The Voting Problem:
A Lesson in Multiagent System
Based on Jose Vidal’s book
Fundamentals of Multiagent Systems
Henry Hexmoor
SIUC
Voting Problem
• Plurality Vote:
e.g. General election
– Vote for favorite option
– Tally votes for most popular
• Runoff election:
e.g. The primaries
– Vote for favorite set of options
– Vote within dominant sets until a winner is determined
• Pair wise election:
– Vote for favorite set of two
– Vote within winning sets
Symmetry
• Refectional Symmetry:
– If one agent prefers A over B and another prefers B
over A, their votes should cancel each other out
• Rotational Symmetry:
– If one agent prefers A,B,C
– Another prefers B,C,A
– Third prefers C,A,B
Then Cancel votes.
All previous votes violate symmetry
The Borda Count
• With x choices, each agent awards x points to
her first choice, x-1 points to her second
choice, and so on.
• The candidate with the most points wins.
• The Borda Count satisfies both symmetry
properties.
Borda Count
• The Borda count is a single-winner election method in
which voters rank candidates in order of preference.
The Borda count determines the winner of an election
by giving each candidate a certain number of points
corresponding to the position in which he or she is
ranked by each voter.
• Once all voter have been counted the candidate with
most points is the winner. Because it sometimes elect
broadly acceptable candidates, rather than those
preferred by the majority, the Borda count is often
described as a consensus-based electoral system, rather
than a majoritarian one.
The Voting problem
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A -Set of agents
O -Set of outcomes
i -Preference function of agent i over outcome
* -Global set of social preferences (i.e., social
contract)
The problem:
What should the final vote reflect and how can
an election system preserve social preferences?
Desirable Voting conditions
1.
2.
3.
* exists for all inputs i
* exist for every pair of outcomes
* is symmetric and transitive over the set of
outcomes
4. * should be pareto efficient . If all agents have
desired outcomes, * should coincide with that.
5. * should be independent of irrelevant alternatives
6.
No dictators
* Kenneth Joeseph Arrow (Noble Prize in
Economics)
Arrow’s Impossibility Theorem
(1951)
There is no social choice rule that satisfies all six
conditions.
Condorcet method
• A Condorcet method is any single-winner election method
that meets the Condorcet criterion.
• The criterion always selects the candidate who would beat
each of the other candidates in a run-off election
– Rank the candidates in order of preference. Tie rankings are
allowed, which express no preference between the tied
candidates.
– Comparing each candidate on the ballot to every other, one at a
time (pair wise), tally a “win” for the victor in each match.
– Sum these wins for all ballots cast. The candidate who has won
every one of their pair wise contests is the most preferred, and
hence the winner of the election.
– In the event of a tie, use a resolution method.
Mechanism design
• Designing rules of a game or system to achieve
a specific outcome, even though each agent
may be self-interested.
• It tries to achieve four outcomes:
1.
2.
3.
4.
truthfulness,
individual rationality,
budget balance, and
social welfare.
Mechanism design
• Let N be the number of players/participants.
• Each player i can have a type/signal/valuation ti  Ti
e.g. in an auction the type of player would be his
valuation/reservation price for the good(s) offered.
• Depending on her type, the player will pick an action
si (ti )  Ai , where Ai is the set of possible actions for
player i offered by the mechanism, e.g. an auction
would be a bid of a certain amount.
• Each player has utility ui : Ti  O  R ,where O is
the outcome generated by the mechanism. In auction,
the outcome would be the final allocation of goods and
the payments each player has to make.
Mechanism design
• A mechanism M is defined to be a pair (A,g), where A  A1  ... An
is the set of action offered to the players/participants and g : A  O
is the function that maps the player’s actions to an outcome o.
• A mechanism is direct, if the set of actions equals the set of
types for each player, i.e.
i  N : Ai  Ti
This is true for auctions, where each player’s action is to
announce their valuation of the product. However, there is no
need to announce the true valuation if a different strategy
yields better utility.
Mechanism design
• A mechanism is direct truthful, incentive
compatible, if it is the dominant strategy to
take. All of the participants fare best when they
truthfully reveal any private information asked
for by the mechanism.
• A function f : T1 ... TN  O a social choice
function f in dominant strategies, if the set of
strategies that lets M generate the same output
as f is a dominant Nash Equilibrium
The House Painting example
S
V(S)
Alice
Yes
Bob
No
Caroline
Yes
Donald
Yes
Emily
Yes
• Alice lives in a house with four other housemates.
• The set of people who vote for painting will share
equally in the cost of the painters, as long as two or
more people vote Yes.
• The people who voted against painting will pay
nothing.
The House Painting example
q = {WantPaint, DontNeedPaint}
O = {Paint,NoPaint}
The cost of painting the house is 20
The agents that want the house painted would get a value of 10
from seeing it painted and 0 if it does not get
Painted.
• Those who think the house is fine as it is get a value of 0 either
way.
• Assume that we want to maximize social welfare.
• Vickrey solution solves this problem
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