Preference Elicitation in Single and Multiple User Settings

1
Preference Elicitation in Single and Multiple
User Settings

• Darius Braziunas, Craig Boutilier, 2005
• (Boutilier, Patrascu, Poupart, Shuurmans, 2003,
  2005)
• Nathanael Hyafil, Craig Boutilier, 2006a, 2006b
• Department of Computer Science
• University of Toronto

2
Overview

• Preference Elicitation in A.I.
• Single User Elicitation
• Foundations of Local queries BB-05
• Bayesian Elicitation BB-05
• Regret-based Elicitation BPPS-03,05
• Multi-agent Elicitation (Mechanism Design)
• One-Shot Elicitation HB-06b
• Sequential Mechanisms HB-06a

3
Preference Elicitation in AI
Luggage Capacity? Two Door? Cost? Engine
Size? Color? Options?
Shopping for a Car
4
The Preference Bottleneck

• Preference elicitation the process of
  determining a users preferences/utilities to the
  extent necessary to make a decision on her behalf
• Why a bottleneck?
• preferences vary widely
• large (multiattribute) outcome spaces
• quantitative utilities (the numbers) difficult
  to assess

5
Automated Preference Elicitation

• The interesting questions
• decomposition of preferences
• 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?
• what decision criterion to use given partial
  utility info?

6
Overview

• Preference Elicitation in A.I.
• Constraint-based Optimization
• Factored Utility Models
• Types of Uncertainty
• Types of Queries
• Single User Elicitation
• Multi-agent Elicitation (Mechanism Design)

7
Constraint-based Decision Problems

• Constraint-based optimization (CBO)
• outcomes over variables X X1 Xn
• constraints C over X spell out feasible decisions
  
• generally compact structure, e.g., X1 X2 ?
  X3
• add a utility function u Dom(X) ? R
• preferences over configurations

8
Constraint-based Decision Problems

• Must express u compactly like C
• generalized additive independence (GAI)
• model proposed by Fishburn (1967) and BG95
• nice generalization of additive linear models
• given by graphical model capturing independence

9
Factored Utilities GAI Models

• Set of K factors fk over subset of vars Xk
• local utility for each local configuration
• Fishburn67 u in this form exists iff
• lotteries p and q are equally preferred whenever
  p and q have the same marginals over each Xk

10
Optimization with GAI Models
f1(A) a 3 a 1
f2(B) b 3 b 1
A
B
C
f3(BC) bc 12 bc 2

• Optimize using simple IP (or Var Elim, or)
• number of vars linear in size of GAI model

11
Difficulties in CBO

• Constraint elicitation
• generally stable across different users
• Utility (objective) representation
• GAI solidifies many of intuitions in soft
  constraints
• Utility elicitation how do we assess individual
  user preferences?
• need to elicit GAI model structure (independence)
• need to elicit (constraints on) GAI parameters
• need to make decisions with imprecise parameters

12
Strict Utility Function Uncertainty

• Users actual utility u unknown
• Assume feasible set F? U 0,1n
• allows for unquantified or strict uncertainty
• e.g., F a set of linear constraints on GAI terms
• How should one make a decision? elicit info?

u(red,2door,280hp) gt 0.4 u(red,2door,280hp) gt
u(blue,2door,280hp)
13
Strict Uncertainty Representation
f1(L) l 7,11 l 2,5
L
N
P
f2(L,N) l,n 2,4 l,n 1,2
Utility Function
14
Bayesian Utility Function Uncertainty

• Users actual utility u unknown
• Assume density P over U 0,1n
• Given belief state P, EU of decision x is EU(x
  , P) ?U (px . u) P( u ) du
• Decision making is easy, but elicitation harder?
• must assess expected value of information in query

15
Bayesian Representation
f1(L) l l
L
N
P
f2(L,N) l,n l,n
Utility Function
16
Query Types

• Comparison queries (is x preferred to x ?)
• impose linear constraints on parameters
• Sk fk(xk) gt Sk fk(xk)
• Interpretation is straightforward

U
17
Query Types

• Bound queries (is fk(xk) gt v ?)
• response tightens bound on specific utility
  parameter
• can be phrased as a local standard gamble query

U
18
Overview

• Preference Elicitation in A.I.
• Single User Elicitation
• Foundations of Local queries
• Bayesian Elicitation
• Regret-based Elicitation
• Multi-agent Elicitation (Mechanism Design)
• One-Shot Elicitation
• Sequential Mechanisms

19
Difficulties with Bound Queries

• Bound queries focus on local factors
• but values cannot be fixed without reference to
  others!
• seemingly different local prefs correspond to
  same u

u(Color,Doors,Power) u1(Color,Doors)
u2(Doors,Power) u(red,2door,280hp)
u1(red,2door) u2(2door,280hp) u(red,4door,280h
p) u1(red,4door) u2(4door,280hp)
10
6
4
1
9
6
3
3
20
Local Queries BB05

• We wish to avoid queries on whole outcomes
• cant ask purely local outcomes
• but can condition on a subset of default values
• Conditioning set C(f) for factor fi(Xi)
• variables that share factors with Xi
• setting default outcomes on C(f) renders Xi
  independent of remaining variables
• enables local calibration of factor values

21
Local Standard Gamble Queries

• Local std. gamble queries
• use best and worst (anchor) local outcomes
  -- conditioned on default values of conditioning
  set
• bound queries on other parameters relative to
  these
• gives local value function v(xi) (e.g., v(ABC)
  )
• Hence we can legitimately ask local queries
• But local Value Functions not enough
• must calibrate requires global scaling

22
Global Scaling

• Elicit utilities of anchor outcomes wrt global
  best and worst outcomes
• the 2m best and worst outcomes for each
  factor
• these require global std gamble queries (note
  same is true for pure additive models)

23
Bound Query Strategies

• Identify conditioning sets Ci for each factor fi
• Decide on default outcome
• For each fi identify top and bottom anchors
• e.g., the most and least preferred values of
  factor i
• given default values of Ci
• Queries available
• local std gambles use anchors for each factor,
  C-sets
• global std gambles gives bounds on anchor
  utilities

24
Overview

• Preference Elicitation in A.I.
• Single User Elicitation
• Foundations of Local queries
• Bayesian Elicitation
• Regret-based Elicitation
• Multi-agent Elicitation (Mechanism Design)
• One-Shot Elicitation
• Sequential Mechanisms

25
Partial preference informationBayesian
uncertainty

• Probability distribution p over utility functions
• Maximize expected (expected) utility
• MEU decision x arg maxx Ep u(x)
• Consider
• elicitation costs
• values of possible decisions
• optimal tradeoffs between elicitation effort and
  improvement in decision quality

26
Query selection

• At each step of elicitation process, we can
• obtain more preference information
• make or recommend a terminal decision

27
Bayesian approachMyopic EVOI
MEU(p)
...
q1
q2
r1,1
r2,1
r1,2
r2,2
...
MEU(p1,2)
MEU(p1,1)
MEU(p2,1)
MEU(p2,2)
28
Expected value of information
MEU(p)
q1
...
q2
r1,1
r2,1
r1,2
r2,2
...
MEU(p1,1)
MEU(p1,2)
MEU(p2,1)
MEU(p2,2)

• MEU(p) Ep u(x)
• Expected posterior utility EPU(q,p) Erq,p
  MEU(pr)
• Expected value of information of query q
• EVOI(q) EPU(q,p) MEU(p)

29
Bayesian approachMyopic EVOI

• Ask query with highest EVOI - cost
• Chajewska et al 00
• Global standard gamble queries (SGQ) Is u(oi) gt
  l?
• Multivariate Gaussian distributions over
  utilities
• Braziunas and Boutilier 05
• Local SGQ over utility factors
• Mixture of uniforms distributions over utilities

30
Local elicitation in GAI models Braziunas and
Boutilier 05

• Local elicitation procedure
• Bayesian uncertainty over local factors
• Myopic EVOI query selection
• Local comparison query
• Is local value of factor setting xi greater than
  l?
• Binary comparison query
• Requires yes/no response
• query point l can be optimized analytically

31
Experiments

• Car rental domain 378 parameters Boutilier et
  al. 03
• 26 variables, 2-9 values each, 13 factors
• 2 strategies
• Semi-random query
• Query factor and local configuration chosen at
  random
• Query point set to the mean of local value
  function
• EVOI query
• Search through factors and local configurations
• Query point optimized analytically

32
Experiments
Percentage utility error (w.r.t. true max
utility)
No. of queries
33
Bayesian Elicitation Future Work

• GAI structure elicitation and verification
• Sequential EVOI
• Noisy responses

34
Overview

• Single User Elicitation
• Foundations of Local queries
• Bayesian Elicitation
• Regret-based Elicitation
• why MiniMax Regret (MMR) ?
• Decision making with MMR
• Elicitation with MMR
• Multi-agent Elicitation (Mechanism Design)

35
Minimax Regret Utility Uncertainty

• Regret of x w.r.t. u
• Max regret of x w.r.t. F
• Decision with minimax regret w.r.t. F

36
Why Minimax Regret?

• Appealing decision criterion for strict
  uncertainty
• contrast maximin, etc.
• not often used for utility uncertainty
  BBB01,HS010

x
x
x
x
x
Better
x
x
x
x
x
x
x
u1
u2
u3
u4
u5
u6
37
Why Minimax Regret?

• Minimizes regret in presence of adversary
• provides bound worst-case loss
• robustness in the face of utility function
  uncertainty
• In contrast to Bayesian methods
• useful when priors not readily available
• can be more tractable see CKP00/02, Bou02
• effective elicitation even if priors available
  WB03

38
Overview

• Single User Elicitation
• Foundations of Local queries
• Bayesian Elicitation
• Regret-based Elicitation
• why MiniMax Regret (MMR) ?
• Decision making with MMR
• Elicitation with MMR
• Multi-agent Elicitation (Mechanism Design

39
Computing Max Regret

• Max regret MR(x,F) computed as an IP
• number of vars linear in GAI model size
• number of (precomputed) constants (i.e., local
  regret terms) quadratic in GAI model size
• r( xk , xk ) u(xk ) u(xk )

40
Minimax Regret in Graphical Models

• We convert minimax to min (standard trick)
• obtain a MIP with one constraint per feasible
  config
• linearly many vars (in utility model size)
• Key question can we avoid enumerating all x ?

41
Constraint Generation

• Very few constraints will be active in solution
• Iterative approach
• solve relaxed IP (using a subset of constraints)
• Solve for maximally violated constraint
• if any add it and repeat else terminate

42
Constraint Generation Performance

• Key properties
• aim graphical structure permits practical
  solution
• convergence (usually very fast, few constraints)
• very nice anytime properties
• considerable scope for approximation
• produces solution x as well as witness xw

43
Overview

• Single User Elicitation
• Foundations of Local queries
• Bayesian Elicitation
• Regret-based Elicitation
• why MiniMax Regret (MMR) ?
• Decision making with MMR
• Elicitation with MMR
• Multi-agent Elicitation (Mechanism Design)

44
Regret-based ElicitationBoutilier, Patrascu,
Poupart, Schuurmans IJCAI05 AIJ 06

• Minimax optimal solution may not be satisfactory
• Improve quality by asking queries
• new bounds on utility model parameters
• Which queries to ask?
• what will reduce regret most quickly?
• myopically? sequentially?
• Closed form solution seems infeasible
• to date weve looked only at heuristic elicitation

45
Elicitation Strategies I

• Halve Largest Gap (HLG)
• ask if parameter with largest gap gt midpoint
• MMR(U) maxgap(U), hence n?log(maxgap(U)/e)
  queries needed to reduce regret to e
• bound is tight
• like polyhedral-based conjoint analysis THS03

f1(a,b)
f1(a,b)
f1(a,b)
f1(a,b)
f2(b,c)
f2(b,c)
f2(b,c)
f2(b,c)
46
Elicitation Strategies II

• Current Solution (CS)
• only ask about parameters of optimal solution x
  or regret-maximizing witness xw
• intuition focus on parameters that contribute to
  regret
• reducing u.b. on xw or increasing l.b. on x
  helps
• use early stopping to get regret bounds (CS-5sec)

f1(a,b)
f1(a,b)
f1(a,b)
f1(a,b)
f2(b,c)
f2(b,c)
f2(b,c)
f2(b,c)
47
Elicitation Strategies III

• Optimistic-pessimistic (OP)
• query largest-gap parameter in one of
• optimistic solution xo
• pessimistic solution xp
• Computation
• CS needs minimax optimization
• OP needs standard optimization
• HLG needs no optimization
• Termination
• CS easy
• Others ?

48
Results (Small Random)
10vars lt 5 vals 10 factors, at most 3
vars Avg 45 trials
49
Results (Car Rental, Unif)
26 vars 61 billion configs 36 factors, at most
5 vars 150 parameters Avg 45 trials
50
Results (Real Estate, Unif)
20 vars 47 million configs 29 factors, at most
5 vars 100 parameters Avg 45 trials
51
Results (Large Rand, Unif)
25 vars lt 5 vals 20 factors, at most 3 vars A
45 trials
52
Summary of Results

• CS works best on test problems
• time bounds (CS-5) little impact on query
  quality
• always know max regret (or bound) on solution
• time bound adjustable (use bounds, not time)
• OP competitive on most problems
• computationally faster (e.g., 0.1s vs 14s on
  RealEst)
• no regret computed so termination decisions
  harder
• HLG much less promising

53
Interpretation

• HLG
• provable regret reduced very quickly
• But
• true regret faster (often to optimality)
• OP and CS restricted to feasible decisions
• CS focuses on relevant parameters

54
Conclusion Single User

• Local parameter elicitation
• Theoretically sound
• Computationally practical
• Easier to answer
• Bayesian EVOI / Regret-based elicitation
• Good guides for elicitation
• Integrated in computationally tractable
  algorithms
• Future Work
• Sequential reasoning

55
Questions?

• References
• D. Braziunas and C. Boutilier
• Local Utility Elicitation in GAI Models, UAI
  2005
• C. Boutilier, R. Patrascu, P. Poupart, D.
  Shuurmans
• Constraint-based Optimization and Utility
  Elicitation using the Minimax Decision
  Criterion, Artificial Intelligence, 2006
• (CP-2003 IJCAI 2005)

56
Preference Elicitation in Single and Multiple
User Settings Part 2

• Darius Braziunas, Craig Boutilier, 2005
• (Boutilier, Patrascu, Poupart, Shuurmans, 2003,
  2005)
• Nathanael Hyafil, Craig Boutilier, 2006a, 2006b
• Department of Computer Science
• University of Toronto

57
Overview

• Single User Elicitation
• Foundations of Local queries
• Bayesian Elicitation
• Regret-based Elicitation
• Multi-agent Elicitation
• Background Mechanism Design
• Partial Revelation Mechanisms
• One-Shot Elicitation
• Sequential Mechanisms

58
Bargaining for a Car
Luggage Capacity? Two Door? Cost? Engine
Size? Color? Options?
59
Multiagent PE Mechanism Design

• Incentive to misrepresent preferences
• 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
• Applications
• Auctions, multi-attribute Negotiation,
  Procurement problems,
• Network protocols, Autonomic computing, ...

60
Basic Social Choice Setup

• Choice of x from outcomes X
• Agents 1..n type ti ?Ti and valuation vi(x,
  ti)
• Type vectors t?T and t-i ?T-i
• Goal optimize social choice function f T ? X
• e.g., social welfare SW(x,t) ? vi(x, ti)
• Assume payments and quasi-linear utility
• ui(x, ?i ,ti ) vi(x, ti ) - ?i
• Our focus SW maximization, quasi-linear utility

61
Basic Mechanism Design

• A mechanism m consists of three components
• actions Ai
• allocation function O A? X
• payment functions pi A? R
• m induces a Bayesian game
• m implements social choice function f if
• in equilibrium ? O(?(t)) f(t) for all t?T

62
Incentive Compatibility (Truth-telling)

• Dominant Strategy IC
• No matter what agent i should tell the truth
• Bayes-Nash IC
• Assume others tell the truth
• Assume agent i has Bayesian prior over others
  types
• Then, in expectation, agent i should tell the
  truth
• Ex-Post IC
• Assume others tell the truth
• Assume agent i knows the others types
• Then agent i should tell the truth

63
Properties

• Mechanism is Efficient
• maximizes SW given reported types
• ? -efficient within ? of optimal SW
• Ex Post Individually Rational
• No agent can lose by participating
• ?-IR can lose at most ?

64
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
• implements SWM in dominant strategies
• incentive compatible, efficient, individually
  rational

65
Groves Schemes

• Strong results Groves is basically the only
  choice for dominant strategy implementation
• Roberts (1979) only social choice functions
  implementable in dominant strategies are affine
  welfare maximizers (if all valuations possible)
• Green and Laffont (1977) must use Groves
  payments to implement affine maximizers
• Implications for partial revelation

66
Issues with Classical Mechanism Design

• Computation Costs
• e.g., Winner Determination
• Revelation Costs
• Communication
• Computation
• Cognitive
• Privacy

67
Issues with Classical Mechanism Design

• Full Revelation and Quality
• trade-off revelation costs with Social Welfare
• Full Revelation and Incentives
• very dependent
• need new concepts

68
Overview

• Single User Elicitation
• Foundations of Local queries
• Bayesian Elicitation
• Regret-based Elicitation
• Multi-agent Elicitation
• Background Mechanism Design
• Partial Revelation Mechanisms
• One-Shot Elicitation
• Sequential Mechanisms

69
Partial Revelation Mechanisms

• Full revelation unappealing
• A partial type is any subset ?i ? Ti
• A one-shot (direct) partial revelation mechanism
• each agent reports a partial type ?i ? ?i
• typically ?i partitions type space, but not
  required
• A truthful strategy report ?i s.t. ti ? ?i
• Goal minimize revelation, computation,
  communication by suitable choice of partial types

70
Implications of Roberts

• Partial revelation means we cant generally
  maximize social welfare
• must allocate under type uncertainty
• But if SCF is not an affine maximizer, we cant
  expect dominant strategy implementation
• What are some solutions?
• relax solution concept to BNE / Ex-Post
• relax solution concept to approx incentives
• incremental and hope for less than full
  elicitation
• relax conditions on Roberts results

71
Existing Work on PRMs Conen,Hudson,Sandholm,
Parkes, NisanSegal, BlumrosenNisan

• Most Approaches
• require enough revelation to determine optimal
  allocation and VCG payments
• hence cant offer savings in general
  NisanSegal05
• Sequential, not one-shot
• specific settings (1-item, combinatorial
  auctions)
• Priority games BlumrosenNisan 02
• genuinely partial and approximate efficiency
• but very restricted valuation space (1-item)

72
Preference Elicitation in MechDes

• We move beyond this by allowing approximately
  optimal allocation
• specifically, regret-based allocation models
• Avoid Roberts by relaxing solution concept
• Bayes-Nash equilibrium?
• NO! HB-06b
• Ex-Post IC?
• NO ! HB-06b
• approximate, ex-post implementation

73
Partial Revelation MD Impossibility Results

• Bayes-Nash Equilibrium
• Theorem HB-06b
• Deterministic PRMs are Trivial
• Randomized PRMs are Pseudo-Trivial
• Consequences
• max expected SW same as best trivial
• max expected revenue same as best trivial
• ? Useless
• Ex-Post Equilibrium
• Same

74
Overview

• Single User Elicitation
• Foundations of Local queries
• Bayesian Elicitation
• Regret-based Elicitation
• Multi-agent Elicitation
• Background Mechanism Design
• Partial Revelation Mechanisms
• One-Shot Elicitation
• Sequential Mechanisms

75
Regret-based PRMs

• In any PRM, how is allocation to be chosen?
• x(?) is minimax optimal decision for
• A regret-based PRM O(?)x(?) for all ? ? ?

76
Regret-based PRMs Efficiency

• Efficiency not possible with PRMs (unless MR0)
• but bounds are quite obvious
• Prop If MR(x(?),?) ? ? for all ? ??, then
  regret-based PRM m is ?-efficient for
  truthtelling agents.
• thus we can tradeoff efficiency for elicitation
  effort

77
Regret-based PRMs Incentives

• Can generalize Groves payments
• let fi (?i) be an arbitrary type in ?i
• Thm Let m be a regret-based PRM with
• partial types ? and a
• partial Groves payment scheme.
• If MR(x(?),?) ? ? for all ? ??, then m is
  ?-ex post incentive compatible

78
Regret-based PRMs Rationality

• Can generalize Clark payments as well
• 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
  ?-ex post individually rational.
• A Clark-style regret-based PRM gives approximate
  efficiency, approximate IC (ex post) and
  approximate IR (ex post)

79
Approximate Incentives and IR

• Natural to trade off efficiency for elicitation
  effort
• Is approximate IC acceptable?
• computing a good lie?
• Good?
• Huge computation costs
• if incentive to deviate from truth is small
  enough, then formal, approximate IC ensures
  practical, exact IC
• Is approximate IR acceptable?
• Similar argument
• Thus regret-based PRMs offer scope to tradeoff
• as long as we can find a good set of partial types

80
Computation and Design/Elicitation

• Minimax optimization given partial type vector ?
• same techniques as for single agent
• varies with setting (experiments CBO with GAI)
• Designing the mechanism
• one-shot PRM must choose partial types for each
  i
• sequential PRM need elicitation strategy
• we apply generalization of CS to each task

81
(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

82
The Mechanism Tree
83
A More Refined Approach

• Simple model has drawbacks
• exponential blowup (naïve partitioning)
• split of ?i useful in reducing regret in one
  partial type vector ?, but is applied at all
  partial type vectors
• Refinement
• apply split only at leaves where it is useful
• keeps tree from blowing up, saves computation
• new splits traded off against cached splits
• once done, use either naïve/variable resolution
  types for each agent

84
Naïve vs. Variable Resolution
p2
p2
p1
p1
?i
?i
85
Heuristic for Choosing Splits

• Adopt variant of current solution strategy
• Let ? be partial type vector with max MMR
• optimal solution x regret-maximizing witness xw
• only split on parameters of utility functions of
  optimal solution x or regret-maximizing witness
  xw
• intuition focus on parameters that contribute to
  regret
• reducing u.b. on xw or increasing l.b. on x
  helps
• pick agent-parameter pair with largest gap

86
Preliminary Empirical Results

• Very preliminary results
• use only very naïve algorithm
• single buyer, single seller
• 16 goods specified by 4 boolean variables
• valuation/cost given by GAI model
• two factors, two vars each (buyer/seller factors
  are different)
• thus 16 values/costs specified by 8 parameters
• no constraints on feasible allocations

87
Preliminary Empirical Results
88
Overview

• Single User Elicitation
• Foundations of Local queries
• Bayesian Elicitation
• Regret-based Elicitation
• Multi-agent Elicitation
• Background Mechanism Design
• Partial Revelation Mechanisms
• One-Shot Elicitation
• Sequential Mechanisms

89
Sequential PRMs

• Optimization of one-shot PRMs unable to exploit
  conditional queries
• e.g., if seller cost of x greater than your upper
  bound, neednt ask you for your valuation of x
• Sequential PRMs
• incrementally elicit partial type information
• apply similar heuristics for designing query
  policy
• incentive properties somewhat weaker opportunity
  to manipulate payments by altering the query path
• thus additional criteria can be used to optimize

90
Sequential PRMs Definition

• Set of queries Qi
• response r?Ri(qi) interpreted as partial type ?i
  (r) ? Ti
• history h sequence of query-response pairs
  possibly followed by allocation (terminal)
• Sequential mechanism m maps
• nonterminal histories to queries/allocations
• terminal histories to set of payment functions pi
• Revealed partial type ?i(h) intersect. ?i (r), r
  in h
• m is partial revelation if exists realizable
  terminal h s.t. ?i(h) admits more than one type
  ti

91
Sequential PRMs Properties

• Strategies ?i(hi ,qi ,ti) selects responses
• ?i is truthful if ti ? ?i (?i(hi ,qi ,ti))
• truthful strategies must be history independent
• (Determ.) strategy profile ? induces history h
• if h is terminal, then quasi-linear utility
  realized
• if history is unbounded, then assume utility 0
• Regret-based PRM allocation defined as in one-shot

92
Max VCG Payment Scheme

• Assume terminal history h
• let ? be revealed PTV at h, x(?) be allocation
• Max VCG payment scheme
• where VCG payment is

93
Incentive Properties

• Suppose we elicit type info until MMR allocation
  has max regret ? ? and we use max VCG
• Define
• Thm m is ?-efficient, ?-ex post IR and
  (??(x(?)))-ex post IC.
• weaker results due to possible payment
  manipulation

94
Elicitation Approaches

• Two Phases
• Standard max regret based approaches
• give us bounds on efficiency ?, no a priori ?
  bounds
• Regret-based followed by payment elicitation
• once ? small enough, elicit additional payment
  information until max ? is small enough

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Elicitation Approaches

• Direct optimization
• global manipulability u(best lie) - u(truth)
• ask queries that directly reduce global
  manipulability
• can be formulated as regret-style optimization
• analogous query strategies possible

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Test Domains

• Car Rental Problem
• 1 client , 2 dealers
• GAI valuation/costs 13 factors, size 1-4
• Car 8 attributes, 2-9 values
• Total 825 parameters
• Small Random Problems
• supplier-selection, 1 buyer, 2 sellers
• 81 parameters

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Results Car Rental
Initial regret 99 of opt SW Zero-regret 71/77
queries Avg remaining uncertainty 92 vs 64 at
zero-manipulability Avg nb params queried 8

• relevant parameters
• reduces revelation
• improves decision quality

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Results Random Problems
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Contributions

• Theoretical framework for Partial Revelation Mech
  Design
• One-shot mechanisms
• generalize VCG to PRMs (allocation payments)
• v. general payments secondary objectives
• algorithm to design partial types
• Sequential mechanisms
• slightly different model, but similar results
• algorithm to design query strategy
• Viewpoint why approximate incentives are useful
• Approximate decision ? trade off cost vs.
  quality
• Formal, approximate IC ensures practical, exact
  IC
• Applicable to general Mechanism Design problems
• Empirically very effective

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PRMs Future Work

• Further investigate splitting / elicitation
  heuristics
• More experimentation
• Larger problems
• Combinatorial Auctions
• Formal model manipulability cost ? formal,
  exact IC
• Formal model revelation costs ? explicit
  revelation vs efficiency trade-off
• Sequentially optimal elicitation

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Questions ?

• References
• Nathanael Hyafil and Craig Boutilier
• Regret-based Incremental Partial Revelation
  Mechanisms, AAAI 2006
• One-shot Partial Revelation Mechanisms, Working
  Paper, 2006

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Extra Slides - Part 1
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Fishburn 1967 Default Outcomes

• Define default outcome
• For any x, let xI be restriction of x to vars
  I, with remaining replaced by default values
• Utility of x can be written F67
• sum of utilities of certain related key outcomes

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Key Outcome Decomposition

• Example GAI over IABC, JBCD, KDE

• u(x) u(xI) u(xJ) u(xK)
• - u(xI?J) - u(xI?K) - u(xJ?K)
• u(xI?J?K)
• u(abcde) u(xabc) u(xbcd) u(xde)
• - u(xbc) - u(x) - u(xd)
• u(x)
• u(abcde) u(abcd0e0) u(a0bcde0) u(a0b0c0de)
• - u(a0bcd0e0) - u(a0b0c0de0)

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Canonical Decompostion F67

• This leads to canonical decompostion of u

u1(x1, x2)
u2(x2, x3)
e.g., IABC, JBCD, KDE

• u(abcde) u(abcd0e0)
• u(a0bcde0) - u(a0bcd0e0)
• u(a0b0c0de) - u(a0b0c0de0)

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Local Queries

• Thus, if for some y (where Y X \ Xi \ C(Xi) )
• then for all y
• hence we can legitimately ask local queries

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Implications for Minimax Regret

• Complicates MMR
• utility of outcome depends linearly on GAI
  parameters
• but GAI parameters depend on bounds induced by
  two types of queries quadratic constraints
• Local pairwise regret notion can be modified
• To compute rxkxk
• set values vxk to u.b. and vxk to l.b.
• If vxk? gt vxk? max u(xTk) and min
  u(x?k)
• otherwise do the opposite

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Bayesian Utility Function Uncertainty

• Users actual utility u unknown
• Assume density P over U 0,1n
• Given belief state P, EU of decision x is
• Decision making is easy, but elicitation harder?
• must assess expected value of information in query

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Extra Slides - Part 2