Chapter summary

Voting is the most commonly employed method of resolving a diversity of views or eliciting expressions of preference. It determines the outcome of elections from local to supranational level and governs the decision-making of those who are elected. The chapter begins with a formalisation of majority voting and a statement of the Median voter theorem which provides a prediction of the outcome of voting. The properties of several alternative methods of voting are then analysed. This analysis highlights a number of limitations of voting which are then given general expression in
Arrow’s impossibility theorem.

10.1 Majority rule

In any situation involving only two options, majority rule involves selecting the option with a majority of votes. Unless unanimity is possible, asking that the few give way to the many is a very natural alternative to dictatorship.

A first desirable property is that all the voters should be treated in the same way. This is called Anonymity since it implies that voting is independent of individual identities. A consequence of Anonymity is that there can be no dictator. A desirable second property is that voting
should treat all possible options alike with no bias in favour of one option or another. This symmetric treatment of the various options is called Neutrality. A voting method should have the property that it always selects a winner. This is the property of Decisiveness. A final desirable property to impose is Positive responsiveness so that increasing the vote for the winning option should not result in another option replacing it as winner.

May’s theorem: When choosing from only two options, majority rule is the only voting rule that satisfies the properties of Anonymity, Neutrality, Decisiveness and Positive responsiveness.

• Desirable properties of a voting mechanism
• Majority voting satisfies properties for choice between two alternatives

10.2 Condorcet winner

When a choice must be made from more than two options the principle of majority voting can still be applied. The process to do this is to employ a series of binary agendas which reduce the problem of choosing among many options to a sequence of votes over pairs of alternatives. With a range of alternatives, there will be numerous agendas which differ in the order in which the votes are taken.

For example, one simple binary agenda for choosing among the three options {a, b, c} is as follows. First, a majority vote is held to choose between a and b. Then, a second vote is held between the winner of this first vote and option c. The winner of this second vote is the chosen option.

This is not the only possible agenda. The most famous pair-wise voting method is the Condorcet method. This consists of a complete round-robin of majority votes with each option being judged against all of the others. For example, with the options {a, b, c} the Condorcet method would involve majority votes of a against b, a against c and b against c. If one of the options defeats all others in such pair-wise majority voting it is called the Condorcet winner.

The shortcoming of this method is that the existence of a Condorcet winner requires very special configurations of individual preferences. Often there is no winner (e.g., a defeats b, c defeats a, b defeats c). This lack of a winner is not a special result but, as will be seen, is the first indication of a general problem.

• Binary agendas
• Condorcet winner
• Possibility of no winner

10.3 Median voter theorem

Although the Condorcet winner does not always exist, there are special circumstances in which it does. When the options differ in a single dimension, sufficient conditions for the existence of a Condorcet winner are given by the Median voter theorem. Formally, it applies only when
the votes are taken over all alternatives but it is usually adopted as the solution to any majority voting problem.

The essential features that lie behind the reasoning of the example is that each consumer has single-peaked preferences and that the decision is one-dimensional. Preferences are termed single-peaked when the voter has a single most-preferred option. In the bus stop example, each consumer most prefers their own location and ranks the others according to how close they are to the ideal.

The general statement of the theorem is:

Median voter theorem: Assume there is an odd number of voters and that the policy space is one-dimensional. If the voters have single-peaked preferences, then the preferred option of the median voter is the Condorcet winner.

This idea of median voting has been applied to the analysis of politics. This agglomeration at the centre is called Hotelling’s principle of minimal differentiation and has been influential in political modelling.

An important aspect of the Median voter theorem is that it does not depend on the intensity of preferences, and so there is no incentive for misrepresentation. Sincere voting is the best strategy for everyone.

Having seen how the Median voter theorem leads to a clearly predicted outcome, it is now possible to enquire whether this outcome is efficient. The chosen outcome reflects the preferences of the median voter, so the efficient choice will only be made if this is the most preferred alternative for the median voter. Obviously, there is no reason why this should be the case. Therefore the Median voter theorem will not in general produce an efficient choice.

• One-dimensional alternatives
• Single-peaked preferences
• Choice of median voter is Condorcet winner

10.4 Borda voting

If one considers the principle of majority rule to be attractive, then the failure to select the Condorcet winner may be regarded as a serious weakness of a voting procedure. This is very relevant because many of the most popular alternatives to majority rule do not always choose the Condorcet winner when one does exist, although they always pick a winner, even when a Condorcet winner does not exist. One such alternative to majority rule is Borda voting.

Borda voting (or weighted voting) is a scoring rule which awards points to each option. With n options each voter’s first choice gets n points, second choice gets n–1 point and so forth, down to a maximum of 1 point for the worst choice. Then the scores are added up, and the highest score wins. It is a very simple method, and almost always picks a winner (even if there is no Condorcet winner).

So a fair question is where does the method fail? Firstly, it does not guarantee a Condorcet winner. Further, the introduction of a new option can change the outcome. It is important to make a voting method secure against such irrelevant alternatives. Without this requirement it would be easy to manipulate the voting outcome by adding or removing irrelevant alternatives that have no real chances of winning the election in order to alter the chances of real contenders.

• Scoring rules
• Borda winner even if no Condorcet winner
• Winner affected by irrelevant alternatives

10.5 The paradox of voting

The working assumption employed in analysing voting so far has been that all voters choose to cast their votes. It is natural to question whether this assumption is reasonable. Although in some countries voting is a legal obligation, in others it is not. The observation that many of the latter countries frequently experience low voter turnouts in elections suggests that the assumption is unjustified.

Participation in voting almost always involves costs. Intuition suggests that the probability of being pivotal decreases with the size of the voting population and increases with the predicted closeness of the election.

To give an alternative perspective, when the population is large the probability of having a fatal accident on the way to the polling station exceeds that of affecting the election. Even if accidental death is ignored, it only requires voting to involve very minor costs for these to exceed the benefit of voting. Clearly, it is not rational to vote. This paradox of voting raises serious questions about why so many people do actually vote. Potential explanations of voting could include mistaken beliefs about the chance of affecting the outcome or feelings of social obligation.

• Benefits and costs of voting
• Small probability of affecting outcome
• Cost exceeds expected benefit

10.6 Arrow’s impossibility theorem

The tradition of economics is to take individual preferences as given and beyond dispute. Determining the preferences of a group of people is not such a simple matter. What social choice theory attempts to do (as illustrated by the voting rules we have been discussing) is to produce from a collection of individual preferences a social preference.

Some of the difficulties of moving from individual to social preferences have been identified in the analysis of voting. Arrow’s impossibility theorem shows that such difficulties will arise for any method of aggregating individual into social preferences. In particular, the theorem shows that there is no way to devise a collective decision-making process that satisfies some few compelling requirements. Despite the number of media references to the ‘will of the people’, it is not easy to determine what that will is. This is the remarkable fact of Arrow’s impossibility theorem.

Individual preferences over social states can be summarised as a ranking of the alternatives. Equally, a social preference is also a ranking of the alternatives. The general problem addressed by Arrow in 1951 was to find a way to aggregate individual rankings over options into a collective ranking that satisfied a set of requirements. Any reasonable aggregation method must satisfy the following requirements as laid down by Arrow.

(I) Independence of irrelevant alternatives: Adding a new option should not affect the previous rankings over the old options.
(N) Non dictatorship: There must not be a dictator whose preferences determine the social preferences.
(P) Pareto criterion: If everybody agrees on the ranking of the options, so should the group and the collective ranking should coincide with the common individual ranking.
(U) Unrestricted domain: The collective decision method should accommodate any possible individual ranking of options.
(T) Transitivity: If the group prefers a to b and b to c; then this group cannot prefer a to c.

These are the conditions that are imposed and individually none is disagreeable. Yet the remarkable result that Arrow proved is that there is no way to devise a social decision making procedure that satisfies them all. Arrow’s impossibility theorem: When choosing among more than two options, there exists no social decision-making procedure that satisfies the conditions I.N.P.U.T.

• Aggregating individual into social preferences
• I.N.P.U.T. conditions
• Impossibility of finding a process

Reading: Jean Hindriks and Gareth D. Myles (2004), Intermediate Public Economics

Chapter 4: Voting

Voting is the most commonly employed method of resolving a diversity of views or eliciting expressions of preference. The natural question to ask of voting is whether it is a good method of making decisions. There are two major properties to look for in a good method. First is the success or failure of the method in achieving a clear-cut decision. Second is the issue of whether voting always produces an outcome that is efficient. Voting would be of limited value if it frequently left the choice of outcome unresolved or lead to a choice that was clearly inferior to other alternatives.
Whether voting satisfies these properties is shown to be somewhat dependent upon the precise method of voting adopted. p49

The central result of the theory of social choice, Arrow’s Impossibility Theorem, says that there is no way to devise a collective decision-making process that satisfies a few commonsense requirements and works in all circumstances. If there are only two options, majority voting works just fine, but with more than two we can get into trouble. Despite all the talk about the ”will of the people”, it is not easy - in fact the theorem proves it impossible - to always determine what that will is. This is the remarkable fact of Arrow’s Impossibility Theorem. p51

... intransitivity of group preferences can arise even when individual preferences are transitive. This generation of social intransitivity from individual transitivity is called the Condorcet Paradox. p51

Condition 1 (I) Independence of Irrelevant Alternatives. Adding new options should not affect the initial ranking of the old options; so the collective ranking over the old options should be unchanged. p51

Condition 2 (N) Non dictatorship. The collective preference should not be determined by the preferences of one individual. p52

Condition 3 (P) Pareto criterion: If everybody agrees on the ranking of all the possible options, so should the group; the collective ranking should coincide with the common individual ranking. p52

Condition 4 (U) Unrestricted domain: The collective choice method should accommodate any possible individual ranking of options. p52

Condition 5 (T) Transitivity: If the group prefers A to B and B to C; then this group cannot prefer C to A. p52

Theorem 1 (Arrow’s Impossibility Theorem) When choosing among more than two options, there exists no collective decision-making process that satisfies the conditions I.N.P.U.T. p52

When there are only two options, majority rule is a simple and compelling method for social choice. When there are more than two options to be considered at a time, we can still apply the principle of majority voting by using binary agendas which allow us to reduce the problem of choosing among many options to a sequence of votes each of which is binary. For example, one simple binary agenda for choosing among the three options
{a, b, c} in the Condorcet Paradox is as follows. First, there is a vote on a against b. Then, the winner of this first vote is opposed to c. The winner of this second vote is the chosen option. The most famous pair-wise voting method is the Condorcet method. It consists of a complete round-robin of majority votes, opposing each option against all of the others. The option which defeats all others in pair-wise majority voting is called a Condorcet winner.

Theorem 2 (May’s theorem) When choosing among only two options, there is only one collective decision-making process that satisfies the requirements of Anonymity, Neutrality, Decisiveness and Positive Responsiveness. This process is majority rule. p54

The problem is that the existence of a Condorcet winner requires very special configurations of individual preferences. For instance, with the preferences given in the Condorcet paradox, there is no Condorcet winner. p54

When the policy space is one-dimensional, sufficient (but not necessary) conditions for the existence of a Condorcet winner are given by the Median Voter Theorems. One version of these theorems refers to single-peaked preferences, while the other version refers to single-crossing preferences. The two conditions of single-peaked and single-crossing are logically independent but both conditions give the same conclusion that the median position is a Condorcet winner. p54

Theorem 3 Median Voter Theorem I (Single-peaked version) Suppose there is an odd number of voters and that the policy space is one-dimensional (so that the options can be put in a transitive order). If the voters have single-peaked preferences, then the median of the distribution of voters’ preferred options is a Condorcet winner. p55

This agglomeration at the centre is called Hotelling’s principle of minimal differentiation and has been influential in political modelling. The reasoning underlying it can be observed in the move of the Democrats in the United States and labor [sic] in the United Kingdom to the right in order to crowd out the Republicans and Conservatives respectively. The result also shows how ideas developed in economics can have useful applications elsewhere. p56

The single-crossing version of the Median Voter Theorem assumes not only that the policy space is transitively ordered, say from left to right (and thus one-dimensional), but also that the voters can be transitively ordered, say from left to right in the political spectrum. The interpretation is that voters at the left prefer left options more than voters at the right. This second assumption is called the single-crossing property of preferences. Formally, Definition 1 (Single-crossing property) For any two voters i and j such that
i < j (voter i is to the left of voter j), and for any two options x and y such
that x < y (x is to the left of y):
(i) If u j (x) > u j (y) then u i (x) > u i (y);
and
(ii) If u i (y) > u i (x) then u j (y) > u j (x).
p57

Theorem 4 Median Voter Theorem II (Single-crossing version) Suppose there is an odd number of voters and that the policy space is one-dimensional (so that the options can be put in a transitive order). If the voters’ preferences together satisfy the single-crossing property, then the preferred option of the median voter is a Condorcet winner. p57

An attractive aspect of the Median Voter Theorem is that it does not depend on the intensity of preferences, and thus nobody has an incentive to misrepresent their preferences. p57

it is now possible to show that the Median Voter Theorem can fail with majority voting failing to generate a transitive outcome... the theorem does not extend beyond one-dimensional choice problems. p59

In a situation in which there is no Condorcet winner, the door is opened to agenda manipulation. This is because changing the agenda, meaning the order in which the votes over pairs of alternatives are taken, can change the voting outcome. p60

Such outcomes are called sophisticated outcomes of binary agendas, because voters anticipate what the ultimate result will be, for a given agenda, and vote optimally in earlier stages.

A remarkable result, due to Miller, is that strategic voting (relative to sincere voting) does not alter the set of outcomes that can be achieved by agenda-manipulation when the agenda-setter can design any binary-agendas, provided only that every option must be included in the agenda. Miller called the set that can be achieved the top cycle. p60

When there exists a Condorcet winner, the top cycle reduces to that single option. With preferences as in the Condorcet Paradox, the top cycle contains all three options {a, b, c}. For example, option b can be obtained by the following agenda (different from the agenda under sincere voting): at the first stage, a is opposed to b, then the winner is opposed to c p61

Even if one considers the principle of majority rule to be attractive, the failure to select the Condorcet winner when one exists may be regarded as a serious weakness of majority rule as a voting procedure. This is very relevant because many of the most popular alternatives to majority rule also do not always choose the Condorcet winner when one does exist, although they always pick a winner even when a Condorcet winner does not exist. This is the case for all the scoring rule methods, like plurality voting, approval voting and Borda voting. p63

Each scoring rule method selects as a winner the option with the highest aggregate score. The difference is in the score voters can give to each option. Under plurality voting, voters give 1 point to their first choice and 0 points to all other options. Thus only information on voters’ most preferred option is used. Under approval voting, voters can give 1 point to more than one option, in fact to as many or as few options as they want. Under Borda voting, voters give the highest possible score to their first choice, then progressively lower scores to worse choices. p63

Under Plurality voting only the first choice of each voter matters and is given one point. Choices other than the first do not count at all. These scores are added and the option with the highest score is the plurality winner. Therefore, the Plurality winner is the option which is ranked first by the largest number of voters. The reason for this is that plurality voting dispenses with all information other than about the first choices.

One problem with plurality rule is that voters don’t always have an incentive to vote sincerely. Any rule that limits each voter to cast a vote for only one option forces the voters to consider the chance that their first-choices will win the election. If the first choice option is unlikely to win, the voters may instead vote for a second (or even lower) choice to prevent the election of a worse option.

In response to this risk of misrepresentation of preferences (i.e., strategic voting), Brams and Fishburn have proposed the approval voting procedure.
They argue that this procedure allows voters to express their true preferences. Under approval voting, each voter may vote (approve) for as many options as they like. Approving one option does not exclude approving any other options.

There is no cost in voting for an option which is unlikely to win. The winning option is the one which gathers the most votes. This procedure is simpler than Borda voting because instead of giving a score for all the possible options, voters only need to separate the options they approve of from those they do not. Approval voting also has the advantage over pair-wise voting procedures that voters need only vote once, instead of engaging in a repetition of binary votes (as in the Condorcet method). The problem with approval voting is that it may fail to pick the Condorcet winner when one exists.

The runoff is a very common scheme used in many presidential and parliamentary elections. Under this scheme only first-place votes are counted; and if there is no majority, there is a second runoff election involving only the two strongest candidates. The purpose of a runoff is to eliminate the least-preferred options. Runoff voting seems fair, and is very widely used. However it has two drawbacks. First, it may fail to select a Condorcet winner when it exists; second, it can violate positive responsiveness which is a fundamental principle of democracy.

... the probability that someone’s vote will change the outcome is essentially zero when the voting population is large enough and so if voting is costly, the cost-benefit model, should imply almost no participation. The small probability of a large change is not enough to cover the cost of voting. Each person’s vote is like a small voice in a very large crowd.