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
1
Overview
Preference Elicitation in A.I.
Single User Elicitation
• Foundations of Local queries
• Bayesian Elicitation
[BB-05]
• Regret-based Elicitation
[BPPS-03,05]
[BB-05]
Multi-agent Elicitation (Mechanism Design)
• One-Shot Elicitation
• Sequential Mechanisms
[HB-06b]
[HB-06a]
2
Preference Elicitation in AI
Shopping for a Car:
Luggage Capacity?
Two Door? Cost?
Engine Size?
Color? Options?
3
The Preference Bottleneck
Preference elicitation:
the process of determining a user’s
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
4
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?
5
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)
6
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
7
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
8
Factored Utilities:
GAI Models
Set of K factors fk over subset of vars X[k]
• “local” utility for each local configuration
• u (x) 
 f k (x[k ])
k K
f1(A)
a: 3
a: 1
A
u(abc) = f1(a)+ f2(b)+ f3(bc)
B
f2(B)
b: 3
b: 1
C
f3(BC)
bc: 12
bc: 2
…
[Fishburn67] u in this form exists iff
• lotteries p and q are equally preferred whenever p
and q have the same marginals over each X[k]
9
Optimization with GAI Models
u (x) 
f1(A)
a: 3
a: 1
k
f
 (x[k ])
k K
A
C
B
f2(B)
b: 3
b: 1
f3(BC)
bc: 12
bc: 2
…
Optimize using simple IP (or Var Elim, or…)
• number of vars linear in size of GAI model

{I x[ k ] , X i }
max
k

ux[k ] I x[ k ] subj. to A, C
x[ k ]Dom( X[ k ])
10
Difficulties in CBO
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
11
Strict Utility Function Uncertainty
User’s actual utility u unknown
Assume feasible set F U = [0,1]n
• allows for unquantified or “strict” uncertainty
• e.g., F a set of linear constraints on GAI terms
u(red,2door,280hp) > 0.4
u(red,2door,280hp) > u(blue,2door,280hp)
How should one make a decision? elicit info?
12
Strict Uncertainty Representation
f1(L)
l: [7,11]
l: [2,5]
L
P
N
f2(L,N)
l,n: [2,4]
l,n: [1,2]
Utility Function
13
Bayesian Utility Function Uncertainty
User’s actual utility u unknown
Assume density P over U = [0,1]n
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
14
Bayesian Representation
f1(L)
l:
l:
L
P
N
f2(L,N)
l,n:
l,n:
…
Utility Function
15
Query Types
Comparison queries (is x preferred to x’ ?)
• impose linear constraints on parameters
Sk fk(x[k]) > Sk fk(x’[k])
• Interpretation is straightforward
U
16
Query Types
Bound queries (is fk(x[k]) > v ?)
• response tightens bound on specific utility parameter
• can be phrased as a local standard gamble query
U
17
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
18
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)
10
6
1
4
9
u(red,2door,280hp) = u1(red,2door) + u2(2door,280hp)
6
3
3
u(red,4door,280hp) = u1(red,4door) + u2(4door,280hp)
19
Local Queries [BB05]
We wish to avoid queries on whole outcomes
• can’t 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
20
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(x[i]) (e.g., v(ABC) )
•
•
Hence we can legitimately ask local queries:
But local Value Functions not enough:
• must calibrate: requires global scaling
21
Global Scaling
Elicit utilities of anchor outcomes wrt global
best and worst outcomes
• the 2*m “best” and “worst” outcomes for each
factor
• these require global std gamble queries
(note: same is true for pure additive models)
22
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
23
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
24
Partial preference information
Bayesian 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
25
Query selection
 At each step of elicitation process, we can
• obtain more preference information
• make or recommend a terminal decision
26
Bayesian approach
Myopic EVOI
MEU(p)
...
q1
r1,1
q2
r1,2
r2,1
r2,2
...
MEU(p1,1)
MEU(p1,2)
MEU(p2,1)
MEU(p2,2)
27
Expected value of information
MEU(p)
q1
r1,1
q2
r1,2
r2,1
...
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) = Er|q,p [MEU(pr)]
Expected value of information of query q:
EVOI(q) = EPU(q,p) – MEU(p)
28
Bayesian approach
Myopic EVOI
 Ask query with highest EVOI - cost
 [Chajewska et al ’00]
• Global standard gamble queries (SGQ) “Is u(oi) > l?”
• Multivariate Gaussian distributions over utilities
 [Braziunas and Boutilier ’05]
• Local SGQ over utility factors
• Mixture of uniforms distributions over utilities
29
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
30
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
31
Experiments
16
Mixture of Uniforms
Uniform
Gaussian
14
12
Percentage
utility error
(w.r.t. true
max utility)
10
8
6
4
2
0
0
10
20
30
40
50
60
70
80
90
100
No. of queries
32
Bayesian Elicitation: Future Work
 GAI structure elicitation and verification
 Sequential EVOI
 Noisy responses
33
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)
34
Minimax Regret: Utility Uncertainty
Regret of x w.r.t. u:
R( x, u )  EU ( xu* , u )  EU ( x, u )
Max regret of x w.r.t. F:
MR ( x, F )  max R ( x, u )
uF
Decision with minimax regret w.r.t. F:
xF*  arg min MR ( x, F ) ;
xFeas( X )
MMR ( F )  MR ( xF* , F )
35
Why Minimax Regret?*
Appealing decision criterion for strict uncertainty
• contrast maximin, etc.
• not often used for utility uncertainty [BBB01,HS010]
Better
x’
x’
x
x’
x’
x
x
x
x
x’
u1
u2
u3
u4
u5
x’
x
u6
36
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]
37
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
38
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( x[k] , x’[k] ) = uT (x’[k] ) – u (x[k] )


{I x '[ k ] , X 'i }
max
k x'[ k ]
rx[k ]x'[k ] I x'[k ] subj. to A, C
39
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’ ?
40
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
41
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
42
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)
43
Regret-based Elicitation
[Boutilier, 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 we’ve looked only at heuristic elicitation
44
Elicitation Strategies I
Halve Largest Gap (HLG)
• ask if parameter with largest gap > 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)
45
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)
46
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 ?
47
Results (Small Random)
10vars; < 5 vals
10 factors, at
most 3 vars
Avg 45 trials
48
Results (Car Rental, Unif)
26 vars; 61
billion configs
36 factors, at
most 5 vars; 150
parameters
Avg 45 trials
49
Results (Real Estate, Unif)
20 vars; 47
million configs
29 factors, at
most 5 vars; 100
parameters
Avg 45 trials
50
Results (Large Rand, Unif)
25 vars; < 5 vals
20 factors, at
most 3 vars
A 45 trials
51
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
52
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
53
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
54
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)
55
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
56
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
57
Bargaining for a Car
$$
$$
Luggage Capacity?
Two Door? Cost?
Engine Size?
Color? Options?
$$
$$
$$
$$ $$
$$
58
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, ...
59
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
60
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
61
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
62
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 
63
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
64
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
65
Issues with Classical Mechanism
Design
Computation Costs
• e.g., Winner Determination
Revelation Costs
• Communication
• Computation
• Cognitive
• Privacy
66
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
67
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
68
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
69
Implications of Roberts
Partial revelation means we can’t generally
maximize social welfare
• must allocate under type uncertainty
But if SCF is not an affine maximizer, we can’t
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
70
Existing Work on PRMs
[Conen,Hudson,Sandholm, Parkes, Nisan&Segal, Blumrosen&Nisan]
Most Approaches:
• require enough revelation to determine optimal
•
•
allocation and VCG payments
hence can’t offer savings in general [Nisan&Segal05]
Sequential, not one-shot
specific settings (1-item, combinatorial auctions)
Priority games [Blumrosen&Nisan 02]
• genuinely partial and approximate efficiency
• but very restricted valuation space (1-item)
71
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 !
• approximate, ex-post implementation
[HB-06b]
72
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
73
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
74
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   
75
Regret-based PRMs: Efficiency
Efficiency not possible with PRMs (unless MR=0)
• but bounds are quite obvious
Prop: If MR(x*(),)  e for all  , then regretbased PRM m is e-efficient for truthtelling agents.
• thus we can tradeoff efficiency for elicitation effort
76
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*(),)  e for all  , then m is
e-ex post incentive compatible
77
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*(),)  e for all  , then m is
e-ex post individually rational.
A Clark-style regret-based PRM gives approximate
efficiency, approximate IC (ex post) and approximate IR
(ex post)
78
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
79
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
80
(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 e is acceptable
81
The Mechanism Tree
(1,… i,…n )
(’1,… i,…n )
(’1,… ’i,… )
(’’1,… i,…n )
(’1,… ’’i,… ) (’’1,… ’i,… )
(’’1,… ’’i,… )
82
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
83
Naïve vs. Variable Resolution
p2
p2
p1
p1
i
i
84
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
85
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
86
Preliminary Empirical Results
87
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
88
Sequential PRMs
Optimization of one-shot PRMs unable to exploit
conditional “queries”
• e.g., if seller cost of x greater than your upper bound,
needn’t 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
89
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
90
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 oneshot
91
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:
92
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
(+e(x*()))-ex post IC.
• weaker results due to possible payment manipulation
93
Elicitation Approaches
Two Phases:
Standard max regret based approaches
• give us bounds on efficiency , no a priori e bounds
Regret-based followed by payment elicitation
• once  small enough, elicit additional payment
information until max e is small enough
94
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
95
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
96
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
97
Results: Random Problems
98
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
99
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
100
Questions ?
References:
Nathanael Hyafil and Craig Boutilier
• “Regret-based Incremental Partial Revelation
Mechanisms”, AAAI 2006
• “One-shot Partial Revelation Mechanisms”,
Working Paper, 2006
101
Extra Slides - Part 1
102
Fishburn [1967]: Default Outcomes
Define default outcome:
For any x, let x[I] 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
103
Key Outcome Decomposition
Example: GAI over I={ABC}, J={BCD}, K={DE}
u(x) = u(x[I]) + u(x[J]) + u(x[K])
- u(x[IJ]) - u(x[IK]) - u(x[JK])
+ u(x[IJK])
u(abcde)
= u(x[abc]) + u(x[bcd]) + u(x[de])
- u(x[bc]) - u(x[]) - u(x[d])
+ u(x[])
u(abcde)
= u(abcd0e0) + u(a0bcde0) + u(a0b0c0de)
- u(a0bcd0e0) - u(a0b0c0de0)
104
Canonical Decompostion [F67]
This leads to canonical decompostion of u:
u1(x1, x2)
u2(x2, x3)
e.g., I={ABC}, J={BCD}, K={DE}
u(abcde) =
u(abcd0e0)
+ u(a0bcde0) - u(a0bcd0e0)
+ u(a0b0c0de) - u(a0b0c0de0)
105
Local Queries
Thus, if for some y (where Y =X \ Xi \ C(Xi) )
then for all y’
hence we can legitimately ask local queries:
106
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 rx[k]x’[k]
set values: vx’[k] to u.b. and vx[k] to l.b.
If vx’[k]↑ > vx[k]↓: max u(xT[k]) and min u(x[k])
otherwise do the opposite
107
Bayesian Utility Function Uncertainty
User’s actual utility u unknown
Assume density P over U = [0,1]n
Given belief state P, EU of decision x is:
EU ( x, P)  
U
  
px u P(u )
Decision making is easy, but elicitation harder?
• must assess expected value of information in query
108
Extra Slides - Part 2
109