Agent-Based Systems Agent-Based Systems Where are we? Last time . . . Agent-Based Systems • Discussed simple, abstract models of multiagent encounters • Utilities, preferences and outcomes Michael Rovatsos • Game-theoretic models and solution concepts mrovatso@inf.ed.ac.uk • Examples: Prisoner’s Dilemma, Coordination Game • Axelrod’s tournament its conclusions and critique Today . . . Lecture 9 – Social Choice • Social Choice 1 / 19 2 / 19 Agent-Based Systems Agent-Based Systems Making Group Decisions Preference Aggregation • Setting: • Previously we looked at agents acting strategically - Ag = { 1 , . . . , n } voters (finite, odd number) • Outcome in normal-form games follows immediately from agents’ - Ω = { ω 1 , ω 2 , . . . } possible outcomes or candidates choices - ̟ i ∈ Π(Ω) , preference ordering for agent i (e.g. ω ≻ i ω ′ ) • Often a mechanism for deriving group decision is present • Preference Aggregation: • What rules are appropriate to determine the joint decision given How do we combine a collection of potentially different individual choices? preference orders in order to derive a group decision? • Social Choice Theory is concerned with group decision making • Voting Procedures: (basically analysis of mechanisms for voting) - Social Welfare Function : f : Π(Ω) × . . . × Π(Ω) → Π(Ω) • Basic setting: - Social Choice Function : f : Π(Ω) × . . . × Π(Ω) → Ω • Agents have preferences over outcomes • Task is either to derive a globally acceptable preference ordering, • Agents vote to bring about their most preferred outcome or determine a winner 3 / 19 4 / 19
Agent-Based Systems Agent-Based Systems Plurality UK Politics Example • Voters submit preference orders • Outcomes: Ω = { ω L , ω D , ω C } , where ω L represents the Labour • The outcome that appears first in most preference orders wins Party, ω D the Liberal Democrats and ω C the Conservative Party • Only submission of the highest-ranked candidate is required • Voters: • Simple majority voting when | Ω | = 2 - 43 % of | Ag | are left-wing voters: ω L ≻ ω D ≻ ω C • Advantages: simple to implement and easy to understand - 12 % of | Ag | are centre-left voters: ω D ≻ ω L ≻ ω C • Problems: - 45 % of | Ag | are right-wing voters: ω C ≻ ω D ≻ ω L • Tactical voting • ω C wins with 45 % • Strategic manipulation • Condorcet’s paradox 5 / 19 6 / 19 Agent-Based Systems Agent-Based Systems Anomalies with Plurality Condorcet’s Paradox • Outcomes: Ω = { ω 1 , ω 2 , ω 3 } • Despite not securing majority, ω C wins with 45 % • Voters: Ag = 1 , 2 , 3 with preference orders • Even worse: ω C is the least preferred option for 55 % of voters - ω 1 ≻ 1 ω 2 ≻ 1 ω 3 • Tactical Voting : - ω 3 ≻ 2 ω 1 ≻ 2 ω 2 - ω 2 ≻ 3 ω 3 ≻ 3 ω 1 Centre-left candidates may do better by voting for ω L instead of their actual preference • With plurality voting, we obtain a tie • For every candidate, 2 3 of the voters prefers another outcome • Strategic manipulation : misrepresenting your preferences to • Condorcet’s Paradox : bring about a more preferred outcome There are scenarios in which no matter which outcome • But is lying bad? Not in principle, but it favours computationally we choose the majority of voters will be unhappy with the stronger voters, and wastes computational resources outcome chosen 7 / 19 8 / 19
Agent-Based Systems Agent-Based Systems Sequential Majority Elections Majority Graphs (I) • Instead of one-step protocol, voting can be done in several steps • Candidates face each other in pairwise elections , the winner • Need to introduce better tools for discussing sequential voting progresses to the next election • Election agenda is the ordering of these elections (e.g. • A majority graph is a succinct representation of voter preferences ω 2 , ω 3 , ω 4 , ω 1 ) • Nodes correspond to outcomes, e.g. ω 1 , ω 2 , . . . • Can be organised as a binary voting tree • There is an edge from ω to ω ′ if a majority of voters rank ω above ω ′ ? ? ω 1 ω 2 ω 1 ω 2 ω 1 ω 2 ? ω 1 ? ? ω 3 ω 3 ω 4 ω 3 ω 4 ? ω 4 ω 3 ω 1 ω 2 ω 4 a b c ω 2 ω 3 • Key Problem: The final outcome depends on the election agenda 9 / 19 10 / 19 Agent-Based Systems Agent-Based Systems Majority Graphs (II) The Borda Count • Tournament: complete, assymetric and irreflexible majority graph • In simple mechanisms above, only top-ranked candidate taken into (produced with odd number of voters) account, rest of orderings disregarded • Possible winner: There is an agenda that leads the outcome to • Borda count looks at entire preference ordering, counts the win strength of opinion in favour of a candidate - Every outcome in graphs a and b • For all preference orders and outcomes ( | Ω = k | ) • Condorcet winner: overall winner for every possible agenda if ω i is l th in a preference ordering, increment its strength by k − l - Outcome ω 1 in graph c • Politics example: • Strategic manipulation: fixing the election agenda - 43 of | Ag | are left-wing voters: ω L ≻ ω D ≻ ω C - 12 of | Ag | are centre-left voters: ω D ≻ ω L ≻ ω C ω 1 ω 2 ω 1 ω 2 ω 1 ω 2 - 45 of | Ag | are right-wing voters: ω C ≻ ω D ≻ ω L ω L : 43 ∗ ( 3 − 1 ) + 12 ∗ ( 3 − 2 ) + 45 ∗ ( 3 − 3 ) = 86 + 12 = 98 ω 3 ω 3 ω 4 ω 3 ω 4 ω D : 43 ∗ ( 3 − 2 ) + 12 ∗ ( 3 − 1 ) + 45 ∗ ( 3 − 2 ) = 43 + 24 + 45 = 112 ω C : 43 ∗ ( 3 − 3 ) + 12 ∗ ( 3 − 3 ) + 45 ∗ ( 3 − 1 ) = 90 a b c 11 / 19 12 / 19
Agent-Based Systems Agent-Based Systems The Slater Ranking Desirable Properties (I) • Idea: how can we minimise disagreement between the majority graph and the social choice? • For each possible ordering measure the degree of disagreement • Pareto Condition with the majority graph - If every voter ranks ω i above ω j then ω i ≻ ∗ ω j • Degree of disagreement = edges that need to be flipped (NP-hard - Satisfied by plurality and Borda, but not by sequential majority to compute) • Condorcet winner condition • Example: - The outcome would beat every other outcome in a pairwise election - Satisfied only by sequential majority elections Consider ω 1 ≻ ∗ ω 2 ≻ ∗ ω 4 ≻ ∗ ω 3 ω 1 ω 2 cost is 2, we have to flip the edges ( ω 3 , ω 4 ) and ( ω 4 , ω 1 ) Consider ω 1 ≻ ∗ ω 2 ≻ ∗ ω 3 ≻ ∗ ω 4 cost is 1, we have to flip the edge ( ω 4 , ω 1 ) ω 3 ω 4 this is the ordering with the lowest disagreement 13 / 19 14 / 19 Agent-Based Systems Agent-Based Systems Desirable Properties (II) Arrow’s Theorem • Overall vision in social choice theory: identify “good” social choice • Independence of irrelevant alternatives (IIA) procedures - The social ranking of two outcomes ω i and ω j should exclusively • Unfortunately, a fundamental theoretical result gets in the way depend on their relevant ordering in the preference orders • Arrow’s Theorem: - Plurality, Borda and sequential majority elections do not satisfy IIA For elections with more than two outcomes, the only • Non-Dictatorship voting procedures that satisfy the Pareto condition and - A social welfare function f is a dictatorship if for some voter i IIA are dictatorships f ( ̟ 1 , . . . , ̟ n ) = ̟ i - Dictatorships satisfy Pareto condition and IIA • Disappointing, basically means we can never achieve combination of good properties without dictatorship 15 / 19 16 / 19
Agent-Based Systems Agent-Based Systems Strategic Manipulation Complexity of Manipulation • As stated above, while lying could be allowed as part of rational • So we have another negative result: strategic manipulation is behaviour, it is unfair and wasteful possible in principle in all desirable mechanisms • Can we engineer voting procedures immune to manipulation? • But how easy is it to manipulate effectively? • A social choice function f is manipulable if, for a collection of • Distinction between being easy to compute and easy to preference profiles there exists ̟ ′ i such that manipulate f ( ̟ 1 , . . . , ̟ ′ i , ̟ n ) ≻ i f ( ̟ 1 , . . . , ̟ i , ̟ n ) • Mechanisms can be designed for which manipulation is very computationally complex (but often only in the worst case) • Gibbard-Satterthwaite Theorem: For elections with more than two outcomes, the • Are there non-dictactorial voting procedures that are easy to only non-manipulable voting method satisfying compute but not easy to manipulate? the Pareto property is a dictatorship • Yes, for example second-order Copeland 17 / 19 18 / 19 Agent-Based Systems Summary • Discussed procedures for making group decisions • Plurality, Sequential Majority Elections, Borda Count, Slater Ranking • Desirable properties • Dictatorships • Strategic manipulation and computational complexity • Next time: Coalition Formation 19 / 19
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