Transcript Document

Social Choice Theory: Responses to Impossibility
Scott Blaha
Mathematics & Statistics Department, Swarthmore College
Introduction
• Social choice theory is the mathematical study
of voting systems.
• A voting system consists of a set N = {1, …, n}
of voters, a set X = {x1, …, xk} of three or more
alternatives, and a social choice rule.
• An individual’s preference relation p is a
transitive ordering of the alternatives, with ties
possible. A profile pn is an assignment of a
preference relation onto each voter.
– Transitivity means that there are no “loops” in a
person’s preferences: if a voter prefers A to B and B
to C, then they must prefer A to C, not C to A.
• Our basic object of study is a social choice
rule, C: pn  p. Such a rule maps voters’
preferences onto a social preference relation.
Individuals
Social
Choice
Rule
Social Preference
• Some simple examples of social choice rules:
– Pick one individual to be the “dictator”. Their
preference relation is the social choice.
– Total up the first-choice votes for each alternative
and order them from highest to lowest.
Arrow’s Theorem
• Some preliminary voting system definitions:
– Universality: Voters may rank alternatives in any
order they choose, as long as the order is transitive.
– Unanimity (Pareto condition): If every voter prefers
A to B, then society prefers A to B.
– Independence of Irrelevant Alternatives (IIA): A
society’s preference for A or B depends only on the
voters’ relative preferences for A and B.
– Dictatorship: There is a voter V such that if V
prefers A to B, then society always prefers A to B.
• Theorem (Arrow; 1951). The only voting
systems satisfying universality, unanimity, and
IIA are dictatorships.
• The three conditions of the theorem all sound
very fair – this surprising theorem says we can’t
have them all in a democratic system!
Responses to Arrow
• The only way to avoid Arrow’s Impossibility
Theorem is to weaken one of our criteria:
– Non-dictatorship: Let’s leave this alone.
– Unanimity: A system where a unanimous decision
is not followed is extremely unfair; we will keep this.
– IIA: This criterion is the most unrealistic. In practice,
most real-life voting systems choose to violate this
condition. See Borda Count below for a weaker but
consistent form of IIA.
– Universality: We can get some good voting
systems by limiting the input to our social choice
rule. See Approval Voting below.
Borda Count
• The Borda count is the sum, over all voters, of
points for a alternative, with k points given for a
first place, down to 1 point for a last place. The
alternatives are then ranked by their counts.
• Let’s define a reasonable IIA-like condition:
– The intensity of the preference between A and B is
the number of alternatives between A and B in a
complete transitive preference relation.
– Intensity of Binary Independence (IBI): A society’s
preference for A or B depends only on the voters’
intensities of preference between A and B.
• Theorem. The Borda count satisfies nondictatorship, universality, unanimity, and IBI.
Approval Voting
• Approval voting lets voters approve of as many
alternatives as they choose. The alternatives
are then ranked by number of approvals.
• This voting system avoids Arrow’s Theorem by
violating universality: voters may only have up to
two “places” in their preference rankings.
• Theorem. Approval voting satisfies nondictatorship, unanimity and IIA.
• However, approval voting can fail the majority
criterion: if there is a majority first choice,
approval voting cannot guarantee that it wins.
Voting Strategy
• Consider this breakdown of voters:
A > B > C (40%) B > C > A (35%) C > A > B (25%)
– Note that there is no clear winner in this situation.
• Suppose that all our voters know each others’
preferences ahead of the election:
– Since A will win if everyone votes honestly, rational
B supporters will dishonestly swap B and C in their
preference lists to ensure that C wins.
– In response, rational A supporters will swap B and
A to entice B supporters to honestly vote B over C.
– In response, rational C supporters will…
• As we can see, this voting system is extremely
manipulable. That is, some voters achieve a
better outcome by being dishonest.
• Can voting systems be made strategy-proof?
Gibbard-Satterthwaite Theorem
(Note: For the purposes of this section, the output of a social choice rule
is a single winner, rather than a complete preference list.)
• Some more voting system properties:
– Onto: Any alternative can be the winner given a
certain arrangement of voter preferences.
– Strategy-proof: No voter may cause a more favored
candidate to win by submitting false preferences.
• Theorem (Gibbard and Satterthwaite; 1973):
The only voting systems satisfying onto and
strategy-proof are dictatorships.
• Once again, we have a surprising impossibility
result: no voting system can guarantee to elicit
honest preferences from all voters all the time!
• Unfortunately, unlike with Arrow’s Theorem,
we cannot weaken these criteria as easily.
• Most responses to this focus on minimizing
rather than eliminating strategic voting:
– Limit the spread of information between voters.
– Introduce some randomization to the social choice
rule (it is no longer technically a social choice rule).
– Make your social choice rule so complex that
voters cannot figure out how to manipulate it.
– Perform computer simulations to calculate
“manipulability scores” for different voting systems.
Real World Examples
• Strategy in US Congressional elections:
– Congressional elections in the US are plurality
voting: the candidate with the most votes wins.
– Due to the strong strategic voting present in
plurality systems, the US has a two-party system.
• Unfairness in French Presidential elections:
– In France, if no candidate wins a majority in the first
round of voting, a second round is held between the
top two candidates of the first round.
– Due to a splintering of the liberal vote in 2002, the
second round was conservative Jacques Chirac
against ultra-conservative Jean-Marie Le Pen.
– The majority of those polled liked neither candidate!
Summary
• Think of social choice rules as compression
algorithms: they seek to take everyone’s
opinions and create a logical compromise.
• The two main impossibility theorems of social
choice theory state that:
– No social choice rule is completely fair.
– All social choice rules are open to strategic voting.
• Our real world experiences agree with the
math: elections are a messy business!
• There are better (and more complicated!)
voting systems than what is currently used.
References
• Arrow, Kenneth J., Amartya K. Sen, and Kotaro
Suzumura. Handbook of Social Choice and Welfare.
Vol. 1. 2002.
• Gartner, Wulf. Domain Conditions in Social Choice
Theory. 2001.
• Hodge, Jonathan K., and Richard E. Klima. The
Mathematics of Voting and Elections: a Hands-on
Approach. 2005.
• Kelly, Jerry S. Arrow Impossibility Theorems. 1978.
• Kelly, Jerry S. Social Choice Theory: an
Introduction. 1988.
• Taylor, Alan D. Mathematics and Politics. 1995.