What is the paradox of voting in America

Theorem 7

(Saari 1990a, 1995a) The positional method ws that is least susceptible to a small, successful manipulation is the Borda Count. As the value of |s−12|increases, so does the level of susceptibility of the positional method.

The Paradox of Voting (Arrow's Impossibility Theorem)

In 1951, economist Kenneth Arrow described what he called the “well-known ’paradox of voting.‘” Although he did not claim to have originated it, he is credited with the systematic formulation of what has also come to be known as Arrow's impossibility theorem. The theorem states that given more than two voting alternatives, and absent the assumption that they are “single peaked” (i.e., that an individual's first preferred choice determines his or her second choice), there is “no method of voting … neither plurality voting nor any scheme of proportional representation, no matter how complicated” that will guarantee an unambiguous aggregate preference. For example, suppose that in 1996 three voters were asked to choose between Clinton, Perot, and Dole in that year's presidential contest. Suppose further that their preferences were ordered as follows:

Voter 1: Dole, Perot, Clinton

Voter 2: Perot, Clinton, Dole

Voter 3: Clinton, Dole, Perot

Since there are various considerations that might govern an individual's choices among these options, all three voters may have been acting quite rationally as individuals. Taken collectively, however, any one of the three options would have been rejected by a majority when pitted against only one of the alternatives. A majority would have preferred Dole to Perot, Perot to Clinton, and Clinton to Dole.

Arrow's theorem has given rise to a substantial body of literature across a number of different disciplines, including political science. Jones et al., for example, employed computer simulations to show that when voters are able to order their preferences across all options, the likelihood of producing a majority decision is smaller the larger the number of voters. However, when voters are indifferent among some choices, the problem is more serious for small groups, such as committees, than with a large electorate.

5.2.1 Ex-ante elections

We start by assuming that elections take place at the beginning of period 1, before private agents have chosen the amount to save in period 1. The platform of the winning candidate is enacted without further re-optimization. A different timing assumption is discussed below.

To characterize the voters’ policy preferences, we follow the same approach as in Section 2. Let Wi(τ) be the indirect utility function of individual i:

Wiτ=U(1−KτK)+V(1−L(τL))+1−τLL(τL)+(1−τK)K(τK)+(τK−τL)ei=Wτ+(τK−τL)ei.

Then, maximize this function with regard to the two tax rates, subject to the government budget constraint and the supply functions defined above. Combining the resulting first-order conditions, we get:

(5.4)KτKi−eiKτKi1+τLi1−τLiηLτLi=LτLi+eiLτLi1+τKi1−τKiηKτKi,

where ηyx≡dydxxy<0denotes the elasticity of y with regard to x. Together with the government budget constraint (5.3), this condition defines the tax policy τi preferred by voter i.

The individual thus wants taxes to be set according to a modified “Ramsey Rule”. Consider first the policy preferred by the individual with average relative income from labor and capital. This policy also has some normative appeal; due to quasi-linear preferences, it coincides with the utilitarian optimum. Clearly, with ei = e = 0, the condition reduces to the familiar inverse elasticity formula of optimal commodity taxation, showing that capital should indeed be taxed more lightly than labor, if its supply is more elastic. Intuitively, the average individual does not care about redistribution, only about efficiency: thus his favored tax policy just minimizes the deadweight loss associated with taxation. We refer to this Ramsey policy as τ*.

When ei ≠ 0, redistributive preferences modify this pure efficiency condition in a predictable way. That is, individuals with more labor than capital income (ei > 0) want the tax rate on capital to be higher and the rate on labor income to be lower, and vice versa if ei < 0 (recall that elasticities are defined to be negative):

(5.5)τKi≦τK∗τLi⋛τL∗asei⋛0.

The monotonicity of these preferences implies that τm, the tax policy preferred by the median voter with endowment em, is a unique Condorcet winner24. As em > 0, the implied equilibrium tax policy τm has a higher taxation of capital and a lower taxation of labor than our normative benchmark policy τ*. In this sense, there is thus overtaxation of capital, due to the skewed distribution of wealth, which implies that the pivotal voter relies relatively more on labor income than on capital income.

1.4 Results for Committees and for the General Model

Black (1948a, pp. 27–28; 1948b, pp. 249–251; 1958, pp. 16–18) proved the following results under the assumptions for his committee model, which are stated above. First, a particular motion cannot beaten (i.e., no motion gets a simple majority over that particular motion) if and only if that particular motion is a median for the distribution of most-preferred motions (with respect to the natural linear order). Second, a particular motion gets a simple majority over every other motion if and only if that particular motion is the unique median for the distribution of most-preferred motions (with respect to the natural linear order).

The following (analogous) theorem holds under the assumptions for the general model (see, for instance, Denzau and Parks 1975, Theorem 4).

For some committees, an absolute majority (i.e., more than half of the entire set of voters) is required for social preference—instead of a simple majority being required. More specifically, for some committees: (i) one alternative beats another one if and only if an absolute majority prefer it, and (ii) otherwise, the two alternatives tie one another.

For any given (X,(≽1,…,≽#Ω)), the relation “beats or ties” for absolute majority rule will be denoted by ≽A. More specifically, for each x, y ∈ X, x≽Aymeans #{ω∈Ω: x≽ωy}≥#Ω∕2. The asymmetric part of ≽Awill be denoted by ≻A. The symmetric part will be denoted by ~A. The definition of ≽Aimplies (i) for each x, y ∈ X, x≻Ayif and only if #{ω∈Ω: x≻ωy}>#Ω∕2; (ii) for each x, y ∈ X, x∼Ayif and only if #{ω∈Ω: x≻ωy}≤#Ω∕2and #{ω∈Ω: y≻ωx}≤#Ω∕2. For any given (X,(≽1,…,≽#Ω)), saying that x ∈ X is a “(weak) absolute majority Condorcet winner” means x≽Ay,∀y∈X. Saying that x ∈ X is a “(strong) absolute majority Condorcet winner” means x≻Ay,∀y∈X−{x}.

The definitions clearly imply the following: (1) if an alternative is a (strong) absolute majority Condorcet winner, then it is a (strong) simple majority Condorcet winner; (2) if an alternative is a (weak) absolute majority Condorcet winner, then it is a (weak) simple majority Condorcet winner. The definitions also clearly imply that the converses of those statements are not true in general. So, in general, results for simple majority Condorcet winners cannot be expected to hold for absolute majority Condorcet winners. Nonetheless, the following variation on the previous theorem does hold (see, for instance, Denzau and Parks 1975, Theorem 4).

It should be noted that there is an important connection between (a) the set of (weak) absolute majority Condorcet winners, and (b) a solution concept that has been applied when committee decisions have been modeled as cooperative games, as in the models developed by Kramer and Klevorick (1974), and Nakamura (1979). If a committee uses absolute majority rule, then a “winning coalition” is a set of voters that contains more than half of the committee members. Saying that alternative x“dominates” alternative y means there exists a winning coalition, W, where x≻ωy,∀ω∈W. Using this dominance relation, we can now easily state the definition of the relevant solution concept: The “core” is the set {y ∈ X: ∄x∈Xsuch that x dominates y}. Using this definition, it follows that when a committee uses absolute majority rule, an alternative is in the core if and only if it is a (weak) absolute majority Condorcet winner. Discussions of part 2 of the above theorem in terms of the core are in McKelvey (1990, Sections 2 and 3.1), Straffin (1994, Section 6), Saari (2004, Sections 3.1 and 4.1) and other references.

Theorems 1 and 2(stated above) are very significant for settings where either simple majority rule or absolute majority rule is used and the supposition for the theorems is satisfied.

To begin with, the theorems are positive results for social choice. More specifically, there is at least one median—so there is at least one (weak) Condorcet winner. So it is possible to select an alternative that cannot be beaten.

If #Ω is odd, there is a unique median. Hence, if #Ω is odd, there is a (strong) Condorcet winner—which, by definition, is also a unique (weak) Condorcet winner. For the cases where #Ω is even: (1) if there is a unique median for the distribution of most-preferred alternatives (with respect to ≤o), then there is a unique (weak) Condorcet winner and it is also a (strong) Condorcet winner; (2) if there is not a unique median for the distribution of most-preferred alternatives (with respect to ≤o), then there is more than one (weak) Condorcet winner and there is no (strong) Condorcet winner.

The ordering relation referred to in the theorems reflects “a similar attitude toward the alternatives” among the voters, since when this ordering relation is used, every voter has the view that moving away from his most-preferred alternative in either “direction” leads to worse and worse alternatives. Therefore, since the Condorcet winners are medians with respect to that ordering relation, they match with a reasonable measure for the center of the distribution of most-preferred alternatives. Hence they are, in this sense, “centrist” social choices.

The theorems also tell us that one would not have to actually make all the majority rule comparisons to find the appropriate choice(s). Instead, one could find the appropriate choice(s) by finding the median(s) of the distribution of most-preferred alternatives.

An individual whose most-preferred alternative is a median for the distribution of most-preferred alternatives (with respect to a given ordering relation ≤o) is called a “median voter” (with respect to ≤o). When there is a unique median for the distribution of most-preferred alternatives (with respect to ≤o), there will be at least one median voter—and there will be more than one if and only if the median is the most-preferred alternative for more than one individual. In the cases where there is a unique median, the theorems imply that an alternative is a Condorcet winner if and only if it is a median voter's most-preferred alternative.

When there is more than one median for the distribution of most-preferred alternatives (with respect to ≤o), there will be at least two median voters—and there will be more than two if and only if there is a median that is the most-preferred alternative for more than one individual. In the cases where there is more than one median, the theorems imply that an alternative is a (weak) Condorcet winner if it is a median voter's most-preferred alternative. In these cases, there will always be two distinct (weak) Condorcet winners that are most-preferred alternatives for median voters. In addition, in some instances, there will also be at least one (weak) Condorcet winner that is not anyone's most-preferred alternative. More specifically, let α be the index number for an individual who has one of the two distinct most-preferred alternatives that are (weak) Condorcet winners as his most-preferred alternative, let β be the index number for an individual who has the other of the two alternatives as his most-preferred alternative, and have the index numbers be such that m(α)< om(β): the theorems imply that any alternative, y, that satisfies m(α)< oy< om(β)will be a (weak) Condorcet winner.

Related discussions of single-peakedness and majority rule are in Arrow (1963, Section 2 in Chapter VII), Fishburn (1973, Sections 9.1–9.3), Denzau and Parks (1975, Sections 1 and 2), Enelow and Hinich (1984, Sections 2.1 and 2.2), Mueller (2003, the last paragraph in Section 5.2 and all of Section 5.3), Mas-Colell, Whinston, and Green (1995, Section 21.D), Shepsle and Bonchek (1997, pp. 83–90 in the section “The Simple Geometry of Voting”) and other references.

21.2.1 The Redistributive and Equalizing Effects of Democracy

We start with the standard “equalizing effect” of democracy, first emphasized formally in Meltzer and Richard's (1981) seminal study (see also Acemoglu and Robinson, 2006). Democratization, by extending political power to poorer segments of society, will increase the tendency for pro-poor policy naturally associated with redistribution, and thus reduce inequality.

Suppose that society consists of agents distinguished only with respect to their endowment of income, denoted by yi for agent i, with the distribution of income in the society denoted by the function F(y) and its mean by y¯. The only policy instrument is a linear tax τ imposed on all agents, with the proceeds distributed lump-sum again to all agents. We normalize total population to 1 without loss of any generality.

The government budget constraint, which determines this lump-sum transfer T, takes the form

(21.1)T≤τy¯−Cτy¯,

where the second term captures the distortionary costs of taxation. C(τ) is assumed to be differentiable, convex and nondecreasing, with C′(0) = 0.

Each agent's post-tax income and utility is given by

(21.2)yˆi=1−τyi+τy¯−Cτy¯.

This expression immediately makes it clear that preferences over policy—represented by the linear tax rate τ—satisfy both single crossing and single-peakedness (e.g., Austen-Smith and Banks, 1999). Hence the median voter theorem, and its variants for more limited franchises (see e.g., Acemoglu et al., 2012) hold.5

Suppose, to start with, that there is a limited franchise such that all agents with income above yq, the qth percentile of the income distribution, are enfranchised and the rest are disenfranchised. Consider a “democratization,” which takes the form of yq decreasing, say to some yq′<yq, so that more people are allowed to vote. Let the equilibrium tax rate under these two different political institutions be denoted by τq and τq′, and the resulting post-tax income distribution by Fq and Fq′. Then from the observation that the median of the distribution truncated at yq′is always less than the median for the one truncated above yq>yq′, the following result is immediate:

A few comments about this proposition are useful. First, this result is just a restatement of Meltzer and Richard's (1981) main result. Second, the first part of the conclusion is stated as τq′≥τq, since if both yq and yq′are above the mean, with standard arguments, τq′=τq=0. Third, the second part of the conclusion does not state that Fq is a mean-preserving spread of, or is second-order stochastically dominated by Fq′, because higher taxes may reduce mean post-tax income due to their distortionary costs of taxation. Instead, the statement is that Fq′is more concentrated around its mean than Fq, which implies the following: if we shift Fq′so that it has the same mean as Fq, then it second-order stochastically dominates Fq (and thus automatically implies that standard deviation and other measures of inequality are lower under Fq′than under Fq).

Finally, the result in the proposition should be carefully distinguished from another often-stated (but not unambiguous) result, which concerns the impact of inequality on redistribution. Persson and Tabellini (1994) and Alesina and Rodrik (1994), among others, show that, under some additional assumptions, greater inequality leads to more redistribution in the median voter setup (which in these papers is also embedded in a growth model). This result, however, is generally not true.6 It applies under additional assumptions on the distribution of income, such as a log normal distribution, or when the gap between mean and median is used as a measure of inequality (which is rather nonstandard). In contrast, the result emphasized here is unambiguously true.

This result of Meltzer and Richard (1981) is the basis for the hypothesis that democracy should increase taxation and income redistribution and reduce inequality. In the model, the only way that redistribution can take place is via a lump-sum transfer. This is obviously restrictive. For example, it could be that individuals prefer the state to provide public goods (Lizzeri and Persico, 2004) or public education. Nevertheless, the result generalizes, under suitable assumptions, to the cases in which the redistribution takes place through public goods or education.

We next discuss another possible impact of democracy and why its influence on redistribution and inequality may be more complex than this result may suggest.

3.1.1 The Political Support for Early Retirement Provisions

In the last 30 years, most OECD countries have experienced a dramatic drop in the labor force participation of their middle-aged and elderly workers. In the OECD countries, the average labor force participation rate of male workers aged between 55 and 64 years has decreased from 84.2% in 1960 to 66.8% in 2015. The extent to which male elderly workers have decreased their participation in the labor market may also be captured by the reduction in the average retirement age.

A comprehensive study on 11 OECD countries edited by Gruber and Wise (1999) suggests that generous early retirement provisions are largely responsible for this drop in the (male) participation rates. Gruber and Wise (1999) and a parallel study by Blondal and Scarpetta (1998) identify two features of the early retirement provisions, which display a strong correlation with the departure of the elderly workers from the labor force: the early (and normal) retirement age and the tax burden which is imposed on the labor income of the individuals who continue to work after reaching the early retirement age. Gruber and Wise (1999) and Blondal and Scarpetta (1998) argue that individuals are often induced to retire early because of the large implicit tax imposed on continuing to work after early retirement age. Individuals’ early retirement decision thus represents the optimal response to the economic incentives provided by the social security system.

Conde-Ruiz and Galasso (2003) develop a positive theory of why early retirement age provisions have been introduced in the first place and sustained over time. They analyze a majority voting game over two dimensions: the payroll tax rate and the decision to introduce or not an early retirement provision. Focusing on the (simultaneous) issue-by-issue voting equilibrium,q they show that early retirement is sustained by a coalition of the young poor and of the old with an incomplete earning history. The latter obviously sustain the system as it makes them eligible for pension benefits. The former vote for early retirement because, due to substitution effects between leisure and consumption, they tend to retire earlier than the rich. It should be noted that the equilibrium is self-sustained over time (or subgame perfect) as, by retiring earlier, the current poor young will continue to sustain the system when becoming old.

In a subsequent paper, Conde-Ruiz and Galasso (2004) note the (negative) macroeconomic consequences of early retirement provisions which, by inducing individuals to retire earlier, depress human capital accumulation and growth.

6.1 Preference aggregation

The theory of preference aggregation in the long and established tradition of Condorcet and Arrow addresses the following question: How can a group of individuals arrive at a collective preference ordering on some set of alternatives on the basis of the group members' individual preference orderings on them? Condorcet's classic paradox illustrates some of the challenges raised by this problem. Consider a group of individuals seeking to form collective preferences over three alternatives, x, y and z, where the first individual prefers x to y to z, the second y to z to x, and the third z to x to y. It is then easy to see that majority voting over pairs of alternatives fails to yield a rational collective preference ordering: there are majorities for x over y, for y over z, and yet for z over x — a ‘preference cycle’. Arrow's theorem [1951/1963] generalizes this observation by showing that, when there are three or more alternatives, the only aggregation rules that generally avoid such cycles and satisfy some other minimal conditions are dictatorial ones. Condorcet's paradox and Arrow's theorem have inspired a massive literature on axiomatic social choice theory, a review of which is entirely beyond the scope of this paper.

How is the theory of preference aggregation related to the theory of judgment aggregation? It turns out that preference aggregation problems can be formally represented within the model of judgment aggregation. The idea is that preference orderings can be represented as sets of accepted preference ranking propositions of the form ‘x is preferable to y’, ‘y is preferable to z’, and so on.

To construct this representation formally (following [Dietrich and List, 2007], extending [List and Pettit, 2004]), it is necessary to employ a specially devised predicate logic with two or more constants representing alternatives, denoted x, y, z and so on, and a two-place predicate ‘_is preferable to_’. To capture the standard rationality conditions on preferences (such as asymmetry, transitivity and connectedness), we define a set of propositions in our predicate logic to be ‘consistent’ just in case this set is consistent relative to those rationality conditions. For example, the set {x is preferable to y, y is preferable to z} is consistent, while the set {x is preferable to y, y is preferable to z, z is preferable to x} — representing a preference cycle — is not. The agenda is then defined as the set of all propositions of the form ‘v is preferable to w’ and their negations, where v and w are alternatives among x, y, z and so on. Now each consistent and complete judgment set on this agenda uniquely represents a rational (i.e., asymmetric, transitive and connected) preference ordering. For instance, the judgment set {x is preferable to y, y is preferable to z, x is preferable to z} uniquely represents the preference ordering according to which x is most preferred, y second-most preferred, and z least preferred. Furthermore, a judgment aggregation rule on the given agenda uniquely represents an Arrovian preference aggregation rule (i.e., a function from profiles of individual preference orderings to collective preference orderings).

Under this construction, Condorcet's paradox of cyclical majority preferences becomes a special case of the problem of majority inconsistency discussed in section 2 above. To see this, notice that the judgment sets of the three individuals in the example of Condorcet's paradox are as shown in Table 5. Given these individual judgments, the majority judgments are indeed inconsistent, as the set of propositions accepted by a majority is inconsistent relative to the rationality condition of transitivity.

Table 5. Condorcet's paradox translated into jugdment aggregation

x is preferable to yy is preferable to zx is preferable to zIndividual 1 (x ≻ y ≻ z)TrueTrueTrueIndividual 2 (y ≻ z ≻ x)FalseTrueFalseIndividual 3 (z ≻ x ≻ y)TrueFalseFalseMajorityTrueTrueFalse

More generally, it can be shown that, when there are three or more alternatives, the agenda just defined has all the complexity properties introduced in the discussion of the impossibility theorems above (i.e., non-simplicity, even-number-negatability, and total blockedness / path-connectedness), and thus those theorems apply to the case of preference aggregation. In particular, the only aggregation rules satisfying universal domain, collective rationality, independence and unanimity preservation are dictatorships [Dietrich and List, 2007; Dokow and Holzman, forthcoming]; for a similar result with an additional monotonicity condition, see [Nehring, 2003]. This is precisely Arrow's classic impossibility theorem for strict preferences: the conditions of universal domain and collective rationality correspond to Arrow's equally named conditions, independence corresponds to Arrow's so-called ‘independence of irrelevant alternatives’, and unanimity preservation, finally, corresponds to Arrow's ‘weak Pareto principle’.

1.3 Consequences of FPTP

The great virtue of FPTP is its simplicity. Voters make a single choice and the candidate with the most votes wins. However, FPTP has fewer desirable properties than many other methods in multicandidate elections. Compared to other rules, plurality is the least likely to pick a Condorcet winner, i.e. a candidate who is able to beat all other candidates in pairwise contests. Also, simulations show that the plurality rule results in average social utilities that are lower than those under other rules (Merrill 1988). Such drawbacks may not be very consequential when there are only two candidates. Furthermore, when there are only two options, as in a two-party system, there is no need for more complex electoral systems.

When three or more candidates compete in a FPTP election, some of the candidates may divide the votes of some segment that might otherwise agree on a winning candidate. This could result in a victorious candidate who is less preferred by the electorate than one of the losers. If voters are aware of this eventuality they may be able to coordinate in order to elect a more preferred candidate. FPTP encourages voters to cast their ballots strategically and consider the behavior of other voters. If a voter's first-choice candidate has little chance of getting a plurality of votes, the voter may find it advantageous to vote for a less preferred but more viable candidate instead of ‘wasting’ his or her vote on a candidate with little chance. In contrast, multi-member districts and even single-member districts with decision rules other than plurality may reduce the incentives for voters to deviate from their first choice.

Strategic behavior and an aversion to wasting one's vote means that FPTP elections are typified by contests between two major candidates, whereas alternative systems often exhibit several viable candidates. Extending this result to parties, FPTP systems routinely have only two major parties, a phenomenon that has been coined Duverger's law, named after the French scholar who extensively described the relationship (Duverger 1954). Proportional representation systems, however, usually have more than two parties, a result that has come to be known as Duverger's hypothesis (Riker 1982, Cox 1997).

When there are only two viable candidates in an FPTP election, no abstentions, and the candidates are choosing their positions to maximize their electoral support, the candidate ideologically closest to the median voter is advantaged (Downs 1957). FPTP systems thus tend to produce more moderate outcomes than alternative systems. The incentive for candidates to take moderate positions can produce candidates who are only minimally differentiated, leading some to conclude that the choices in FPTP are often of little consequence. One potential advantage to such a system, however, is that the outcomes from one election to the next tend to be more ideologically consistent, resulting in small, incremental policy changes over time.

Generalized Median Social Choice Function

A social choice function C:AI×SPn→Xis a generalized median social welfare function if there exists a profile RP=(Rn+1P,…,R2n−1P)∈SPn−1such that for all A∈AIand all R∈SPn,

C(A,R)=PrA  median{π(R1),…,π(Rn),π(Rn+1P),…,π(R2n−1P)}.

As in the construction of a generalized median social welfare function in Example 1, fixed single-peaked preferences for n − 1 phantom individuals are specified. For each profile R∈SPn, the median of the peaks of the 2n − 1 real and phantom individuals is determined. For any agenda A∈AI, the choice set C(A,R)is the projection of this median peak to A. Equivalently, C(A,R)is the best alternative in A for the individual (either real or phantom) with the median peak. It is straightforward to verify that C(A,R)can also be determined by first projecting all 2n − 1 peaks onto A and then computing the median of these projected peaks.

Recall from Section 3 that the binary relation ≤(resp. ≥) on X declares x to be weakly preferred to y if and only if x≤y(resp. x ≥ y). If n is odd, half of the phantoms have the preference ≤, and the other half have the preference ≥, then the phantom individuals are irrelevant and C(A,R)is obtained by maximizing the preference of the (real) individual with the median peak. If there are n − k phantoms with preference ≤ and k − 1 with preference ≥, C(A,R)maximizes the preference of the individual with the kth smallest peak.

Suppose the C is a generalized median social choice function. Because the number of phantom individuals is less than the number of real individuals, the median peak in (R,RP)must lie in the interval defined by the smallest and largest peaks in R. As a consequence, C satisfies SP.97 For a fixed profile, the choices from different agendas are determined by maximizing the same preference, so C satisfies ACA. If the profiles R1 and R2 coincide on the agenda A, then the projections of the individual peaks coincide as well. Hence, C satisfies IIF. It is clear that a generalized median social choice function also satisfies ANON and IC. Moulin (1984) has shown that these five axioms characterize the class of generalized median social choice functions. To facilitate the comparison of this result with the other theorems in Part II, we state Moulin's theorem as a theorem about social choice correspondences.

By Lemma 5, SV, ACA, and IC imply that a social choice correspondence C:AI×SPn→Xcan be rationalized by a social welfare function F:SPn→SP. For k=1,…,n−1, let Rkbe the profile in which Rik=≤for i=1,…,kand Rik=≥for i=k+1,…,nand let βk=π(F(Rk)). The sufficiency part of the proof of Theorem 25 involves showing that C is the generalized median social choice function defined by the profile of phantom preferences R¯P=(F(R1),…,F(Rn−1)). To do this, it is sufficient to show that for all R∈SPn, π(F(R))= median{π(R1),…,π(Rn),β1,…,βn−1}. See Moulin (1984) for the details of the proof.

Moulin (1984) has also considered the domains of single-plateaued and quasiconcave preferences. A preference R on X=[x¯,x¯]is single-plateaued if there exist β1,β2∈X(not necessarily distinct) such that (i) xPy whenever β1≥x>yor β2≤x<yand (ii) xIy whenever x,y∈[β1,β2]. A preference R on X=[x¯,x¯]is quasiconcave if there exists a β∈Xsuch that xRy whenever β≥x>yor β≤x<y.98 A single-peaked preference is single-plateaued and a single-plateaued preference is quasiconcave. Moulin has shown that a version of Theorem 25 holds for single-plateaued preferences and that his axioms are incompatible when the preference domain includes all profiles of quasiconcave preferences.

For the domain of Theorem 25, Ehlers (2001) has considered the problem of choosing exactly m alternatives from each agenda, where m < n(so that it is not possible always to pick everyone's preferred alternative). Thus, a social alternative consists of m(not necessarily distinct) points in X. Preferences need to be extended from X to the set of subsets of X of cardinality at most m. Ehlers assumes that each individual orders subsets by comparing his or her most-preferrred alternatives in these sets. For m = 2, he has shown that the only social choice correspondence satisfying ACA, IIF, SP, and IC is the extreme peaks social choice correspondence. For each profile, this solution identifies the individuals with the smallest and largest peaks and then maximizes their preferences on each agenda. For m > 2, Ehlers has shown that SP and IC are incompatible.

The civic returns to educational attainment

Civic engagement refers to a diverse set of behaviors, attitudes, and knowledge that constitute effective citizenship.

For example, one particularly prominent dimension of civic engagement involves the allocation of time: participation in voting, volunteering, membership in civic organizations, and engagement with elected representatives. A second key component of civic engagement is a belief in the validity and desirability of democratic, pluralistic institutions, and related values such as tolerance and respect. A third dimension involves having both the requisite cognitive skills and an awareness of current events that make an informed deliberation on complex social and technological issues possible.

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