social choice

Social Choice Theory

Enrico Franchi [email protected]

Group Decisions }  Agents are required to choose among a set of outcomes

}  Agents can choose one outcome in Ω }  Agents can express a preference of outcomes }  Let the set of preference orderings of outcomes }  We also write to express that agent i prefers ω1

to ω2

{ }1 2, ,ω ωΩ = …

( )Π Ω

ω1 i ω 2

2 Enrico Franchi ([email protected])

Social Welfare }  A social welfare function takes the voter preferences and

produces a social preference order: or in the slightly simplified form:

}  We write to express that the first outcome ranked above the second in the social outcome

( ) ( ): Nf Π Ω →Π Ω

( ): Nf Π Ω →Ω

ω1 * ω 2


Plurality }  Simplest voting procedure: used to select a single

outcome (candidate) }  Everyone submits his preference order, we count how

many times each candidate was ranked first }  Easy to implement and to understand }  If the outcomes are just 2, it is called simple majority voting }  If they are more than two, problems arise


Voting in the UK }  Three main parties:

}  Labour Party (left-wing) }  Liberal Democrats (center-

left) }  Conservative Party (right-

wing)

}  Left-wing voter: }  Center voter: }  Right-wing voter: }  Tactical Voting }  Strategic Manipulation

Labour Party 44%

Liberal Democ

rats 12%

Conservative Party 44%

Voters

ω L ω D ωC

ω D ω L ωC

ωC ω D ω L


Condorcet’s Paradox }  Consider this election:

}  No matters the outcome we choose: two thirds of the electors will be unhappy

Ω = ω1,ω 2 ,ω3{ } Ag = 1,2,3{ }

ω1 1 ω 2 1 ω3

ω3 2 ω1 2 ω 2

ω 2 3 ω3 3 ω1


Sequential Majority }  Series of pair-wise elections, the winner will go on to the

next election }  An agenda is the strategy we choose to order the

elections (linear, binary tree) }  An outcome is a possible winner if there is some agenda

which would make that outcome the overall winner }  An outcome is a Condorcet winner if it is the overall

winner for every possible agenda }  Can we choose the agenda to choose a winner?


Borda Count and Slater Ranking }  Borda Count

}  We have K outcomes }  Each time an outcome is in the j-th position for some agent, we

increment its counter by K-j }  We order the outcomes according to their counter }  Good for single candidates

}  Slater ranking }  Tries to be as close to the majority graph as possible }  Unfortunately, is NP-hard


Properties }  Pareto condition: if every agent ranks ωi above ωj,

then }  Plurality, Borda

}  Condorcet winner: if an outcome is a Condorcet winner, then it should be ranked first }  Sequential majority elections

}  Independence of Irrelevant Alternatives (IIA): social ranking of two outcomes should only be affected by the way that they are ranked in their preference orders }  Almost no protocol satisfies IIA

ω i * ω j


Properties }  Dictatorship: a social welfare function f is a dictatorship

if for some voter j we have that:

}  Unrestricted Domain: for any set of individual voter preferences, the social welfare function should yield a unique and complete ranking of societal choices. }  E.g., not random, always answers, does not “loop”

( )1, , N jf ω ω ω… =


Arrow’s Theorem

}  There is no voting procedure for elections with more than two outcomes that satisfies }  Non-dictatorship }  Unrestricted Domain }  Pareto }  Independence of Irrelevant Alternatives


Gibbard-Satterthwaite’s Theorem }  Sometimes voters “lie” in order to obtain a better

outcome }  Is it possible to devise a voting procedure that is not

subject to such manipulation?

}  Manipulation (i prefers ωi):

}  The only procedure that cannot be manipulated and satisfies the Pareto condition is dictatorship

f ω1,…,ω i '…,ω n( ) i f ω1,…,ω i…,ω n( )


Complexity and Manipulation

Enrico Franchi ([email protected]) 13

}  Even if all procedures can be manipulated, can we devise procedures which are hard to manipulate?

}  Hard means “difficult to compute” in an algorithmic sense, e.g., NP-Hard procedures

}  These procedures are easy (polynomial) to compute? }  Second-order Copeland may be “difficult” to manipulate

}  In theory it is NP-Hard }  However, it is only a worst case complexity

References

Enrico Franchi ([email protected]) 14

1.  Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations; Yoav Shoham and Kevin Leyton-Brown; Cambridge Press

2.  Game Theory: Analysis of Conflict; Roger B. Myerson; Harvard Press

3.  An Introduction to Multi-Agent Systems; Michael Wooldridge; Wiley Press

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