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Voting and Democracy


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Title: Voting and Democracy

Voting and Democracy
What is Democracy?
  • Democracy is the worst form of government except
    for all those others that have been tried
    Winston Churchill
  • What, exactly, is so great about democracy?

Democracy and its follies
  • Not always stable
  • Relatively new form of government
  • Catch-all term, does it mean anything?
  • Reduces economic growth

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Sacred Cows
  • What is so special about democracy?
  • What is so special about majority rule (501) in

Choice of voting rule
  • How to determine the optimal majority?
  • Two costs to consider
  • Decision making costs
  • External costs (tyranny of the majority)

There are two general rules First, the more
grave and important the questions discussed, the
nearer should the opinion that is to prevail
approach unanimity. Secondly, the more the matter
in hand calls for speed, the smaller the
prescribed difference in the numbers of votes may
be allowed to become where an instant decision
has to be reached, a majority of one vote should
be enough. The first of these two rules seems
more in harmony with the laws, and the second
with practical affairs. In any case, it is the
combination of them that gives the best
proportions for determining the majority
necessary. --Jean Jacques Rousseau (The Social
Calculus of Consent
With unanimity rule, any law that passes is
necessarily Pareto optimal. As we require fewer
people to agree with a law, the expected external
cost rises
The optimal voting majority is the majority that
minimizes the sum of the expected bargaining
costs and expected external costs.
Expected Costs
Expected total costs
With rule of one, there are no costs of getting a
majority. As we require more people to agree with
a law, the expected bargaining cost rises
External Costs
Bargaining Costs
of individuals whose agreement is required for
collective action
Multiple optimal majorities
  • The optimal majority will vary depending on what
    type collective action we are talking about.
  • So what is so special about 501?
  • Bargaining and external costs are probably
    discontinuous at 50

Spatial Competition
  • Hotelling (1929)
  • Assume a town is composed of one street with 100
    evenly-spaced, identical households
  • Assume that two petrol stations are going to
    enter the city, and that residents will go to
    whichever station is closest to them

Spatial Competition
Where will the two petrol stations set up?
B, in turn, would want to move just to the left
of A
If they set up like this, they would each get
half the market, and nobody would be too far from
the station, which is good
This leapfrogging would continue until both were
in the centre of town
But this is not an equilibriumwhy not?
Which centre? Turns out the median is where they
want to be
Either station can get more of the market by
moving next to their competitor!
Spatial Competition
  • What does this model explain?
  • Why shopping centres develop
  • Why fast food restaurants are often next door to
    each other
  • Why political parties are often very similar!

Spatial Competition in Voting
  • If
  • voter ideology can be arrayed along a single
    dimension, and
  • Individuals have single-peaked preferences, then
  • Median Voter Theorem--Politicians will seek to
    put the policies desired by the median voter into

Median Voter Theorem
  • What does it mean for preferences to be single
  • For any good, Individuals must
  • Always prefer more of that good than less, or
  • Always prefer less of that good than more, or
  • Have a bliss point. A bliss point is an optimal
    quantity of a good such that as the actual
    quantity consumed deviates from that bliss point,
    the utility of the individual declines

Single Peaked Preferences-1 dimensional case
Preferences are said to single peaked if
An individual always prefers more of a good to
less of that good, or
Bliss Point
An individual always prefers less of a good to
more of that good, or
An individual has a bliss point
Median Voter Theorem
It is only efficient if the median valuation is
equal to the mean valuation. In this case, it is
not. If the median voter valued the public good
at 70, however, wed get the efficient result
The Median Voter Theorem says that 90 will be
the amount of the public good provided if the
decision is left to democracyregardless of
preference intensity, dispersion of bliss points,
or efficiency. Is this equilibrium efficient?
Consider 5 individuals with different bliss
points over the quantity of a public good Person
1-10 Person 2-20 Person 3-90 Person
4-100 Person 5-150
10 20 90 100
Median Voter Theorem
  • Choosing public goods through voting and
    democracy does not by any means ensure the
    efficient quantityneed considerable luck for the
    mean voter to be the same as the median voter.
  • Not only are you likely to get inefficient
    results, if we relax the assumptions of
    single-dimensionality or single-peakedness,
    voting becomes very unstable!

Multiple Peaked Preferences
  • Sometimes single peaked preferences make little
  • Options might be discrete
  • All or nothing response is appropriate

All or Nothing
  • Consider a country with 3 citizens and three
    options all-out war (W), limited involvement
    (L), or no war (P).
  • You might have Hawks (WgtLgtP), Doves (PgtLgtW), and
    All-or-Nothing types (WgtPgtL)
  • How do we compare these three options with voting?

Condorcet Voting
  • Evaluate all possible pairwise votes and look for
    a Condorcet Winner
  • Condorcet Winner-An option that beats all other
    options in pairwise voting
  • Condorcet Loser-An option that loses to all other
    options in pairwise voting

Condorcet Voting
  • Back to our Hawks (WgtLgtP), Doves (PgtLgtW), and
    All-or-Nothing types (WgtPgtL)
  • We have everybody vote on the 3 possible pairwise
    votes W v. L, W v. P, and P v.L
  • If we assume there are the same number of hawks
    as doves as all-or-nothing types
  • W v. L, W wins
  • W v. P, W wins
  • P v. L, L wins

Condorcet Voting
  • W v. L, W wins
  • W v. P, W wins
  • P v. L, L wins
  • W is our Condorcet winner
  • P is our Condorcet loser
  • We can construct a social preference ordering of

Condorcet Voting
  • What if our example had Doves (PgtLgtW),
    All-or-Nothing types (WgtPgtL), but instead of
    Hawks you have people who feel we need to do
    something, but would prefer being reasonable
  • W v. L, L wins
  • W v. P, W wins
  • P v. L, P wins
  • What is the social ordering? PgtLgtWgtP

  • This constellation of preferences yields what is
    called a voting cycle.
  • How likely are cycles? Very
  • More voters ? increased likelihood of cycles
  • Relax assumption of single dimensional policy
    space ? cycles are almost certain!

The inevitability of cycles in 2-dimensions
A, B, and C represent the bliss points in 2
dimensional policy space of voters A, B, and C
Lets start by choosing the median point
We can draw indifference circles around each
bliss point that pass through the median
Any point in these three areas
Would beat the median point in a pairwise vote
The inevitability of cycles in 2-dimensions
Lets choose some point within one of the winsets.
Now, compare the blue point with the orange
point. A and C prefer the orange point to the
blue point, while B prefers blue to orange. We
can make this more obvious by getting rid of all
of the extra indifference curves
In a vote between the orange and pink points, A
prefers orange and B and C prefer pink. Lets
draw new indifference curves for the pink point.
The pink point also has a winset. Lets pick
another point that would beat pink in a pairwise
We have generated a cycle in 2 a two-dimensional
policy space. It is almost impossible for a
cycle NOT to exist!
In a vote between the blue and pink points, A and
B like blue and C prefers pink. Lets draw new
indifference curves for the blue point.
The inevitability of cycles in 2-dimensions
  • The likelihood of avoiding cycles in a
    2-dimensional issue space is remote at best
  • As the number of dimensions increases, cycles
    become more likely as well
  • How likelyArrows impossibility theorem! It is
    impossible to construct a rational social ordering

Back to voting
  • As said before, there is no reason to suspect
    that voting on public goods will get you the
    efficient quantity or Pareto Optimal solutions
  • Moreover, voting will often lead to Kaldor-Hicks
    inefficient results as well!

Consider an example
Individual Cost to Taxpayer Benefit to Taxpayer Net Benefit
Taxpayer 1 100 40 -60
Taxpayer 2 100 110 10
Taxpayer 3 100 140 40
Total 300 290 -10
Example, contd
  • From a cost benefit standpoint, this project is
  • Still, however, it would garner a majority of

Another Example
Individual Cost Benefit Net Benefit
1 100 140 40
2 100 90 -10
3 100 160 60
4 100 70 -30
5 100 80 -20
Total 500 540 40
Example, contd
  • From a cost benefit standpoint, this project is
  • However, it would be voted down in a majority
  • Both of these examples illustrate the fact that
    the nature of cost benefit analysis has
    changedrather than can the winners compensate
    the losers, question is whether there are more
    winners than losers, irrespective of magnitude of
    win or loss!

Vote selling
  • All of these problems would be solved if
    individuals were able to sell their votes.
  • Ethical issues
  • Despite illegality of vote selling, vote trading
    does take place (and often!)

Coalitions and Logrolling
  • Logrollingtwo parties agree to vote for the
    preferred project of the other.
  • Does logrolling improve efficiency? We know vote
    buying would, but what about vote trading?

Project Politician1 Politician2 Politician3 Net Benefit
A 120 -30 -40 50
B -20 80 -20 40
C -50 -20 130 70
Total 50 30 70
  • Each of these policies is efficient.
  • A vote for policy A, B, or C on its own would not
  • A coalition/logroll between politicians 12, 13,
    or 23 would be feasible, however, with the
    coalition between 12 the most likely.
  • Why is a coalition between 123 infeasible?

Project Politician1 Politician2 Politician3
AB 100 50 -60
AC 70 -50 90
BC -70 60 110
ABC 50 30 70
  • In this case, vote trading has led to a Pareto
    improvement (although we are not at a Pareto
    efficient outcome).
  • Vote selling would have led to an efficient
    outcome (all three projects getting funded)
  • However, it is not always (typically?) the case
    that logrolling leads to Pareto improvements.

Project Politician1 Politician2 Politician3 Net Benefit
A 60 -30 -50 -20
B -40 70 -40 -10
C -50 -60 80 -30
Total -30 -20 -10
  • Each of these policies is inefficient.
  • A vote for policy A, B, or C on its own would not
  • However, A coalition/logroll between politicians
    12, 13, or 23 would be feasible, and again the
    coalition between 12 is the most likely.

Project Politician1 Politician2 Politician3
AB 20 40 -90
AC 10 -90 30
BC -90 10 40
ABC -30 -20 -10
  • Here, vote trading has led to a reduction in the
    general welfare of society.
  • Again, vote selling would have led to an
    efficient outcome (none of the projects going
  • Is logrolling used for efficient or inefficient
    projects? Or is all logrolling just Pork-Barrel

Representative Democracy
  • Thus far, we have been talking mostly about
    direct democracy, but most decisions are made via
    representative democracy.
  • Biggest benefit of direct democracy is that there
    is no principal-agent dilemma between voters and
  • Biggest costs are costliness of always voting and
    rational ignorance (or even rational

Back to the Median Voter Model
  • If
  • voter ideology can be arrayed along a single
    dimension, and
  • Individuals have single-peaked preferences, then
  • Median Voter Theorem--Politicians will seek to
    put the policies desired by the median voter into

Duvergets Law
  • What would happen if we attempted to add a third
    candidate to our model?
  • The third candidate would get squeezed out
  • Duvergets Law In a majoritarian political
    system, there is a tendency for party structure
    to be a two party system.

  • Why dont we see full convergence toward the
  • Asymmetric informationcandidates may not know
    the median
  • Campaign contributionscandidates may want to
    move away from the median to get contributions
  • Ideology
  • Cohesionvoters may be less likely to vote for a
    politician who is known to waffle

Flavours of Democracy
  • Plurality Rule
  • Majority Rule (with and without runoffs)
  • Proportional Representation
  • Borda Count
  • Transferable-Vote
  • Point Voting
  • Approval Voting

Flavours of Democracy
  • What do they all have in common?
  • All methods of voting lead to inefficient
    outcomes (especially the most common ones!)
  • Each is unlikely to pick a Condorcet winner if
    one exists.
  • Each is susceptible to strategic or sophisticated

Condorcet Efficiency
Utilitarian Efficiency
Efficiency and voting
  • No voting rule leads to efficient results
  • Why? Voting is, in effect, like doing CBA
    without taking into effect magnitudes of gains
    and losses.
  • There is no reason to assume that policies
    selected by voting will be efficient, especially
    in large societies with many options/candidates!
  • So how are we to get efficient results?

  • Once collective decisions have been made, the
    implementation of these decisions is left to
    government bureaucracy.
  • Somebody needs to administer policies related to
    taxation spending.
  • Problem Principal-Agent Dilemma!
  • Problem Rents!

Principal-Agent Problems and Bureaucracy
  • AgentsBureaucrats, civil servants, government
    officials, individuals who have a career in the
    government bureaucracy.
  • Principalsvoters or their representatives
  • P-A problems arise due to information asymmetry
    between Principal and Agent

  • Most government agencies act as monopolists,
    which generate monopolistic rents
  • Private monopolists receive these rents as
  • Governments (or non-profit agencies) can not
    receive these as profitswhere do they go?
  • Typically, these rents are dissipated away
    through increased costsGovernment run projects
    are far less efficient than prive projects

Efficiencythe Australian Experience
  • Davies (1971, 1974, 1977)Australias private
    airlines are 12-100 more efficient than its
    public airlines
  • Davies (1981)Australia private banks are more
    productive and profitable than public banks
  • Davies and Brucato (1987)Private banks hold
    riskier assets and are more profitable than
    public banks.

Efficiencythe International Experience
  • Massive body of literature, about half looking at
    the US.
  • In nearly all cases in this literature,
    bureaucratic decision-making is far more
    inefficient than private decision making.
  • Why is this?

The folly of bureaucracy
  • What prevents bureaucrats from being as efficient
    as private firms?
  • Budget-maximization behaviour and rent seeking
  • Creating demand for their own output
  • Information asymmetry between bureau and sponsors
  • Difficulty in measuring output and substitution
    into visible activities
  • Biased expert advice

Budget-Maximization Behaviour and Rent Seeking
  • If bureaucrats are self interested, what do they
    want (what is in their utility function?)
  • Power, prestige, income, stuff like that.
  • All of these things will be positively related to
    the size of their budget
  • While private firms have incentives to cost
    minimize, bureaucracies have incentives to cost

Creating Demand for their Own Output
  • One way of budget maximizing is to create demand!
  • Militaries might manufacture enemies, for

Information asymmetry
  • Bureaucrats know what their costs are, but
    sponsors do not!
  • Sometimes it is even illegal to distribute these
    confidential or top secret costs.

Difficulty in measuring output and substitution
into visible activities
  • What is the output of a military?
  • Many of the goods that governments produce are
    unobservable and very difficult to measure.
  • These bureaucracies may try to do fewer of the
    unobservable things (that people want) and do
    more unobservable things (that nobody cares

  • Bureaus often hire external experts to do
    cost-benefit analysis
  • Will they hire somebody who is critical of their
  • Will experts, who want future employment, be

Where do bureaucrats come from?
  • Often political appointmentspositions are
    favours granted, not positions earned based on
    expertise etc.
  • Bureaucratic advancement often a function of
    currying favour with elected officialsdoes this
    lead to efficient results?

Bureaucracies and Public Goods
  • All of these add up to two basic principles
  • Bureaucracies tend to oversupply public goods
  • Bureaucracies tend to supply public goods at
    inefficiently high cost

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