VOTING

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VOTING

• Lecturer Prof.Dr. Fuad Aleskerov

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Plutarch Collective decision making on the
basis of aggregation procedures (in particular,
on basis of voting procedures) dates from
antiquity. Plutarch tells in his ' Lives' about
two rules used at the times of the legendary
Spartan king and legislator Lycurgus (IX --
VIII cc. B.C.). One rule was used, in modern
terms, for electing to a club 'And they say that
a candidate for membership in one of these messes
underwent the following ordeal. Each of the
mess-mates took in his hand a bit of soft
bread, and when a servant came along with a bowl
upon his head, then they cast it into this
without a word, like a ballot, leaving it just as
it was if he approved of the candidate, but if
he disapproved, squeezing it tight in his hand
first. For the flattened piece of bread had the
force of a perforated, or negative, ballot. And
if one such is found in the bowl, the candidate
is not admitted to the mess, because they wish
all its members to be congenial. Only the
members of a club participated in the election.
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Plutarch
Another procedure was used for electing the
'Council of the Elders,' where all Spartan
population took part 'The election was made in
the following manner. An assembly of the people
having been convened, chosen men were shut up in
a room near by so that they could neither see nor
be seen, but only hear the shouts of the
assembly. For as in other matters, so here, the
cries of the assembly decided between
competitors. These did not appear in the body,
but each one was introduced separately, as the
lot fell, and passed silently through the
assembly. Then the secluded judges, who had
writing-tablets with them, recorded in each case
the loudness of the shouting, not knowing for
whom it was given, but only that he was
introduced first, second, or third, and so on.
Whoever was greeted with the most and loudest
shouting, him they declared elected.

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How the voters opinion is described?
1. Pointing out chosen alternative(s)
2. Constructing a ranking
3. Pointing out intensity of preference
Voters opinion
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How the voters opinion is described?
4. Voter can be asked to approve or reject some
project or decision. Then the question can be
stated as
Do you approve the project of the
Constitution? Yes No
Point out the
desirable answer.
In this ballot two alternatives are considered
Accept the project of the Constitution, Reject
it, and voters opinion is described as a choice
(say, I choose a)).
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Simple majority rule
Each voter points out her most preferable
candidate, and that alternative wins which
obtains more than 50 of votes. If the number of
alternatives is greater than 2, such winner may
not exist. Consider an example with 21 voters
(simple majority is 11 votes)
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Condorcet Paradox (Paradox of Voting)
?
?
?
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Arrows Impossibility Theorem
Kenneth Arrow did not analyze different
procedure, but he tried to formulate different
conditions to which any reasonable aggregation
rule should satisfy to.
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Condition 1. The number of voters is not less
than 2, and the number of alternatives is not
less than 3.
Condition 2. Voting rule process voters
rankings, and any ranking is admissible as
individual opinion. Each voter chooses her own
ranking independently of other voters opinions.
Social decision is a ranking as well.
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Condition 3 (IIA). This condition Arrow called
Independence of Irrelevant Alternatives
Aggregation Rule in the construction of the
decision about which alternative is better, x or
y, takes into account voters opinions only about
those alternatives and does not take into account
the opinions about other alternatives, say, x and
z or w and z.
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Condition 4 (Unanimity). If all voters
unanimously consider that x is better than y,
social decision must be the same.
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Let us introduce one more rule. Assume that in
the set of voters one voter is designated, say,
2, and the social decision is always coincide
with the opinion of that voter independently of
others opinions. This rule can naturally be
called the dictatorial rule.
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Arrows Theorem. The only rule satisfying the
above four conditions is the dictatorial one. In
other words, under those conditions, a social
decision coincides with the opinion of one of the
voters.
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A special case. In Codition1 it was assumed that
the number of alternatives is greater than 2.
Let there are only two alternatives. These
alternatives might be a) elect the candidate, b)
not elect him or a) approve the project of the
constitution, b) reject it. Assume also that the
number of voters is odd. Arrow showed that the
only rule satisfying Conditions 1-4 under these
additional assumptions is the simple majority
rule.
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It looks like that the simple majority rule is
the best procedure. However, let us note that
the number of candidates can be more than one.
And in the case of one candidate, if we do not
elect him, the office cannot be vacant, so we
should consider another candidate, the third one,
etc. Who can guarantee that the candidate
rejected at one of the stages would not be better
comparing with the candidates on the next stages?
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Arrows Theorem deals with such basic notions as
public interests, the best for the society,
maximum of social welfare. Under the
conditions stated above these notions can be
defined exactly if the definition is given by a
dictator. This situation can not be accepted by
many people since a dictator does not take into
account the preferences of other members of the
society.