+Francesca RossiUniversity of Padova, Italy                              Preference reasoning                             ...
+ My wordlePRICAI 2012 - Kuching, Malaysia
+ Outline   n    Preferences   n    Collective decision making in multi-agent systems   n    Social choice   n    Comp...
+ Why preferences?  ¡  An   intelligent system must be able to handle       soft information        l  different levels ...
+ Preferences     n    Ubiquitous in real life           n  I prefer Venice to Rome     n    A more tolerant way to set...
Example: University timetabling  Professor	                      Constraints                    Administra/on	            ...
+ Several kinds of preferences   n  Positive        (degrees of acceptance)       n  I   like ice cream   n  Negative  ...
+ Two main ways to model preferences   n    Quantitative         n    Numbers, or ordered set of objects               n...
+ Desiderata for an AI preference framework   ¡  Expressive   power   ¡  Compactness   ¡  Efficiency   ¡  Suitability ...
+ Preferences for collective decision making in multi-agent systems   n  Several      agents   n  Common          set of...
+ Example    n    Three friends need to decide what to cook for dinner      n  4   items (pasta, main, dessert, drink), ...
+ Another example:    n          Several time slots under consideration    n          Partecipants accepts or reject eac...
+ Collective decision scenarios    n          IT enabled social environments          n     People are connected all the...
+ How to compute a collective decision?    n    Let the agents vote by expressing their          preferences over the pos...
+ Voting theory (Social choice)  n    Voters  n    Candidates  n    Each voter expresses its preferences over the candi...
+               Some voting rules   n    Plurality         n    Voting: one most preferred decision         n    Select...
+    Plurality     n  Ballot: 1        alternative     n  Result: alternative(s)         with the most vote(s)     n  E...
+Borda                                                                  Winner                                            ...
+ Some desirable properties (1)    n    Condorcet-consistency          n  If there is an option who beats every other op...
+ Some desirable properties(2)      n  Non-dictatorship         n  Thereis no voter such that his top choice always     ...
+   Different properties for different voting   rules   n    For the voting rules in our list:         n    All are anon...
+ Two classical impossibility results   n    Arrow’s theorem         n    it is impossible to have unanimity + IIA + non...
+    Social choice    scenarios vs. multi-    agent systems
+   Is social choice all we need for   collective decision making in multi-   agent systems?   After all …   n  Voters   ...
+ Main differences   n  In    multi-agent AI scenarios, we usually have       n  Large sets of candidates (w.r.t. number...
+ Large set of candidates   n  In   AI scenarios, usually the set of decisions is much       larger than the set of agent...
+ Combinatorial structure for the set of decisions    n  Combinatorial             structure for the set of decisions    ...
+ Formalisms to model preferences compactly   n    Preference ordering over a large set of decisions è need to         m...
+ Incomparability   n  Preferences          do not always induce a total order   n  Some    items are naturally incompar...
+ Uncertainty, vagueness   n    Missing preferences         n    Too costly to compute them         n    Privacy concer...
+ Computational concerns   n    We would like to avoid very costly ways to         n    Model the preferences         n...
+ Computational social choice   n    Between multi-agent systems and social choice         n    AI, economics, mathemati...
+    Computational    concerns
+ Computational concerns about voting rules   n    We want to avoid spending too much time to         n    Elicit prefer...
+ Manipulation   n  A    rule is manipulable if an agent can gain by lying       about its preferences       n  Gain = o...
+ Intractable manipulation   n  Ifmanipulation is computationally intractable for F,       then F might be considered res...
+ Manipulability as a decision problem           Manipulability(F)           Instance: Set of ballots for all but one vote...
+     Plurality is easy to manipulate   Manipulability(Plurality)ε P   n    Simply vote for x, the alternative to be made...
+ Manipulating Borda is easy   MANIPULABILITY(Borda) ε P   n    Place x (the alternative to be made winner through       ...
+ Manipulating STV is difficult   n    MANIPULABILITY(STV) ε NP-complete   n    NP-membership is clear: checking whether...
+ Coalitional constructive  manipulation         n    Manipulation by a coalition of agents         n    Constructive: t...
+Weighted-coalitional constructive   manipulationPRICAI 2012 - Kuching, Malaysia
+ Destructive manipulation        n     The goal is to make sure that a candidate does not win               (whoever els...
+Weighted-coalitional destructive    manipulationPRICAI 2012 - Kuching, Malaysia
+    Two examples:    •  Soft constraints    •  CP-nets                          Formalisms to model                      ...
+ Modelling preferences compactly   n    Preference ordering: an ordering over the whole set of         solutions (or can...
+ Soft Constraints  (the c-semiring framework)   n  Variables       {X1,…,Xn}=X   n  Domains         {D(X1),…,D(Xn)}=D  ...
+ Soft Constraintsn    Soft constraint: a pair c=<f,con> where:      n    Scope: con={Xc1,…, Xck} subset of X      n   ...
Example: fuzzy constraints       Preference of a decision: minimal preference of its parts       Aim: to find a decision w...
+ Soft Constraints  n    A soft CSP induces an ordering over the solutions, from the        ordering of the preference se...
Qualitative and conditionalpreferencesn Softconstraints model quantitatively unconditional  preferencesn Many        pro...
+                                                                               Op/mal	  solu/on	    Solution ordering    ...
Soft constraints vs. CP-nets                                  Soft CSPs   Tree-like        CP-nets   Acyclic CP-          ...
+    Aggregating    combinatorially-    structured preferences
+    Multiple issues    n    Example:          n    3 referendum (yes/no)          n    Each voter has to give his pref...
+ Maybe Plurality on tuples? n    Ask each voter for her most preferred combination and       apply the Plurality rule   ...
+ Other voting rules on tuples n    Vote on combinations and use other voting rules that use the       whole preference o...
+ Sequential voting n    Main idea: Vote separately on each issue, but do so       sequentially n    This gives voters t...
+Sequential voting and Condorcet losersn    Condorcet loser (CL): candidate that loses against any other      candidate i...
+    A sequential approach    to aggregate    combinatorially-    structured preferences
+ Dinner example, three agents, fuzzy constraints       Pesto 1                  Pesto 0.9                Pesto 1       To...
+ The sequential voting approach   n  For      each variable          n    compute an explicit profile over the variable...
+ Dinner example using plurality   Pesto   Pesto      1              1              Pesto 0.9                    Pesto 1  ...
+ Local vs. sequential properties   n    If each ri has the property, does the sequential rule have the         property?...
Properties                                     Local to sequential        Sequential to local             Condorcet consis...
+ The sequential approach behaves like the non-sequential one    n    Independently of the variable ordering    n    Ind...
+    Sequential Voting with CP-nets
+ Profiles via compatible CP-nets n    n voters, voting by giving a CP-net each       n    Same variables, different dep...
Example  3    Rovers must decide:  •    Where to go: Location A or Location B  •    What to do: Analyze a rock or Take a p...
+    Bribing CP-nets
+ Bribery when voting with CP-nets       n  Agents         express their preferences via CP-nets       n  External      ...
+Complexity results for bribery withCP-nets              Sequential     Sequential   Plurality      Plurality             ...
+    Preference elicitation
Preference elicitation         n  Some       preferences may be missing         n  Time      consuming, costly, difficul...
+ Possible and necessary winners     n    Necessary winner           n  However remaining votes are cast, he must win   ...
+ Computing possible and         necessary winners         n    Consider specific voting rules         n    Unweighted v...
+ Manipulation can be allowed: iterative voting   n    Once agents have voted, maybe some would want to change         th...
+ Conclusions   n    Brief introduction to computational social choice:         n    Between multi-agent systems and soc...
+ If you want to know more     “A short introduction to preferences:     between Artificial Intelligence and     Social Ch...
+   Waiting for all of you in   Beijing next year for IJCAI   2013!PRICAI 2012 - Kuching, Malaysia
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Preference reasoning and aggregation: between AI and social choice

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Keynote by Francesca Rossi at the 12th Pacific Rim International Conference on Artificial Intelligence (PRICAI2012) during the Knowledge Technology Week (KTW2012). September 3 - 7, 2012. Kuching, Sarawak, Malaysia. http://ktw.mimos.my/ktw2012/

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Preference reasoning and aggregation: between AI and social choice

  1. 1. +Francesca RossiUniversity of Padova, Italy Preference reasoning and aggregation: between AI and social choice
  2. 2. + My wordlePRICAI 2012 - Kuching, Malaysia
  3. 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 elicitationPRICAI 2012 - Kuching, Malaysia
  4. 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 availablePRICAI 2012 - Kuching, Malaysia
  5. 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. 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. 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. 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 themPRICAI 2012 - Kuching, Malaysia
  9. 9. + Desiderata for an AI preference framework ¡  Expressive power ¡  Compactness ¡  Efficiency ¡  Suitability for multi-agent settingsPRICAI 2012 - Kuching, Malaysia
  10. 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. 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 dinnersPRICAI 2012 - Kuching, Malaysia
  12. 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 participantsPRICAI 2012 - Kuching, Malaysia
  13. 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. 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. 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 resultsPRICAI 2012 - Kuching, Malaysia
  16. 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. 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
  18. 18. +Borda Winner Bordarank rank rank rank rank Count3 3 3 3 3 92 2 2 2 2 81 1 1 1 1 70 0 0 0 0 6 1 voter 1 voter 1 voter 1 voters 1 voter
  19. 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 selectedPRICAI 2012 - Kuching, Malaysia
  20. 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 lyingPRICAI 2012 - Kuching, Malaysia
  21. 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 IIAPRICAI 2012 - Kuching, Malaysia
  22. 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!PRICAI 2012 - Kuching, Malaysia
  23. 23. + Social choice scenarios vs. multi- agent systems
  24. 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. 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 concernsPRICAI 2012 - Kuching, Malaysia
  26. 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 solversPRICAI 2012 - Kuching, Malaysia
  27. 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 optionsPRICAI 2012 - Kuching, Malaysia
  28. 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-netsPRICAI 2012 - Kuching, Malaysia
  29. 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 ordersPRICAI 2012 - Kuching, Malaysia
  30. 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  EstimatesPRICAI 2012 - Kuching, Malaysia
  31. 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. 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 directionsPRICAI 2012 - Kuching, Malaysia
  33. 33. + Computational concerns
  34. 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 manipulatePRICAI 2012 - Kuching, Malaysia
  35. 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. 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. 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. 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. 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. 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. 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
  42. 42. +Weighted-coalitional constructive manipulationPRICAI 2012 - Kuching, Malaysia
  43. 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 coalitionPRICAI 2012 - Kuching, Malaysia
  44. 44. +Weighted-coalitional destructive manipulationPRICAI 2012 - Kuching, Malaysia
  45. 45. + Two examples: •  Soft constraints •  CP-nets Formalisms to model preferences
  46. 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 spacePRICAI 2012 - Kuching, Malaysia
  47. 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. 48. + Soft Constraintsn  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 preferencen  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 forbiddenPRICAI 2012 - Kuching, Malaysia
  49. 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)=1PRICAI 2012 - Kuching, Malaysia
  50. 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. 51. Qualitative and conditionalpreferencesn Softconstraints model quantitatively unconditional preferencesn 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 userPRICAI 2012 - Kuching, Malaysia
  52. 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. 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 easyFind 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 optimalPRICAI 2012 - Kuching, Malaysia
  54. 54. + Aggregating combinatorially- structured preferences
  55. 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. 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. 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. 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. 59. +Sequential voting and Condorcet losersn  Condorcet loser (CL): candidate that loses against any other candidate in a pairwise contest n  Plurality may choose a Condorcet losern  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. 60. + A sequential approach to aggregate combinatorially- structured preferences
  61. 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. 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
  63. 63. + Dinner example using plurality Pesto Pesto 1 1 Pesto 0.9 Pesto 1 Tom Tom 0 0.7 Tom 1 0 Tom 0.3 0 Plurality Pasta = Pasta Pasta Pasta Pesto (Pesto, Beer) 11 (Pesto, Beer) (Pesto, Beer) 11 (Pesto, Beer) (Pesto, Beer) 1 (Pesto, Beer) 1 (Pesto,Wine) 0.5 (Pesto,Wine) 0.5 (Pesto,Wine) 0.9 (Pesto,Wine) 0.9 (Pesto,Wine) 0.3 (Pesto,Wine) 0.3 (Tom ,Beer) 0 (Tom ,Beer) 0.7 (Tom ,Beer) 0 (Tom ,Beer) 0.9 (Tom ,Beer) 0.3 (Tom ,Beer) 0 (Tom,Wine) 0 (Tom,Wine) 0.3 (Tom,Wine) 0 (Tom,Wine) 0.9 (Tom,Wine) 1 (Tom,Wine) 0 Plurality Drink Drink Drink Drink = Beer Beer 1 Beer 1 Beer 1 Wine 0.7 0.5 Wine 1 0.9 Wine 10.3 Winner Agent 1 Agent 2 Agent 3PRICAI 2012 - Kuching, Malaysia
  64. 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. 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 yesPRICAI 2012 - Kuching, Malaysia
  66. 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
  67. 67. + Sequential Voting with CP-nets
  68. 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. 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> AnalyzeImage >Analyze Image >Analyze Analyze >Image Plurality WHAT WHAT WHAT WHAT = ImageSt1 >St2 St2>St1 St2>St1 Plurality DLINK DLINK DLINK DLINK = St2ROVER 1 ROVER 2 ROVER 3
  70. 70. + Bribing CP-nets
  71. 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. 72. +Complexity results for bribery withCP-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]
  73. 73. + Preference elicitation
  74. 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 fixedPRICAI 2012 - Kuching, Malaysia
  75. 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. 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. 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, MaximinPRICAI 2012 - Kuching, Malaysia
  78. 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 directionsPRICAI 2012 - Kuching, Malaysia
  79. 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. 80. + Waiting for all of you in Beijing next year for IJCAI 2013!PRICAI 2012 - Kuching, Malaysia

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