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(One-shot) Mechanism Design with

Partial Revelation

Nathanaël Hyafil, Craig Boutilier

IJCAI 2007

Department of Computer Science

University of Toronto

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Bargaining for a Car

Luggage Capacity?Two Door? Cost?

Engine Size?Color? Options?

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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

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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 ?

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Overview

Mechanism Design Background

Partial Revelation Mechanisms (PRM)

Regret-based PRMs

Partition Optimization

Experimental Results

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Basic Social Choice Setup

Choice of x from outcomes X (e.g. cars)

Agents 1..n: type ti Ti and valuation vi(x, ti)

Type vectors: t TGoal: implement social choice function f: 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

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Basic Mechanism Design

A direct mechanism M 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

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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

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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

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Cost of Full Revelation

Communication costs

Computation costs

Cognitive costs

Privacy costs

INTRACTABLE!

Partial revelation?

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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?

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Overview

Mechanism Design Background

Partial Revelation Mechanisms (PRM)

Regret-based PRMs

Partition Optimization

Experimental Results

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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 i of 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

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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

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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 trivialmax expected revenue = same as best trivial

“Useless”

Ex-Post Equilibrium:Same

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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

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Overview

Mechanism Design Background

Partial Revelation Mechanisms (PRM)

Regret-based PRMs

Partition Optimization

Experimental Results

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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

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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?

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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

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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

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Overview

Mechanism Design Background

Partial Revelation Mechanisms (PRM)

Regret-based PRMs

Partition Optimization

Experimental Results

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(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 :

p1

p2

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The Mechanism Tree

(1,… i,…n )Worst-case

Heuristic: Split 1

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The Mechanism Tree

(’1,… i,…n ) (’’1,… i,…n )

(1,… i,…n )Worst-case

Heuristic: Split i

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The Mechanism Tree

(’1,… i,…n ) (’’1,… i,…n )

(’1,… ’i,… ) (’1,… ’’i,… )

(1,… i,…n )

More details necessary to make it tractable

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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)

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Empirical Results

average = 706.5 vs 11 bits

(40% savings)

worst = 905.5 vs 11 bits (50% savings)

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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%

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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

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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”

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QUESTIONS?

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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.

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(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

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The Mechanism Tree

(’1,… i,…n ) (’’1,… i,…n )

(’1,… ’i,… ) (’1,… ’’i,… ) (’’1,… ’i,… ) (’’1,… ’’i,… )

(1,… i,…n )Worst-case

Heuristic: Split 1Heuristic: Split i

*

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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 leaveskeeps tree from blowing up, saves computation

• new splits traded off against “cached” splits

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Naïve vs. Variable Resolution

i

p1

p2

i

p1

p2

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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