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Social Choice Theory and Voting - Coggle Diagram
Social Choice Theory and Voting
Social Choice Theory (SCT)
: the formal study of aggregation methods.
A society (N, A) is a (finite) set of individuals N and (finite) set of alternatives A.
A profile for a society (N, A) specifies, for each individual i in N, a strict ordering over the alternatives in A.
Social Welfare Function(SWF)
: maps each profile for(N,A) to a weak ordering of A.
Social Choice Function (SCF)
: maps each profile for (N,A) to a non-empty subset of A, producing a winner.
Ordinal aggregation problem
: how to aggregate the performance of alternatives on multiple criteria as specified by ordinal information in order to obtain an overall winner / ranking of the alternatives
Plurality rule
: the winner(s) in a profile is (are) the alternative(s) with the most first places.
Majority winner
: an alternative with a majority of first places
Condorcet winner
: an alternative that beats any other alternative in a pairwise comparison.
Condorcet loser
: an alternative that is beaten by every other alternative in a pairwise comparison.
a Condorcet winner need not exist
The majoritarian preferences are cyclic (and hence intransitive). Such majoritarian preference cycles are also called
Condorcet cycles
.
Some philosopher think (rational) preferences should be transitive. Hence, the existence of Condorcet cycles is also referred to as the
Condorcet paradox
.
The Copeland rule
: Look at the outcome of each pairwise comparison. The winner of the one-on-one comparison receives 1 point. The loser receives 0 points. In the case of a tie, they each receive 1⁄2 of a point. The candidate(s) with the greatest number of points is (are) decreed to be the winner(s).
if a Condorcet winner exists then it is declared to be the (sole) winner according to the Copeland rule.
Borda rule
: When there are k alternatives, an alternative receives: k − 1 points for each time it is ranked #1, ..., 0 points for each time it is ranked number last, The alternative(s) that receives the highest Borda score (total number of points) is (are) declared as winner(s)
Violates IIA
Instant Run-off Voting (IRV) rule
: if an alternative has a majority of first places, declare that alternative the winner. If no alternative has a majority of first places, then remove the alternative(s) with the fewest number of first places. Repeat steps (1) and (2) in turn, until: A unique winner results from an application of step (1), or All remaining alternatives are removed via an application of step (2), in which case those remaining alternatives are the winners
Violates weak monoticity
Axioms
Weak Monotonicity
: If a winning alternative a gets more support from the voters, and each other alternative gets the same support, a should still be a winning alternative. If alternative a is a winning alternative in P, and if R is obtained from a profile P via an
order preserving change
favoring alternative a, then alternative a should be a winning alternative in R also
Arrow’s theorem
: any reasonable method of aggregation should satisfy the three Arrovian axioms but, provably, no method can satisfy all three Arrovian axioms
The Three Arrovian Axioms
(1)
No-dictatorship axiom
: SWF F is not a dictatorship. There is no individual i in society (``the dictator’’) such that for any profile P, the societal ranking F(P) is the same as the dictator’s individual ranking
(2)
Pareto axiom
: if every individual prefers x to y in profile P, then x is also preferred to y according to the societal ranking F(P).
(3)
Independence of Irrelevant Alternatives axiom (IIA)
: If every individual has the same x-y preferences in P as in R, then society has the same x-y preferences in F(P) as its does in F(R).
Manipulation
Agenda Control: committees often decide by taking subsequent pairwise majority votes, the outcome depends on who controls the agenda
Introducing new alternatives: to create Condorcet cycles and manipulate the outcome
Strategic Manipulation and the Borda Rule: by misrepresenting his preferences, one can manipulate the outcome
Gibbard-Satterthwaite theorem
: “(virtually) no SCF is strategy-proof”. A resolute SCF F for a society with 3 or more alternative is unanimous and strategy-proof if and only if F is a dictatorship.
Voting (Riker)
Liberal: voting means that officials will constantly fear being voted out of office. And this fear means that officials will tend not to use the force of government to interfere in people’s lives, negative liberty.
Populist: the “will of the people” is precious: people have liberty if and only if they are governed by laws that embody the “will of the people”, positive liberty.