(One-shot) Mechanism Design with Partial Revelation

(One-shot) Mechanism Design with Partial Revelation Nathanaël Hyafil, Craig Boutilier IJCAI 2007 Department of Computer Science University of Toronto

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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

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 ?

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

Basic Social Choice Setup • Choice of x from outcomes X(e.g. cars) • 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

Basic Mechanism Design • A direct mechanismM consists of three components: • types Ti • allocation function m: T X • payment functions pi : 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

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 

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

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

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?

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

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

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

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 enoughFormal, approximate IC  practical, exact IC

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  

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?

Regret-based PRMs: Incentives • Can generalize Groves payments • fi (-i): arbitrary type in -i and hi (-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

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

(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 p2 i : p1

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

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

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

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)

Empirical Results • worst  = 90 • 5.5 vs 11 bits • (50% savings) average  = 70 • 6.5 vs 11 bits (40% savings)

Empirical Results • Mechanism accounts for all types • Initial regret: 50-146% 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%

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

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”

QUESTIONS?

Regret-based PRMs: Rationality • Can generalize Clark payments as well • fi (-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.

(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

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

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

Naïve vs. Variable Resolution p2 p2 p1 p1 i i

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 xw • intuition: focus on parameters that contribute to regret • reducing u.b. on xw or increasing l.b. on x* helps • But: have to account for both “answers” • Here: also consider second best MR

(One-shot) Mechanism Design with Partial Revelation