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F.P. TarassenkoTomsk State University

Seven Paradoxes of Voting Systems:

Pitfalls in Collective Decision Making

Seven Paradoxes of Voting Systems: Pitfalls in Collective Decision Making

RESULTS OF ELECTIONS

DEPEND ON:

1) Pre-election campaign running

2) Approved election procedure fulfillment

3) Honest counting votes

We will consider peculiarities of collective decision making beyond of all that:

only those that are hidden in the procedure of voting itself.

Seven Paradoxes of Voting Systems: Pitfalls in Collective Decision Making

X = {x1, x2, … ,xk} – a ballot-list.

Ci (X) – a choice of i-th voter on X in a form requested by election rules.

C1(X), C2(X), … ,Ci(X), … ,CN(X) – all ballot-lists filled in.

C0(X) = f {C1(X), C2(X), … ,Ci(X), … ,CN(X)} – a

collective choice.

f(.) – an election procedure, or vote rule.

Seven Paradoxes of Voting Systems: Pitfalls in Collective Decision Making

Paradox 1: COLLECTIVE DECISION IS NOT

OBLIGATORY CORRECT

Voting does NOT guarantee the rightness of the decision made.

Examples:

Galileo, Jordano Bruno, Kopernikus.

Seven Paradoxes of Voting Systems: Pitfalls in Collective Decision Making

Paradox 1: COLLECTIVE DECISION IS NOT (cont.) OBLIGATORY CORRECT

CAUSE: Ability of any human entity (including the electorate) to estimate anything, is accompanied by a

liability to mistakes and errors in its estimation.VIRTUES OF VOTING:

1) Collective decision is STATISTICALLY more reliable than an individual one

2) Its ULTIMATE goal is: to make all, INCLUDING THOSE DISAGREED, to act according to the decision.

Seven Paradoxes of Voting Systems: Pitfalls in Collective Decision Making

Paradox 2: ANY ELECTION PROCEDURE CAN HOVER

Although the voting is intended to arrive to a decision, it may result in NOT making it

CAUSE: A procedure is defined by certain constraints. If any of them is NOT met, - the procedure is

inapplicable: decision is not made.

Examples:Simple majority (50%+1), qualified majority (2/3), etc.

Seven Paradoxes of Voting Systems: Pitfalls in Collective Decision Making

Paradox 3: THE CONDORSET’S PARADOX

CAUSE: When preferences make up a circle

B’sP’s

B1 B2 B3

P1 1 2 3

P2 3 1 2

P3 2 3 1

C1(B)

C2(B)

C3(B)

1

3 2

Graph of Preferences

Seven Paradoxes of Voting Systems: Pitfalls in Collective Decision Making

Paradox 3: THE CONDORSET’S (cont.)PARADOX

Examples: Treatments of the paradox

1

3 2

3

5 4

1

2a) negligible,insignificant

b) Induce some votersto change their

preferences between lower alternatives

(“black” or “white” PR)

Seven Paradoxes of Voting Systems: Pitfalls in Collective Decision Making

Paradox 4: MINORITY CAN WIN IN ANY MAJORITARIAN VOTING SYSTEM

- and there are several possibilities for that

Examples:

1.

2.

voters nonvoters evading the polls, turnout

obligation to vote

30% 30% 40% 100% of attendance!

splitting votes

<50% >50%

60% 40%

Seven Paradoxes of Voting Systems: Pitfalls in Collective Decision Making

Paradox 4: MINORITY CAN WIN IN ANY

(cont.1) MAJORITARIAN VOTING SYSTEM

- and there are several possibilities for that

3. multistage elections

Necessary conditions:

1) Multistage rule for elections

2) Critical proportion of minority (2 - 44%, 3 – 33,7%, …)

3) Discipline of minority voters

Examples: 4 Presidents of USA; soviet leaders, etc.

Seven Paradoxes of Voting Systems: Pitfalls in Collective Decision Making

Paradox 4: MINORITY CAN WIN IN ANY

(cont.2) MAJORITARIAN VOTING SYSTEM

- and there are several possibilities for that

4. 100% attendance, no splitting, one-stage election!

Australian Constitution: Australian voter fills in his/her ballot-paper only once. But he/she must rank the candidates in order of

preference.

Then the procedure uses all abovementioned conditions

Seven Paradoxes of Voting Systems: Pitfalls in Collective Decision Making

Paradox 5: EVEN the OVERWHELMING MAJORITY RULE IS NON DEMOCRATIC

CAUSE: the concept of “US and THEM”, “own and alien” has nothing to do with the concept of democracy.

Example: The voting procedure with two only,

looking very “democratic”, rules:

1) (N-1) – “pro”, 1 – “contra” decision. N is ANY: if 2 or more voters are “against” alternative is rejected.

2) Voter is voting “pro” if the alternative is beneficial or not harmful to him/her.

This is the end of democracy (a case example follows).

VOTING BY MAJORITY AND DEMOCRACY

ARE NOT THE SAME THINGS

Seven Paradoxes of Voting Systems: Pitfalls in Collective Decision Making

Paradox 6: THE ONLY DEMOCRATIC VOTING IS UNANIMOUS ONE. BUT IT IS SUBJECT TO PARADOX TWO,TOO

CAUSE: Defending interest of everyone, this procedure can not avoid possibility of hovering.

Example: some parliament decisions; Security Council of UNO; election of the next Pope; verdict

of jurors; joint-stock companies with unlimited responsibility; idealized design.

Seven Paradoxes of Voting Systems: Pitfalls in Collective Decision Making

Paradox 6: THE ONLY DEMOCRATIC VOTING

(cont.) IS UNANIMOUS ONE. BUT IT IS SUBJECT TO PARADOX TWO,TOO

Paradoxical Situations about Unanimity

1) Substitution of unanimity principle (all are “for”) by principle of consensus (nobody is “against”).

2) Failure to collect 100% of votes.

Ways out:

a) looking for compromise;

b) a “ladder” principle.

Seven Paradoxes of Voting Systems: Pitfalls in Collective Decision Making

Paradox 7: THE ARROW’S THEOREM OF IMPOSSIBILITY

C0(X) = f {C1(X), C2(X), … ,Ci(X), … ,CN(X)}

Conditions of “right”, “just”, “reasonable” choice:1) Condition of general right to vote and undesirability of “the

dictator’s choice” - C0(X)C1(X).

2) Condition of monotony of f(.).

3) Condition of independency of alternatives.

4) Condition of sovereignty.

THEN: The given requirements are incompatible. For any election procedure there exists such a set of alternatives, on which the choice can not be made, but the “dictator’s choice” is an exclusion.

Seven Paradoxes of Voting Systems: Pitfalls in Collective Decision Making

ALL AFORESAID HAS NOTING

TO DO WITH POLITICS:

it is purely logical study of the formula

C0(X) = f {C1(X), C2(X), … ,Ci(X), … ,CN(X)}