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Article Excerpt
Summary. In a two candidate election, it might be that a candidate wins in a majority of districts while he gets less vote than his opponent in the whole country. In Social Choice Theory, this situation is known as the compound majority paradox, or the referendum paradox. Although occurrences of such paradoxical results have been observed worldwide in political elections (e.g. United States, United Kingdom, France), no study evaluates theoretically the likelihood of such situations. In this paper, we propose four probability models in order to tackle this issue, for the case where each district has the same population. For a divided electorate, our results prove that the likelihood of this paradox rapidly tends to 20% when the number of districts increases. This probability decreases with the number of states when a candidate receives significatively more vote than his opponent over the whole country.
Keywords and Phrases: Voting, Paradox, US presidential election, Probability.
JEL Classification Numbers: D71.
1 Introduction
The 2000 US presidential elections remind us that voting paradoxes are not only theoretical issues for economists and political scientists; They sometimes happen! With 48.4% of the popular vote, A. Gore could only win 21 States among 51, for a total of 266 electors in the Electoral College, while G.W. Bush get 271 electors with less support from the popular vote (see Table 1).
This paradox is known in Social Choice literature as the referendum paradox (see Nurmi [15]). It may occur each time the decision is not taken directly by the voters by referenda, but through representatives locally elected. Then, the decision taken by the representatives may not reflect the will of the voters. Note that this paradox is related to other voting paradoxes, such as the Ostrogorski paradox (see Ostrogorski [16], Nurmi [15], Laffond and Laine [12], Saari and Sieberg [18]). If the decision is taken through a more complex hierarchy of committees, the same problem can occur (see Galam [7]).
In fact, the occurrence of the referendum paradox has been observed in many democracies. The US presidential elections displayed the paradox in 1824, 1876, 1888 and 2000 (see Leip [13], Saari [17]). Since World War Two, it happened twice in the United Kingdom (see Table 2). This paradox may become frequent in French local elections: since 1992, many aspects of the local public policies are no longer decided directly by the cities themselves, but through communities of cities. For example, we have been able to identify such a paradox in the "Grand Caen" area, which gathers 18 cities in the surrounding of Caen. Though the left parties get a large majority of votes over the whole area and control 13 cities since the 2001 elections, they have been defeated in the main towns. In turns, they only control 26 seats over 70 in the council of the community (see Table 3). As there are now about 90 communities of cities in France, the case of Caen is probably not a unique one.
Thus two-level voting may clearly lead to undesired results. Moreover, one of the main assumption of Public Economy, the fact that majority voting picks out the preferred policy of the median voter when the preferences are single peaked (see Black [1], Downs [5]) is no longer true; The social compromise may be beaten by a more extremist policy in a federal system. Thus, as many local, federal, national and international (1) bodies are committees of representatives designated by the adequate districts, it is of importance to get a better theoretical knowledge about the occurrence of the referendum paradox. This is what we intend to do in this paper, by adapting models used in statistics, physics, and social choice to treat this issue. Of course, our study is quite preliminary and uses some simplifying assumptions. We propose to study this paradox:
-- When there are two parties. Although possible, the study of the case of more than two parties is left for further research.
-- When all the states have the same population size.
-- For a large number of voters in each state and for small population sizes whenever it is possible.
-- From N = 3 states (or constituencies, districts, cities,...) to N = 100. We get exact values for some simple cases and computer estimations otherwise.
-- With 4 different models. The first two are based upon adapted versions of the Impartial Culture and Impartial Anonymous Culture (respectively abbreviated IC and IAC hereafter) assumptions that are used in Social Choice Theory; They assume that each party has equal chance to win. The other two propose ways to introduce a systematic bias in favor of one candidate.
In Section 2, we adapt the two classical models that are generally used in social choice theory, for the case N = 3. In this simple case, we are able to derive rather easily formulas for the computation of the probability of the referendum paradox and can also illustrate the differences between both models with figures. Section 3 displays the figures under the IC and IAC models for the case N > 3 When N = 4orN = 5, we are still able to derive exact formulas, but as the computations become tedious, the details of the computations are presented in the appendices. For all the other cases, we rely on computer estimations in order to evaluate the likelihood of the referendum paradox. We present two new models in Section 4, by taking into account the fact that one party is more likely to win. We discuss our results in Section 5.
2 Adapting the classical models: the case N = 3
In order to present and compare the different models we shall use, we first present them for the case N = 3 states (or groups, districts, constituencies). Let n be the number of voters in each state. We denote by [n.sub.i] the number of voters who, in state i, vote for candidate A, i = 1, 2,..., N. The other voters are assumed to vote for candidate B; There is no abstention. A voting situation is a vector n = ([n.sub.1], [n.sub.2],..., [n.sub.N]) with [less than or equal to] [n.sub.i] [less than or equal to] n. For N = 3, n = ([n.sub.1], [n.sub.2], [n.sub.3]). A conflict between a decision made by a majority of states and a decision made nationwide through a referendum occurs if, for example, States 1 and 2 vote for candidate A, while a majority of voters prefer B. This situation is described by inequalities (2.1), (2.2) and (2.3):
[n.sub.1] > n/2 (2.1)
[n.sub.2] > n/2 (2.2)
[n.sub.1] + [n.sub.2] + [n.sub.3] < 3n/2 (2.3)
There are five other cases leading to a paradox, similar to this one. Thus, we only need to estimate the probability that inequalities (2.1) to (2.3) are met with an adequate probability model describing the behavior of the voters.
2.1 Impartial culture model
The Impartial Culture condition has been introduced in Social Choice literature by Guilbaut [10], for the study of the Condorcet paradox. It assumes that each voter picks his preference randomly among the possible preference types according to an uniform probability distribution. In our case, each voter has a probability 1/2 to cast his vote in favor of candidate A, and a probability 1/2 to cast his vote for candidate B. The distribution of [n.sub.i] is a binomial law; When the number of voters is large in each state, the distribution of [n.sub.i] tends to a normal law, of mean n/2 and variance [sigma] = [square root of n]/2. For each [n.sub.i], i = 1,..., N, let
[x.sub.i] =[1/[sigma]]([n.sub.i] - [n/2])
The Central Limit Theorem implies the following convergence for the density function as n [right arrow] [infinity]:
f([x.sub.i]) [??] [1/[square root of (2[pi])]][e.sup.[-[x.sub.i.sup.2]/2]].
For the three states case, the joint distribution of x = ([x.sub.1], [x.sub.2], [x.sub.3]) as n [right arrow] [infinity] is given by:
f(x) [??] [1/[([square root of (2[pi])])[.sup.3]]][e.sup.[[-|x|[.sup.2]]/2]]
where |x|[.sup.2] = [x.sub.1.sup.2] + [x.sub.2.sup.2] + [x.sub.3.sup.2]. By subtracting or dividing the number of voters by the same constant, the quantities change but the comparison between them is unchanged, therefore one can claim that n satisfies conditions (2.1),(2.2),(2.3) if and only if x satisfies (2.1)', (2.2)' and (2.3)':
[x.sub.1] > (2.1)'
[x.sub.2] > (2.2)'
-[x.sub.1] - [x.sub.2] - [x.sub.3] > (2.3)'
Let [P.sub.IC]([infinity], 3) be the probability of the referendum paradox for three states of population n [right arrow] [infinity] under the IC condition. Thus, [P.sub.IC]([infinity], 3) is equal to six times:
I = [1/[([square root of (2[pi])])[.sup.3]]][[integral].sub.C.sub.1][e.sup.[-|x|[.sup.2]/2]]d[x.sub.1]d[x.sub.2]d[x.sub.3]
where [C.sub.1] = {x [member of] [R.sup.3] : x satisfies (2.1)', (2.2)' and (2.3)'}. We must integrate I in the triangular cone delimited by the three straight lines Oa, Ob and Oc (see Fig. 1). We write as [r.sup.2]d[OMEGA]dr the volume element d[x.sub.1]d[x.sub.2]d[x.sub.3] where d[OMEGA] is the element of solid angle and r = ([x.sub.1.sup.2] + [x.sub.2.sup.2] + [x.sub.3.sup.2])[.sup.1/2] and the integration on r is straightforward. We observe that
[FIGURE 1 OMITTED]
I =[1/[4[pi]]][integral]d[OMEGA]
Hence, computing the desired probability reduces to find the measure of the cone...
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