Preference Elicitation in Single and Multiple User Settings
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Transcript Preference Elicitation in Single and Multiple User Settings
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
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
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 )
uF
Decision with minimax regret w.r.t. F:
xF* arg min MR ( x, F ) ;
xFeas( 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] ) = u(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 nlog(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: tT 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 tT
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
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Sequential PRMs: Definition
Set of queries Qi
• response rRi(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
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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
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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:
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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
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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
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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 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
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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[IJ]) - u(x[IK]) - u(x[JK])
+ u(x[IJK])
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)
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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)
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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 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
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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
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Extra Slides - Part 2
109