Voting Systems, Direct Democracy, Representative Democracy, Leviathan Hypothesis & Arrow’s Impossibility Theorem

7 months ago

Atiqur Rahman

14 minutes

Introduction

Voting represents collective decisions. While studying the allocation of public goods (Samuelson, 1955), we assume that somehow the policymakers will be able to estimate the marginal benefit curve of the consumers (of public good), which will be nothing but a vertical summation of the demand curve (i.e., marginal benefit curve) of each and every consumer. Such a demand curve is called as pseudo-demand curve.

But at the same time, it is acknowledged by public economists that it is misleading to conclude that social goods can be provided by market mechanisms of demand and supply. As social goods are typically non-rival in consumption (a non-rival good is a good which may be consumed by many at the same time at no additional cost, for example, national defence or a piece of scientific knowledge.), the exclusion is not possible and hence the consumers will not reveal their preferences voluntarily. Furthermore, they will be convinced that their decision not to contribute towards the provision cost will have no effect on the provision because the number of consumers will be very large.

In this type of situation, the use of a political process becomes a must. The political process can help in two ways:

Through the political process, preferences can be revealed, and the government can decide on the provision of public good, i.e., which public good people really want
As provision cost will be involved, the political process can further suggest the course of revenue generation for the provision.

In this context, Musgrave and Musgrave (1973) write, “Individuals, knowing that they must comply with the majority decision, will find it in their best interest to vote for that solution which will move the outcome closer to their own desires, and in this way, they will be induced to reveal their preferences.”

Voting Systems

While casting a vote, people may or may not be aware of the enormous responsibility it involves. According to Hillman (2003), “…because the probability of one person’s vote being decisive in large voting populations is effectively zero, the reason why people vote may not be that they expect to influence the outcome of voting. Voting may be expressive: voters may express themselves by identifying with a policy or with a candidate, as they might, for example, cheer for and identify with a sports team and be happy or sad according to whether “their” team wins; or, by expressing support for a candidate or political party, a voter may be communicating with and seeking approval from other people. People may also vote as an act of civic duty: they have been taught that taking the time to vote is correct pro-social behaviour. Their utility from voting may be from democratic participation.”

But in an ideal set-up, if voting is to be used as an efficient mechanism for preference revelation, the revenue generation decision and the public spending decision both should be linked to it. Voters are then confronted to choose from various budget proposals, each of which will carry a price tag in terms of their own contribution.

Another option can be that instead of democratically choosing to make the collective decisions on public policy and finance through voting (mainly on what to provide and how to finance the provision), we can randomly select a person from the society, and he/she will be expected to make collective decisions on behalf of everyone.

Will such a randomly selected representative person make better decisions as compared to the collective decision of voters? Or is democracy more desirable? The answer to this type of dilemma can be found in the Condorcet Jury Theorem, given by Marquis de Condorcet (1785).

The Condorcet Jury Theorem presupposes that all voters have the same objectives, and the only question concerns how an agreed common objective can best be achieved. For example, there are two alternative proposed means, A and B, of seeking to achieve efficiency. One means is correct, and the other is incorrect. The correct choice is A.

However, there is uncertainty or imperfect information among voters, who only know that A is the correct decision with some probability. Under the conditions of the Condorcet jury theorem, the probability is taken to be the same for everyone and exceeds 0.5. The commonly held probability that A is the correct choice might, for example, be 0.6.

Voters then randomize (as in the case of mixed strategies) and vote for alternative A with a 60 percent probability (and for alternative B with a 40 percent probability). The collective decision is determined as the alternative with the majority (more than 50 percent) of votes.

The Marquis de Condorcet showed that if everyone votes without being influenced by how others vote, then – as the number of voters increases – the likelihood that the correct decision A will have majority support increases. As the number of voters becomes very large, the likelihood that the majority makes the correct decision approaches certainty.

Though as concluded by Hillman (2003), “The Condorcet jury theorem is not applicable to many of the questions that we generally confront in the study of economics. People differ, for example, in preferences regarding public spending on public goods and about who should pay taxes that finance the public spending. They may be affected differently by externalities – or may be the sources of externalities. If the vote is on income redistribution, voters are usually influenced by whether they personally gain or lose from an income-redistribution proposal. They may also be influenced by whether low-income people gain or lose. The Condorcet jury theorem, which is based on a single objective common to all voters, provides no answers about voting outcomes when people have different objectives”.

This theorem shows that in the presence of information limitations, the collective decisions of democracy are preferable to dictatorship – and also to randomly selecting someone to make decisions on behalf of society – and this is so even if the dictator or randomly selected decision maker seeks an objective on which everyone agrees.

As it is clear that the determination of the budget is more of a political process than a market process, let us now examine the political process more closely. We will broadly talk about:

Direct Democracy
Representative Democracy, and
Leviathan Hypothesis

Direct Democracy

Direct democracy is a system in which a referendum among the voters determines the final outcome of fiscal decisions. Whatever decision is reached by voting is considered to be binding on the voters.

Given the non-excludability of a social good and a large number of voters, a rational voter is aware of the fact that his or her individual vote has very less probability of affecting the final outcome. Still, if he or she chooses to vote, it is because of their sense of responsibility.

In this context, Musgrave and Musgrave (1973) observe, “…such action may not reflect a narrowly defined act of ‘rational behaviour’, but happily, human action is not limited to that premise”.

Under direct democracy, there can be alternative voting rules. The majority rule is one where each individual has one vote, and the majority wins.

In the context of majority voting, the median voter theorem emphasizes that such a majority voting will ultimately lead to an outcome which will be the one most preferred by the median voter.

To explain the concept, we can take an example of a hypothetical country or economy where three voters, X, Y and Z, have to make a choice about public spending.

Let X opt for a spending decision of 1000 Rs, Y for 2000 Rs and Z for 4000 Rs. In this example, Y can be called as the median voter (because he is in the middle of X and Z). Each individual will opt for a public spending decision which is closer to their preferred spending amount to those that are farther from it. We can easily conclude that Y, being the median voter, will always cast his vote for the policy that is adopted, and his preferred policy decision will defeat any other.

The median voter theorem explains many intricacies of majority voting, viz.

The decision of politicians to opt for similar campaigns (more towards a middle ground, rather than having extremist views) so that the median voter can be lured;
This theorem also provides a convincing explanation for the fact that in a majority voting system, generally two major political parties who will co-opt the platforms of the minor parties in order to secure more votes.

Representative Democracy

This approach is strongly influenced by the concept of ‘Self Interest’ given by Adam Smith. Both politicians and voters will be guided by self-interest in their behaviour. The politicians will seek to maximise the votes in order to stay in power and secure re-election. The objective of the voter is to maximise the net benefits from any fiscal activity.

Here, by net benefit, voters will imply the difference between the benefit from public expenditure and the cost of taxes to finance this expenditure. In such a situation, the voters will seek to elect those who will best represent their interests, and the politicians will offer promises and programmes that best satisfy the voters.

In representative democracy, the role of coalitions also becomes very important. Some policies may not be supported by voters if they are considered in isolation, but the same may win acceptance if, are considered in coalition.

Leviathan Hypothesis

The Leviathan model of government was given by Brennan and Buchanan (1980). Leviathan is a sea monster referenced in the Testament, the first section of the Christian Bible.

Brennan and Buchanan (1980) assumed that the government acts as a monopolist that maximizes tax revenues. Such representation is given under the assumptions of (1) rational ignorance, (2) fiscal illusion and (3) outright collusion among elected officials, which ultimately deprive voters (who are the taxpayers, too) of any control they may have over government.

According to this hypothesis, monopolization of government activity leads to an ability to extract revenue from citizens. The more decentralized the fiscal relations between levels of government, the less monopoly power can be exerted in tax collection.

Arrow’s Impossibility Theorem

The theorem was given by Kenneth Arrow in 1950. He showed that there is no constitutional rule that will satisfy the list of reasonable ‘conditions’ of correspondence. He enlists five conditions of correspondence, viz.,

Collective nationality, i.e., convexity and transitivity, must be there. Convexity means that the outcomes must be comparable. Transitivity means if X is preferred to Y, Y is preferred to Z, then X must always be preferred to Z.
Pareto Principle, i.e., if all members of a community prefer X to Y, then the collective choice that emerges from a constitutional rule must also indicate that X is preferred to Y.
Independence of irrelevant alternatives, i.e., if the choice has to be made between two alternatives X and Y, this choice should not be influenced by the ranking of either X or Y with other alternatives such as Z or A.
Non-dictatorship
Unrestricted domain, i.e., no individual should be excluded from contributing to the establishment of a collective choice, provided that the individual has a transitive order of preferences.

The Arrow Impossibility theorem shows that no constitutional rule will comply with all these conditions. This can be explained by the forthcoming example given by Cullis and Jones (1998).

Table 1: The Arrow Impossibility Theorem (Majority Voting)

	Options
Voters	X	Y	Z
A	3	2	1
B	1	3	2
C	2	1	3

Table 2: Outcome Dependency on Agenda Setting

Vote between	Vote between	Outcome
Y v. X	X v. Z	Z
X v. Z	Z v. Y	Y
Z v. Y	Y v. X	X

In table 1, three individuals (voters A, B and C) choose between three alternatives, X, Y and Z. The rankings of the individuals are shown as 3, 2, 1 (in order of preference, 3 being the most preferred). It is clear that each voter has a transitive preference order. (Transitivity means if X is preferred to Y, Y is preferred to Z, and then X must always be preferred to Z).

If a collective decision was required as between X and Y, the majority would choose X ( voters A and C); between Y and Z, a majority would opt for Y( voters A and B). As a consequence, to satisfy the condition of collective rationality, majority voting should lead to a preference for X rather than Z. However, in the example, voters B and C constitute a majority in favour of Z rather than X. In this way, the simple majority voting fails the Arrow test.

The final winner is dependent on which vote is taken first. The sequence of votes given in Table 2 indicates alternative winners.

Voting Systems, Direct Democracy, Representative Democracy, Leviathan Hypothesis & Arrow’s Impossibility Theorem

Introduction

Voting Systems

Direct Democracy

Representative Democracy

Leviathan Hypothesis

Arrow’s Impossibility Theorem

Read More in: Theory of Public Finance

Related

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Introduction

Voting Systems

Direct Democracy

Representative Democracy

Leviathan Hypothesis

Arrow’s Impossibility Theorem

Read More in: Theory of Public Finance

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Related

Share Your ThoughtsCancel reply