The document summarizes Francesca Rossi's presentation on preference reasoning and aggregation between artificial intelligence and social choice. Some key points:
1) Rossi discussed modeling and reasoning with preferences in multi-agent systems and the challenges of aggregating preferences from multiple agents to make collective decisions.
2) She drew connections between voting theory/social choice and preference aggregation in multi-agent systems, noting voting rules like plurality and Borda counting.
3) However, she noted some differences in multi-agent systems, like larger sets of possible decisions, combinatorial structure of decisions, and need for compact preference representation formalisms.
3. +
Outline
n Preferences
n Collective decision making in multi-agent systems
n Social choice
n Computational social choice (CSS)
n Some specific issues in CSS
n Computational concerns
n Intractable manipulation
n Two preference formalisms
n soft constraints, CP-nets
n Sequential voting
n Preference elicitation
PRICAI 2012 - Kuching, Malaysia
4. +
Why preferences?
¡ An intelligent system must be able to handle
soft information
l different levels of preference or rejection
l several levels of tolerance
l vagueness
l imprecision
¡ Information may be non-crisp
l intrinsically: the world is not binary
l due to information which is only partially
available
PRICAI 2012 - Kuching, Malaysia
5. +
Preferences
n Ubiquitous in real life
n I prefer Venice to Rome
n A more tolerant way to set some constraints over the possible scenarios
n I prefer a blue car
n Constraints can be used when we know what to accept or reject
n I don’t want to spend more than X
n If all constraints, possibly
n no solution, or
n too many of them, all apparently equally good
n Some problems are naturally modelled with preferences
n I don’t like meat, and I prefer fish to cheese
n Constraints and preferences may be present in the same problem
n Configuration, timetabling, etc.
PRICAI 2012 - Kuching, Malaysia
6. Example: University timetabling
Professor
Constraints Administra/on
Constraints
I cannot teach on Wednesday
afternoon.
I prefer not to teach early in
the morning, nor on Friday Lab C can fit only 120 students.
afternoon.
Better to not leave 1-hour holes in
the day schedule.
Preferences
Preferences
PRICAI 2012 - Kuching, Malaysia
7. +
Several kinds of preferences
n Positive (degrees of acceptance)
n I like ice cream
n Negative (degrees of rejection)
n I don’t like strawberries
n Unconditional
n I prefer taking the bus
n Conditional
n I prefer taking the bus if it s raining
n Multi-agent
n I
like blue, my husband likes green, what color do we
buy the car?
PRICAI 2012 - Kuching, Malaysia
8. +
Two main ways to model
preferences
n Quantitative
n Numbers, or ordered set of objects
n My preference for ice cream is 0.8, and for cake is 0.6
n E.g., soft constraints
n Qualitative
n Pairwise comparisons
n Ice cream is better than cake
n E.g., CP-nets
n Both very natural in some scenarios
n Different expressive power
n Different computational complexity for reasoning with them
PRICAI 2012 - Kuching, Malaysia
9. +
Desiderata for an AI preference
framework
¡ Expressive power
¡ Compactness
¡ Efficiency
¡ Suitability for multi-agent settings
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10. +
Preferences for collective decision
making in multi-agent systems
n Several agents
n Common set of possible decisions
n Each agent has its preferences over the possible
decisions
n Goal: to choose one of the decisions, based on the
preferences of the agents
n Also a set of decisions, or a ranking over the decisions
n AI scenarios add: imprecision, uncertainty, complexity,
etc.
PRICAI 2012 - Kuching, Malaysia
11. +
Example
n Three friends need to decide what to cook for dinner
n 4 items (pasta, main, dessert, drink), 5 options for each
n Each friend has his/her own preferences over the
meals
Agents = friends
Decisions = all possible dinners
PRICAI 2012 - Kuching, Malaysia
12. +
Another example:
n Several time slots under consideration
n Partecipants accepts or reject each time slot
n Very simple way to express preferences over time
slots
n Very little information communicated to the system
n Collective choice: a single time slot
n The one with most acceptance votes from
participants
PRICAI 2012 - Kuching, Malaysia
13. +
Collective decision scenarios
n IT enabled social environments
n People are connected all the time
n Social networks allow us to share a large amount of
information
n More and more, we want to exploit this information to
take collective decisions
n With our friends, colleagues, etc.
n Also committees of agents
n Search engines
n Solvers
n Classifiers
n Product ranking agents, etc.
PRICAI 2012 - Kuching, Malaysia
14. +
How to compute a collective
decision?
n Let the agents vote by expressing their
preferences over the possible decisions
n Aggregate the votes to get a single decision
n Let’s look at voting theory then!
PRICAI 2012 - Kuching, Malaysia
15. +
Voting theory
(Social choice)
n Voters
n Candidates
n Each voter expresses its preferences over the candidates
n Goal: to choose one candidate (the winner), based on the
voters’ preferences
n Also many candidates, or ranking
n Rules (functions) to achieve the goal
n Properties of the rules
n Impossibility results
PRICAI 2012 - Kuching, Malaysia
16. + Some voting rules
n Plurality
n Voting: one most preferred decision
n Selection: the decision preferred by the largest number of agents
n Majority: like plurality, over 2 options
n Borda (m options)
n Voting: rank over all options, m-i score of ith option
n Selection: option with greatest sum of scores
n Approval (m options)
n Voting: approval of between 1 and m-1 options
n Selection: option with most votes
n This is what Doodle uses
n Copeland
n Voting: ranking over all options
n Selection: option which wins most pairwise competitions
n Cup
n Voting: ranking over all options
n Selection: winner in the agenda (tree of pairwise competitions)
PRICAI 2012 - Kuching, Malaysia
17. +
Plurality
n Ballot: 1 alternative
n Result: alternative(s) with the most vote(s)
n Example:
n 6 voters
n Candidates:
Ballot Profile Winner
19. +
Some desirable properties (1)
n Condorcet-consistency
n If there is an option who beats every other option in pairwise
competitions, it is selected
n Anonymity
n Result does not depend on who are the agents
n Neutrality
n Result does not depend on which are the options
n Monotonicity
n If an agent improves his preference for the winning option, this
option is still selected
n Consistency
n If two sets of agents have the same result, this result is also
obtained by joining the two sets
n Participation
n Given an agent, its addition to a profile leads to an equally or
more preferred result for this agent
n Unanimity (efficiency)
n If all agents have the same top choice, it is selected
PRICAI 2012 - Kuching, Malaysia
20. + Some desirable properties(2)
n Non-dictatorship
n Thereis no voter such that his top choice always
wins, regardless of the votes of other voters
n Independence to Irrelevant Alternatives
n If choice X wins given a profile p, then Y cannot
win in any profile where the relation between X
and Y is as in p
n Strategy-proofness
n Thereis no profile and no agent such that the
agent would be better off lying
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21. +
Different properties for different voting
rules
n For the voting rules in our list:
n All are anonymous, neutral, non-dictatorial, and manipulable
n All but Cup are efficient
n Cup and Copeland are Condorcet-consistent
n All but Cup and Copeland are consistent and participative
n Only Approval is IIA
PRICAI 2012 - Kuching, Malaysia
22. +
Two classical
impossibility results
n Arrow’s theorem
n it is impossible to have unanimity + IIA + non-dictatoriality
n Bad news, but IIA is a very strong property
n Gibbard-Sattherwaite’s theorem
n it is impossible to have surjectivity + non-dictatoriality + strategy-
proofness
n This are really bad news: we don’t want to give up
surjectivity, nor non-dictatoriality!
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23. +
Social choice
scenarios vs. multi-
agent systems
24. +
Is social choice all we need for
collective decision making in multi-
agent systems?
After all …
n Voters = agents
n Candidates = decisions
n Preferences
n Winner = chosen decision
But …
PRICAI 2012 - Kuching, Malaysia
25. +
Main differences
n In multi-agent AI scenarios, we usually have
n Large sets of candidates (w.r.t. number of voters)
n Combinatorial structure for candidate set
n Knowledge representation formalisms to model
preferences
n Incomparability
n Uncertainty, vagueness
n Computational concerns
PRICAI 2012 - Kuching, Malaysia
26. +
Large set of candidates
n In AI scenarios, usually the set of decisions is much
larger than the set of agents expressing
preferences over the decisions
n Many web pages, few search engines
n Many solutions of a constraint problem, few
solvers
PRICAI 2012 - Kuching, Malaysia
27. +
Combinatorial structure for the set
of decisions
n Combinatorial structure for the set of decisions
n Car
(or PC, or camera) = several features, each with
some instances
n Dinner example:
n Three friends need to decide what to cook for dinner
n 4 items (pasta, main, dessert, drink)
n 5 options for each è 54 = 625 possible dinners
n In
general: Cartesian product of several variable
domains
n Variables = items of the menu, domain= 5 options
PRICAI 2012 - Kuching, Malaysia
28. +
Formalisms to model preferences
compactly
n Preference ordering over a large set of decisions è need to
model them compactly
n Otherwise too much space and time to handle such preferences
n Two examples:
n Soft constraints
n CP-nets
PRICAI 2012 - Kuching, Malaysia
29. +
Incomparability
n Preferences do not always induce a total order
n Some items are naturally incomparable
n Not because the information is missing
n Itdepends also on the combinatorial structure (multi-
criteria and Pareto dominance)
n To model uncertainty
n As a means to resolve conflicts
n Many AI formalisms to model preferences allow for
partial orders
PRICAI 2012 - Kuching, Malaysia
30. +
Uncertainty, vagueness
n Missing preferences
n Too costly to compute them
n Privacy concerns
n Ongoing elicitation process
n Imprecise preferences
n Preferences coming from sensor data
n Too costly to compute the exact preference
n Estimates
PRICAI 2012 - Kuching, Malaysia
31. +
Computational concerns
n We would like to avoid very costly ways to
n Model the preferences
n Compute the winner
n Reason with the agents’ preferences
n On the other hand,we need a computational barrier against
bad behaviours (such as manipulation)
PRICAI 2012 - Kuching, Malaysia
32. +
Computational social choice
n Between multi-agent systems and social choice
n AI, economics, mathematics, political science, etc.
n New concerns
n Preference modelling
n Algorithms, complexity
n Uncertainty, preference elicitation
n Cross-fertilization in both directions
PRICAI 2012 - Kuching, Malaysia
34. +
Computational concerns about
voting rules
n We want to avoid spending too much time to
n Elicit preferences
n Compute the winner (winners, ranking)
n On the other hand, we want a computational barrier against
manipulation
n Given the impossibility result, we want to avoid rules which are
computationally easy to manipulate
PRICAI 2012 - Kuching, Malaysia
35. +
Manipulation
n A rule is manipulable if an agent can gain by lying
about its preferences
n Gain = obtain a result which is more preferred by
the agent
n Strategy-proof rule: there is no incentive to
misrepresent the preferences
n Gibbard-Sattherwaite impossibility result
n With at least three agents and two candidates, it is
impossible for a voting rule to be at the same time
surjective, strategy-proof, and non- dictatorial
n We cannot give up non-dictatoriality and surjectivity!
PRICAI 2012 - Kuching, Malaysia
36. +
Intractable manipulation
n Ifmanipulation is computationally intractable for F,
then F might be considered resistant (albeit still
not immune) to manipulation
n Mostinteresting for voting procedures for which
winner determination is tractable
n complexity gap between manipulation (undesired
behaviour) and winner determination (desired
functionality)
PRICAI 2012 - Kuching, Malaysia
37. +
Manipulability as a decision
problem
Manipulability(F)
Instance: Set of ballots for all but one voter; alternative x.
Question: Is there a ballot for the final voter such that x wins?
How difficult it is to answer this question?
n A manipulator would have to solve Manipulability(F) for all alternatives, in its
preference ordering, to understand if there is a way he can get something
better
n If Manipulability(F) is computationally intractable, then manipulability may
be considered less of a worry for procedure F.
n Remark: We assume that the manipulator knows all the other ballots
n This unrealistic assumption is reasonable for intractability results: if manipulation is intractable
even under such favorable conditions, then all the better.
PRICAI 2012 - Kuching, Malaysia
38. +
Plurality is easy to manipulate
Manipulability(Plurality)ε P
n Simply vote for x, the alternative to be made winner by
means of manipulation. If manipulation is possible at all, this
will work. Otherwise not.
n In general: Manipulability(F) ε P for any rule F with
polynomial winner determination and polynomial number of
ballots.
[Bartholdi,Tovey,Trick,1989]
PRICAI 2012 - Kuching, Malaysia
39. + Manipulating Borda is easy
MANIPULABILITY(Borda) ε P
n Place x (the alternative to be made winner through
manipulation) at the top of your ballot.
n Then inductively proceed as follows: Check if any of the
remaining alternatives can be put next on the ballot without
preventing x from winning. If yes, do so. If no, manipulation is
impossible.
[Bartholdi,Tovey,Trick,1989]
PRICAI 2012 - Kuching, Malaysia
40. + Manipulating STV is difficult
n MANIPULABILITY(STV) ε NP-complete
n NP-membership is clear: checking whether a given ballot
makes x win can be done in polynomial time (just try it, STV
is polynomial to compute).
n NP-hardness: by reduction from another NP-complete
problem (3-Cover). The basic idea is to build a large election
instance introducing all sorts of constraints on the ballot of
the manipulator, such that finding a ballot meeting those
constraints solves a given instance of 3-Cover.
PRICAI 2012 - Kuching, Malaysia [Bartholdi,Orlin, 1991]
41. + Coalitional constructive
manipulation
n Manipulation by a coalition of agents
n Constructive: to make some candidate
win (not just to get a better result)
PRICAI 2012 - Kuching, Malaysia
43. + Destructive manipulation
n The goal is to make sure that a candidate does not win
(whoever else wins)
n Ifconstructive manipulation is easy then so is
destructive manipulation
n Destructive manipulation can be easy even though
constructive manipulation is hard
n Reverse does not hold
n E.g. Borda is polynomial to manipulate desctructively but NP-
hard constructively for 3 or more candidates for a weighted
coalition
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45. +
Two examples:
• Soft constraints
• CP-nets
Formalisms to model
preferences
46. + Modelling preferences compactly
n Preference ordering: an ordering over the whole set of
solutions (or candidates, or outcomes, …)
n Solution space with a combinatorial structure è preferences
over partial assignments of the decision variables, from which
to generate the preference ordering over the solution space
PRICAI 2012 - Kuching, Malaysia
47. + Soft Constraints
(the c-semiring framework)
n Variables {X1,…,Xn}=X
n Domains {D(X1),…,D(Xn)}=D
n Soft constraints
n each constraint involves some of the variables
n a preference is associated with each assignment of the
variables
n Set of preferences A
n Totally or partially ordered (induced by +)
n a ≤ b iff a+b=b
n Combination operator (x)
n Top and bottom element (1, 0)
n Formally defined by a c-semiring <A,+,x,0,1>
PRICAI 2012 - Kuching, Malaysia
48. + Soft Constraints
n Soft constraint: a pair c=<f,con> where:
n Scope: con={Xc1,…, Xck} subset of X
n Preference function :
f: D(Xc1)x…xD(Xck) → A
tuple (v1,…, vk) → p preference
n Hard constraint: a soft constraint where for each tuple (v1,…, vk)
f (v1,…, vk)=1 the tuple is allowed
f (v1,…, vk)=0 the tuple is forbidden
PRICAI 2012 - Kuching, Malaysia
49. Example: 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
{fish,
meat}
{white,
red}
Decision A
meal
wine
Lunch time= 13
Meal = carne
(fish,
white)
à
1
(meat,white)
à
0.3
Wine = bianco
(fish,
red)
à
0.8
(meat,
red)
à
0.7
Swimming time= 14
pref(A)=min(0.3,0)=0
{12,
13}
{14,
15}
Decision B
Lunch time = 12
Lunch
Swimming
Meal = pesce
/me
/me
Wine = bianco
(13,
14)
à
0
Swimming time = 14
(12,
14)
à
1
(12,
15)
à
1
(13,
15)
à
1
pref(B)=min(1,1)=1
PRICAI 2012 - Kuching, Malaysia
50. + Soft Constraints
n A soft CSP induces an ordering over the solutions, from the
ordering of the preference set
n Totally ordered semiring è total order over solutions (possibly
with ties)
n Partially ordered semiring è total or partial order over solutions
(possibly with ties)
n Any ordering can be obtained!
PRICAI 2012 - Kuching, Malaysia
51. Qualitative and conditional
preferences
n Softconstraints model quantitatively unconditional
preferences
n Many problems need statements like
n I like white wine if there is fish
(conditional)
n I like white wine better than red wine
(qualitative)
n Quantitative è a level of preference for each
assignment of the variables in a soft constraint è
possibly difficult to elicitate preferences from user
PRICAI 2012 - Kuching, Malaysia
52. +
Op/mal
solu/on
Solution ordering
Fish,
white,
peaches
fish>meat
Main
course
Fish,
red,
peaches
Fish,
white,
berries
Main course Wine
fish white > red
Wine
meat red > white
meat,
red,
peaches
Fish,
red,
berries
peaches
>
strawberries
Fruit
meat,
white,
peaches
meat,
red,
berries
PRICAI 2012 - Kuching, Malaysia meat,
white,
berries
53. Soft constraints vs. CP-nets
Soft CSPs Tree-like CP-nets Acyclic CP-
soft CSPS nets
Preference orderings all all some some
difficult easy difficult easy
Find an optimal decision
easy easy difficult difficult
Compare two decisions
Find the next best difficult difficult for
weighted, difficult easy
Decision easy for fuzzy
Check if a decision difficult easy easy easy
is optimal
PRICAI 2012 - Kuching, Malaysia
55. +
Multiple issues
n Example:
n 3 referendum (yes/no)
n Each voter has to give his preferences over triples of yes and no
n Such as: YYY>NNN>YNY>YNN>etc.
n With k issues, k-tuples (2k of such tuples if binary issues)
n Not every voting rule will work well
n Example: 13 voters, 3 binary issues:
n 3 voters each vote for YNN, NYN, NNY
n 1 voter votes for YYY, YYN, YNY, NYY
n No voter votes for NNN
n If we use majority on each issue: the winner is NNN!
n Each issue has 7 out of 13 votes for N
56. +
Maybe Plurality on tuples?
n Ask each voter for her most preferred combination and
apply the Plurality rule
n Avoids the paradox, computationally light
n But … almost random decisions
n Example: 10 binary issues, 20 voters è 210 = 1024 combinations
to vote for but only 20 voters, so very high probability that no
combination receives more than one vote è tie-breaking rule
decides everything
n Similar also for voting rules that use only a small part of the
voters’ preferences (ex.: k-approval with small k)
57. +
Other voting rules on tuples
n Vote on combinations and use other voting rules that use the
whole preference ordering on combinations
n Avoids the arbitrariness problem of plurality
n Not feasible when there are large domains
n Example:
n Borda (needs the whole preference ordering)
n 6 binary issues è 26=64 possible combinations è each voter has
to choose among 64! possible ballots
58. +
Sequential voting
n Main idea: Vote separately on each issue, but do so
sequentially
n This gives voters the opportunity to make their vote for one
issue depend on the decisions on previous issues
59. +
Sequential voting and Condorcet losers
n Condorcet loser (CL): candidate that loses against any other
candidate in a pairwise contest
n Plurality may choose a Condorcet loser
n Thm.: Sequential plurality voting over binary issues never
elects a Condorcet loser
n Proof: Consider the election for the final issue. The winning
combination cannot be a CL, since it wins at least against the
other combination that was still possible after the penultimate
election
n [Lacy, Niou, J. of Theoretical Politics, 2000]
n But no guarantee that sequential voting elects the Condorcet
winner (Condorcet consistency).
60. +
A sequential approach
to aggregate
combinatorially-
structured preferences
61. +
Dinner example, three agents,
fuzzy constraints
Pesto 1 Pesto 0.9 Pesto 1
Tom 0.7 Tom 1 Tom 0.3
Pasta Pasta Pasta
(Pesto, Beer) 1 (Pesto, Beer) 1 (Pesto, Beer) 1
(Pesto,Wine) 0.5 (Pesto,Wine) 0.9 (Pesto,Wine) 0.3
(Tom ,Beer) 0.7 (Tom ,Beer) 0.9 (Tom ,Beer) 0.3
(Tom,Wine) 0.3 (Tom,Wine) 0.9 (Tom,Wine) 1
Drink Drink Drink
Beer 1 Beer 1 Beer 1
Wine 0.7 Wine 1 Wine 1
Agent 1 Agent 2 Agent 3
62. +
The sequential voting approach
n For each variable
n compute an explicit profile over the variable domain
n apply a voting rule to this explicit profile
n add the information about the selected variable value
n Similar approach used for CP-nets in [Lang, Xia, 2009]
PRICAI 2012 - Kuching, Malaysia
64. +
Local vs. sequential properties
n If each ri has the property, does the sequential rule have the
property?
n If some ri does not have the property, does the sequential
rule not have it?
n If the sequential rule has a property, do all the ri have it?
PRICAI 2012 - Kuching, Malaysia
65. Properties
Local to sequential Sequential to local
Condorcet consistency no yes
Anonymity yes yes
Neutrality no yes
Consistency yes yes
Participation no yes
Efficiency yes if single most yes
preferred option for all
agents
Monotonicity yes yes
IIA no yes
Non-dictatorship yes yes
Strategy-proofness no yes
PRICAI 2012 - Kuching, Malaysia
66. +
The sequential approach behaves
like the non-sequential one
n Independently of the variable ordering
n Independently of the amount of consensus among agents
n Also on best and worst cases
n … in our experimental setting
[Dalla Pozza, Pini, Rossi, Venable, IJCAI 2011]
[Dalla Pozza, Rossi, Venable, ICAART 2011]
PRICAI 2012 - Kuching, Malaysia
68. +
Profiles via compatible CP-nets
n n voters, voting by giving a CP-net each
n Same variables, different dependency graph and CP tables
n Compatible CP-nets: there exists a linear order on the variables
that is compatible with the dependency graph of all CP-nets (that
is, it completes the DAG)
n Then vote sequentially in this order
n Thm.: Under these assumptions, sequential voting is Condorcet
consistent if all local voting rules are
n (Lang and Xia, Math. Social Sciences, 2009)
69. Example
3 Rovers must decide:
• Where to go: Location A or Location B
• What to do: Analyze a rock or Take a picture
• Which station to downlink the data to: Station 1 or Station 2
Loc-A >Loc-B Loc-A
Loc-B> Loc-A Loc-A >Loc-B Winner
WHER WHER
E
WHERE E Plurality WHERE
=
Loc-A
Loc-A: Image > AnalyzeLoc-A: Analyze> Image
Loc-B: Analyze> Image Loc-B: Image> Analyze
Image >Analyze Image >Analyze Analyze >Image
Plurality WHAT
WHAT WHAT WHAT =
Image
St1 >St2 St2>St1 St2>St1
Plurality
DLINK
DLINK DLINK DLINK
=
St2
ROVER 1 ROVER 2 ROVER 3
71. +
Bribery when voting with CP-nets
n Agents express their preferences via CP-nets
n External agent with a desired candidate p and a budget B
n Briber can ask voters to change their preferences
n Voters charge the briber
n Cost scheme describing the cost of bribing each voter
n Fixed cost (C-equal), number of flips (C-flip), number of flips with any cost per flip (C-any), flips of
more important vars count more (C-level), distance from current optimal and new optimal (C-dist)
Can the briber make p win by spending within budget B?
How difficult it is for the briber to know this and understand
what to do?
PRICAI 2012 - Kuching, Malaysia
72. +
Complexity results for bribery with
CP-nets
Sequential Sequential Plurality Plurality
Majority Majority Veto Veto
with weights K-Approval K-Approval*
(IV) (DV, IV+DV)
C_EQUAL NP-complete NP-complete P P
C_FLIP P NP-complete P P
C_LEVEL P NP-complete P ?
C_ANY P NP-complete ? ?
C_DIST ? NP-complete P P
[Mattei, Rossi, Venable, Pini, AAMAS 2012]
74. Preference elicitation
n Some preferences may be missing
n Time consuming, costly, difficult, to elicit them all
n Want to terminate elicitation as soon as winner
fixed
PRICAI 2012 - Kuching, Malaysia
75. +
Possible and necessary winners
n Necessary winner
n However remaining votes are cast, he must win
n Possible winner
n There is a way for remaining votes to be cast so that he wins
n Closely connected to manipulation
n A is possible winner iff there is a constructive manipulation for A
n A is a necessary winner iff there is no destructive manipulation for A
n Closely connected to preference elicitation
n Elicitation can only be terminated iff possible winners = necessary winner
n “Deciding elicitation is over” is in P => computing possible (and necessary)
winners is also in P
[Konczak
and
Lang,
2005]
PRICAI 2012 - Kuching, Malaysia
[Walsh,
2008]
76. + Computing possible and
necessary winners
n Consider specific voting rules
n Unweighted votes
n Arbitrary number of candidates
n For STV, computing possible winners is NP-hard, and
necessary winners is coNP-hard
n Even NP-hard to approximate set of possible winners
within constant factor in size
n Easy for many other rules
[Pini,
Rossi,
Venable,
Walsh,
IJCAI
2007]
[Pini,
Rossi,
Venable,
Walsh,
AAMAS
2011]
[Lang,
Pini,
Rossi,
Salvagnin,
Venable,
Walsh,
JAAMAS
2011]
PRICAI 2012 - Kuching, Malaysia [Pini,
Rossi,
Venable,
Walsh,
AIJ
2011]
77. +
Manipulation can be allowed:
iterative voting
n Once agents have voted, maybe some would want to change
their vote because the collective decision is not acceptable
to them
n Doodle allows for that
n A form of legal manipulation
n But we want to make sure this process converges
n At some point, all agent must be either satisfied with the result or
must be unable to change the result
n Define manipulation moves (how to modify one agent’s vote)
to assure convergence
n Plurality and veto converge, Borda does not
n On going work for Copleand, Cup, Maximin
PRICAI 2012 - Kuching, Malaysia
78. +
Conclusions
n Brief introduction to computational social choice:
n Between multi-agent systems and social choice
n AI, economics, mathematics, etc.
n New concerns
n Preference modelling
n Algorithms, complexity
Uncertainty, preference elicitation
n
n Cross-fertilization in both directions
PRICAI 2012 - Kuching, Malaysia
79. +
If you want to know more
“A short introduction to preferences:
between Artificial Intelligence and
Social Choice”
F. Rossi, K. B. Venable, T. Walsh
Morgan & Claypool
2011
http://www.morganclaypool.com/
PRICAI 2012 - Kuching, Malaysia
80. +
Waiting for all of you in
Beijing next year for IJCAI
2013!
PRICAI 2012 - Kuching, Malaysia