRMs Trade-off “quality” with revelation costs  Partial Types Optimization Avoid exponential blow-up Use regret to guide elicitation effectively

31 Current + Future Work  Sequential PRMs (Hyafil Boutilier AAAI 06)  Formal model manipulation and revelation costs  formal, exact IC  explicit revelation/quality trade-off  Partial Revelation Automated Mech Design General objective functions include “execution costs”

32 QUESTIONS?

33 Regret-based PRMs: Rationality  Can generalize Clark payments as well f i (  - i ): arbitrary type in  -I  Thm: Let m be a regret-based PRM with partial types  and a partial Clark payment scheme. If MR(x*(  ),  )   for all  , then m is  -individually rational.

34 (One-shot) Partial Type Optimization  Designing PRM: must pick partial types we focus on bounds on utility parameters  A simple greedy approach Let  be current partial type vectors (initially {T} ) Let  =(  1,…  i,…  n )   be partial type vector with greatest MMR Choose agent i and suitable split of partial type  i into  ’ i and  ’’ i Replace all   [  i ] by pair of vectors:  i   ’ i ;  ’’ i Repeat until bound  is acceptable

35 The Mechanism Tree (  ’ 1,…  i,…  n )(  ’’ 1,…  i,…  n ) (  ’ 1,…  ’ i,… )(  ’ 1,…  ’’ i,… )(  ’’ 1,…  ’ i,… )(  ’’ 1,…  ’’ i,… ) (  1,…  i,…  n ) Worst-case Heuristic: Split  1 Heuristic: Split  i *

36 A More Refined Approach  Simple model has drawbacks exponential blowup (“naïve” resolution) split of  i useful in reducing regret in one partial type vector , but is applied at all partial type vectors  Refinement: variable resolution apply split only at leaves where it is “useful”  Ignore on other leaves  keeps tree from blowing up, saves computation new splits traded off against “cached” splits

37 Naïve vs. Variable Resolution ii p1p1 p2p2 ii p1p1 p2p2

38 Heuristic for Choosing Splits  Adapted from single agent preference elicitation techniques: Current Solution Strategy  Let  be partial type vector with max MR optimal solution x* regret-maximizing witness x w intuition: focus on parameters that contribute to regret  reducing u.b. on x w or increasing l.b. on x* helps But: have to account for both “answers” Here: also consider second best MR