Mathematics and Democracy

Transcription

1 Mathematics and Democracy A. Slinko What is democracy? Democracy is a mechanism of creating the collective wisdom of a society from individual wisdoms of members of that society. However, starting from the famous Arrow s impossibility theorem that there are no reasonably good voting systems, some recent results shows that there are severe theoretical limitations for such a collective wisdom and even the mere possibility of it is somewhat doubtful. In this paper we illustrate these difficulties on a simple model of some hypothetical parliament of some hypothetical country. Let us outline the basic assumptions of this model. The main object of consideration will be political programs which are viewed as packages of bills related to at least two independent issues; it could be, for example, internal and external policy. To measure them we need at least one separate dimension for each issue. Therefore we suggest that: 1. Programs of the government can be characterized by points of some Euclidean space E; 2. Political views of deputies are their notions of an ideal program, and can be also represented by points of E. The space E we shall suppose to be a plane (in fact, the following results do not depend on the dimension of E) and call the Plane of Political Views. In this case the abscissa of a point of E can represent the value of liberalismconservatism of the program (or deputy) in internal policy while the ordinate of that same point can represent the corresponding value in external policy. We assume that the deputy with political views A prefers a program X to a program Y, if X is closer to A, than Y, according to the Euclidean distance. To complete the model description, let us also introduce two more conditions which are not so essential but simplify presentation: 3. The total number of deputies is odd; 1

2 4. Different deputies have different views. Let us begin with a question: whether there exists a political program, which reflects political views of deputies in the best possible way, in the sense that deputies will vote in favor of it against any other program? We shall call such a program (if it exists) the Ideal Political Program. Having offered the Ideal Political Program, the government has no chance to change it in future. Theorem 1:(Plott) The Ideal Political Program exists if and only if there exist a deputy O and a partition of other deputies into pairs A i, B i such that every segment [A i, B i ] contains the point O (See Fig.1). A 1 B 4 A 3 O B 2 B 3 A 2 A 4 B 1 To prove this theorem, we introduce a notion of a median line. This is a straight line such that each of the two closed half-planes, defined by this line, contains more than half of the political views of the deputies. Note that any median line necessarily contains the political views of at least one deputy. Lemma 1: In every direction in the plane E there exists a unique median line. Proof: Let n = 2k + 1 be the total number of deputies. Let us draw a straight line with a given direction, far enough from the points, so that all political views lie in one of the half-planes defined by this straight line. Then let us begin to move it toward the points, keeping the direction unchanged. At the outset the number of points in one half-plane was equal to zero and then it began to grow. Let us stop the line at the moment when the number of the points in the half-plane mentioned, including the points on the line itself, exceeds k for the first time. Note that at this moment the number of the points in the other half-plane, also including the points on the line, exceeds k too. Therefore we obtain at this moment a median line. 2

3 Lemma 2: If at least one median line m does not pass through a point X, then the point X is not an Ideal Political Program. Proof: Let us denote by Y the foot of the perpendicular, drawn from X to m (Fig.2). X m Y The program Y will defeat X, being put on vote, since all deputies on the other side of m will vote in favor of Y and so will do those on the line m. And this is more than a half of all voters. We can now prove the theorem. If O is an Ideal Political Program, then by Lemma 2 all median lines intersect at this point. Conversely, by Lemma 1 we can conclude that every straight line, passing through O, is median. Let m be a line, passing through O and containing points A 1,..., A p, B 1,..., B q, that represent views of those deputies, lying on m as shown in Fig.3. B 1 B q A p O A 1 Let us prove that p = q; this will mean just that the points can be divided into pairs as required. Suppose that one of the half-planes, defined by m, contains x political views while other contains y (views on m are excluded). Let us rotate the line m about the point O through a very small angle, such that this line does not meet any political views during the rotation. We obtain a new median 3

4 line, that contains the point O as the only political view on it. We have x + q = y + p = k. If we rotate m in the opposite direction, we obtain x + p = y + q = k. It follows from the two obtained equalities that p = q. Suppose now that the set of views of all deputies except for the point O can be divided into a set of pairs A i, B i as indicated in the theorem. Then for each program X different from O, in view of the Triangle Inequality, we have A i X + XB i A i O + OB i. Hence either A i or B i prefers the program O to the program X. At last, O himself prefers O and the majority of votes is ensured to O. This proves the theorem. Corollary 1: Suppose that the Ideal Political Program O exists. Then X will defeat Y if and only if the deputy O prefers X to Y. Proof: Suppose that X defeats Y. Then there exists a pair A i, B i, such that both deputies of this pair prefer X. Hence, X lies in the intersection of the circles with centers A and B and radii A i Y and B i Y, respectively (Fig.4). Y A i X O B i But this figure is situated in the circle of radius OY with the center O, and this means that OX < OY. Conversely, if OX < OY, then for each pair A i, B i we have A i X + XB i < A i Y + Y B i, and at least one deputy of this pair will vote in favor of X. Therefore, X defeats Y. What a remarkable deputy corresponds to the point O! In the parliament, torn apart, this member votes exactly as the parliament as a whole. He is a real Father of Peoples. But this extraordinary situation is unstable, because the slightest move of each deputy breaks the harmony, which can be maintained by force only. 4

5 The fact, that the parliament prefers X to Y, we shall denote by X Y or Y X. Corollary 2: If the Ideal Political Program exists, then X Y and Y Z imply X Z. The property of relation, stated in the Corollary, is known as transitivity. Let us consider now the general case. By Lemma 1 no median lines are parallel. Therefore a characteristic feature of the general case is the presence of at least three median lines m 1, m 2, m 3, that have pairwise intersections but have no point in common. These three lines determine a triangle (Fig.5). r I m 1 m 3 m 2 Let I be the incenter of this triangle and r be the inradius. Lemma 3: For each political program X there exists a political program Y that defeats X and IY 2 > IX 2 + r 2. Proof: Since the three strips shaded on Fig.6 5

6 I m 1 m 3 m 2 have only one point P in common, then for at least one median line of the three, let it be m 1, for example, the points X and I are in the same half-plane of E, defined by m 1, and the distance from X to m 1 is not less than that of from I to m 1. X I M m 1 Let us denote a = XM, and draw a perpendicular to m 1, passing through X, and extend it beyond this line to the distance equal to a. Let us denote the endpoint of this segment by Y. Then the distance from Y to m 1 is less than the distance from X to m 1, and hence the program Y defeats X. On the other hand, IY 2 IX 2 = Y M 2 MX 2 = (r + a) 2 a 2 = r 2 + 2ra > r 2. This proves the lemma. This surprising lemma shows that political programs can defeat one another, moving away farther and farther from political views of deputies, as in due course they can leave every circle with I as a centre. 6 Y

7 Theorem 2:(McKelvey) For every two points X and Y of the plane E of political views there exist points Z 1,..., Z n such that X Z 1..., Z n Y. Proof: Let R be the least possible radius of a circle with the centre I, which contains the point Y and all political views of the deputies. Using Lemma 3 we can choose the sequence of points Z 1,..., Z n such that X Z 1..., Z n and IZ n > 3R. But then it is obvious that Z n Y, and the theorem is proved. If we set Y = X in this theorem, we shall obtain a cyclic trajectory of political programs in the sense that X Z 1..., Z n X. In particular, this shows that in the absence of the Ideal Political Program the relation is not transitive. Making use of this intransitivity the government can in due course gain approval for every political program if, of course, it firmly knows what it wants. The situation is also possible when the government, crazy with problems, is wandering around with constant approval of the parliament. Only the president can help here by dissolving such a parliament and the government created by it. Maybe the reader has a tiny hope that the intransitivity of the relation can be avoided, by approving new programs by a majority of 2/3 of all votes or another way. Let us dot all the i s. Imagine yourself a worker s body of 100 workers with wages 1, 2, 3, dollars an hour, respectively. Let us put to the vote the following motion: would it be better to add 1 dollar to the wages of each of the first 99 workers and subtract 99 dollars from the wage of the last? Certainly, it will be accepted, as many times as it will be put to the vote, with 99 percent of votes in the favor the motion. Again we see cyclic trajectories and, hence, intransitivity. REFERENCES 1. Plott C.R. A notion of the equilibrium and its possibility under majority rule, Amer. Econ. Rev., 1967, v.lviii, n.4, p McKelvey R.D. Intransitivities in multidimensional voting models and some implications for agenda control, J. Econ. Theory, 1976, v.12, n.3, p