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Outline
Intelligent Preference
Reasoning for Multi-Agent
Decision Making
Maria Silvia Pini
Department of Information Engineering
University of Padova (Italy)
Colloquia DEI, 25 October 2012
Outline Outline
Preferences
Multi-agent decision making
Social choice (voting theory)
Voting rules
Desirable properties
Impossibility results
Computational social choice
Computational concerns
Large set of candidates
Compact preference formalisms
Missing and imprecise preferences
Stable allocations
Preferences Outline
Outline Preferences
Preferences are ubiquitous in everyday decision making
Essential ingredients in every reasoning tool
Preferences are orderings over possible options
Options: computers, candidates, cars, books, movies …
Preferences can model levels of acceptance, or costs
Preferences are tolerant constraints
Constraints are strict requirements that must be satisfied
Constraints and preferences may be present in the same problem
Configuration, timetabling, etc.
Preferences
Outline Example: University timetabling
Professor Administration
I cannot teach on Wednesday
afternoon.
I prefer not to teach early in
the morning, nor on Friday
afternoon. Lab C can fit only 120 students.
Better to not leave 1-hour holes in
the day schedule.
Constraints
Preferences
Preferences University timetable
Outline Preferences for collective decision making in
multi-agent systems
Several agents
Common set of possible decisions
Each agent has its preferences over the possible decisions
Goal: to choose one of the decisions, based on the preferences
of the agents
Also a set of decisions, or a ranking over the decisions
AI scenarios add: imprecision, uncertainty, complexity, etc.
Outline Applications
Doodle
Several time slots under consideration
Participants accept or reject each time slot
Very simple way to express preferences over time slots
Very little information communicated to the system
Collective choice: a single time slot
The one with most acceptance votes from participants
Other applications
Meta-search engines
Group recommender systems
Outline How to compute a collective decision?
Let the agents vote by expressing their
preferences over the possible decisions
Aggregate the votes to get a single decision
Let’s look at voting theory
Agents = Voters
Decisions = Candidates
Preferences
Chosen decision = winner
Outline Main differences
In multi-agent AI scenarios, we usually have
Incomparability
Computational concerns
Large sets of candidates (w.r.t. number of voters)
Formalisms to compactly represent preferences
Uncertainty, vagueness
Outline Voting theory
(Social choice)
Voters
Candidates
Each voter expresses its preferences over the candidates
Goal: to choose one candidate (the winner), based on the voters’ preferences
Also many candidates, or ranking
Rules (functions) to achieve the goal
Properties of the rules
Impossibility results
Outline Some voting rules
Plurality
Voting: one most preferred decision
Selection: the decision preferred by the largest number of agents
Majority: like plurality, over 2 options
Approval (m options)
Voting: approval of between 1 and m-1 options
Selection: option with most votes
Doodle
Borda
Voting: rank over all options,
Score of an option: number of options that it dominates
Selection: option with greatest sum of scores
Outline
Unanimity (efficiency)
If all voters have the same top choice, it is selected
Non-dictatorship
There is no voter such that his top choice always wins, regardless of the votes of other voters
Non-manipulability
There is no incentive for agents to misrepresent the preferences
Some desirable properties
Outline Two classical impossibility results
Arrow’s theorem (1951)
Totally ordered preferences
it is impossible to find a voting rule with some desirable properties including
unanimity
non-dictatoriality
Gibbard-Sattherwaite’s theorem (1973)
Totally ordered preferences
it is impossible to have a reasonable voting rule that is
non-dictatorial
non-manipulable
These impossibility results holds also when we allow incomparability in preferences
Nobel prize in Economics 1972
Pini, Rossi, Venable, Walsh. Aggregating Partially Ordered Preferences. J. Logic and Computation 19(3): 475-502 (2009)
Outline Computational concerns
Given the impossibility result, we want to avoid rules which are
computationally easy to manipulate
We have studied computational complexity of manipulation/winner
determination for voting rules when some preferences are missing
Some voting rules are difficult to manipulate when we have weighted voters
and incomparable pairs
For some classes of voting rules it is computationally easy to find possible and
necessary winners and terminate preference elicitation
Pini, Rossi, Venable, Walsh: Incompleteness and incomparability in preference aggregation: Complexity results. Artificial Intelligence 175(7-8): 1272-1289 (2011)
Pini, Rossi, Venable, Walsh: Winner determination in voting trees with incomplete preferences and
weighted votes. Autonomous Agents and Multi-Agent Systems 25(1): 130-157 (2012)
Bartholdi, Tovey, Trick. The computational difficulty of manipulating an election. Social Choice and Welfare 1989
Outline Computational Social Choice
It is an interdisciplinary field at the interface of
social choice theory
computer science and AI
Main goals
1. Application of techniques of computer science, such as complexity
analysis or algorithm design, to the study of social choice mechanisms,
such as voting procedures
2. Importing concepts from social choice theory into computing. For
instance, the study of preference aggregation mechanisms is relevant to
multi-agent systems
Chevaleyre, Endriss, Lang, Maudet, A short introduction to Computational Social Choice, 2007
Outline Computational Social Choice
Between multi-agent systems and social choice
AI, economics, mathematics, political science, etc.
Social choice
Voting rules
Desirable properties
Impossibility results
Computational social choice
Incomparability
Computational concern
Compact preference formalism
Uncertainty and preference elicitation
Outline Formalisms to model preferences
compactly
Preference ordering over a large set of decisions (candidates,
outcomes, …) need to model them compactly
Otherwise too much space and time to handle such preferences
An Example: Soft constraint formalism
Preferences over partial assignments of the decision variables, from
which to generate the preference ordering over the solution space
Outline Soft Constraints
(the c-semiring framework)
Variables {X1,…,Xn}=X
Domains {D(X1),…,D(Xn)}=D
Soft constraints each constraint involves some of the variables a preference is associated with each assignment of the
variables
Set of preferences A Totally or partially ordered (induced by +) a ≤ b iff a+b=b
Combination operator (x) Top and bottom element (1, 0) Formally defined by a c-semiring <A,+,x,0,1>
Bistarelli, Montanari, Rossi: Semiring-based constraint satisfaction and optimization. J. ACM 44(2): 201-236 (1997)
Example: fuzzy constraints
Lunch time= 13 Meal = meat Wine = white Swimming time= 14
Decision A
pref(A)=min(0.3,0)=0
Lunch time = 12 Meal = fish Wine = white Swimming time = 14
Decision B
pref(B)=min(1,1)=1
{12, 13} {14, 15}
Lunch time
Swimming time
(12, 15) 1
(12, 14) 1 (13, 14) 0
(13, 15) 1
{fish, meat} {white, red}
meal wine
(fish, red) 0.8
(fish, white) 1 (meat,white) 0.3
(meat, red) 0.7
Example with fuzzy constraints Preference of a decision: minimal preference of its parts Aim: to find a decision with maximal preference Preference values: between 0 and 1
A soft constraint problem induces an
ordering over the solutions
Outline Uncertainty and vagueness
Missing preferences
Too costly to compute them
Privacy concerns
Ongoing preference elicitation process
Imprecise preferences
Preferences coming from sensor data
Too costly to compute the exact preference
Estimates
Compact preference formalisms and solving techniques
to model and solve problems with missing or imprecise
preferences
Gelain, Pini, Rossi, Venable, Wilson. Interval-valued soft constraint problem. Annals Mathematics and Artificial Intelligence 58(3-4): 261-298 (2010)
Gelain, Pini, Rossi, Venable, Walsh: Elicitation strategies for soft constraint problems with missing preferences: Properties, algorithms and experimental studies. Artificial Intelligence 174(3-4): 270-294 (2010)
Outline
Stable allocations
Matching of two sets
Men to women
Doctors to hospitals
Students to schools
Two-sided markets
Kidney donors and patients
Preferences
Each member of one group expresses a total order over all the members of the other group
Stability
not two agents who would prefer each other over their current counterparts
Outline Gale-Shapley algorithm
Gale-Shapley algorithm (1962)
If the number of doctors and hospitals is the same
The algorithms always find a stable allocation
Irrespective of agents’ preferences
It takes O(n^2) time, where n is the number of hospitals/doctors
A. Roth: Gale-Shapely algorithm is manipulable (1984)
Nobel prize in Economics 2012
A. Roth L. Shapely
Applications:
Doctors-Hospitals
USA, Scotland
Students-schools
New York, Boston, Spain, Hungary
Professors-schools
France, UK
Kidney transplants (donors-patients)
Spain, UK, USA, Australia
Outline Stable matching
In practical applications
It is useful to allow for ties and incomparable elements
Hospitals with many applicants have expressed the desire to use ties
It is more natural to express scores than a preference ordering
Score may model profits or costs
My research
New notions of stability and optimality in these scenarios
Algorithms that generalize Gale-Shapely alg. to find matchings that are stable and optimal according to new stability/optimality notions
Computational complexity of manipulating of stable matching procedures
Stable matching procedure based on voting rules difficult to manipulate
Pini, Rossi, Venable, Walsh: Manipulation complexity and gender neutrality in stable marriage procedure,
Autonomous Agents and Multi-Agent Systems 22(1): 183-199 (2011)
Pini, Rossi, Venable, Walsh: Stability in Matching Problems with Weighted Preferences. ICAART 2011
Pini, Rossi, Venable, Walsh: Weights in stable marriage problems increase manipulation opportunities. TARK 2011,
Best poster & presentation award
Outline Conclusions
Intelligent preference reasoning in multi-agent decision making
Computational social choice (CCS)
Between multi-agent systems and social choice
Preference modelling
Incomparability
Uncertainty and preference elicitation
Stable allocations
Cross-fertilization in both directions
Prabhakar Raghavan (Vice President of Strategic Technologies at Google)
DEI - Distinguished Lecturer Series, September 2012
“The academic challenge: How can we combine computer science and social science?”
Preference reasoning in CCS is a first step in this direction
Outline Future work
Voting theory and stable matching
Compact preference formalisms for expressing agents’ preferences
Influence, bribery and control
Stable matching procedures based on voting rules
Uncertainty and preference elicitation in stable matching procedures
Doodle with preferences
Voting rules to aggregate agents’ preferences
Preferences in recommender and reputation systems
Outline Joint work with …
Francesca Rossi
University of Padova
Brent Venable
Tulane University and IHMC
Toby Walsh
NICTA, Australia
Jerome Lang
LAMSADE, Paris
Ulle Endriss
ILLC, University of Amsterdam
Nicolas Maudet University Paris-Dauphine
Mirco Gelain University of Padova
Nic Wilson
4C, Ireland
Nick Mattei University of Kentucky