3.1 Arrow s Theorem. We study now the general case in which the society has to choose among a number of alternatives

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1 3.- Social Choice and Welfare Economics 3.1 Arrow s Theorem We study now the general case in which the society has to choose among a number of alternatives Let R denote the set of all preference relations on X that are complete, reflexive, and transitive Let P be the set of linear orders in R, that is, P = { R x y x = y} Definition. Social Welfare Aggregator A Social welfare aggregator defined on A R I is a rule F : A R that assigns a preference relation R to every possible profile ( 1,..., I ) A Example: F : P I R Notation: In general, ( 1,..., I ) A F ( 1,..., I ) R IDEA - Microeconomics II

2 Properties [P] Pareto A Social welfare aggregator F satisfies the Pareto property if {x, y} X and ( 1,..., I ) A we have that i I x i y x y [PI] Pairwise Independence A Social welfare aggregator F satisfies the Pairwise Independence property if the social preference between any two alternatives depends only on the individual preferences between these two alternatives, that is, {x, y} X and ( 1,..., I ), ( ˆ 1,..., ˆ I ) A such that i I x i y x ˆ i y and y i x y ˆ i x then x y x ˆ y and y x y ˆ x IDEA - Microeconomics II

3 [ND] Non Dictatorship A Social welfare aggregator F satisfies the Non Dictatorship property if there is no agent h I such that {x, y} X and ( 1,..., I ) A x h y x y IDEA - Microeconomics II

4 Example: Borda Count Consider n alternatives, X = {x 1, x 2,..., x n }, and (for simplicity) assume that all preferences are linear orders: i P i I. The Borda Count social welfare aggregator consists of: (i) Each individual assigns n points to the most preferred alternative n 1 points to the second preferred alternative... 1 point to the less preferred alternative Let c i (x) be the points assigned to alternative x X by individual i I (ii) The Social aggregator assigns a social score to each alternative x X according to the following rule c(x) = i I c i (x) (iii) Alternatives in X are socially ordered according to the scores c(x) IDEA - Microeconomics II

5 Ex: X = {x, y, z}, I = {1, 2, 3} X Society x y z Hence, y x z It is clear that the Borda count satisfies the [ND] property Also, it is easy to verify that the Borda count satisfies the [P] property. Indeed, {x, y} X such that x i y i I then c i (x) > c i (y) i I and, hence, i I ci (x) > i I ci (y). Therefore, x y The Borda count, though, does not satisfy the [PI] condition IDEA - Microeconomics II

6 Ex: X = {x, y, z}, I = {1, 2} (a) Suppose preferences i are such that x 1 y 1 z and y 2 z 2 x X 1 2 Society x y z Hence, y x z IDEA - Microeconomics II

7 Ex: (continued) (b) Suppose now that preferences change to ˆ i, where x ˆ 1 z ˆ 1 y and y ˆ 2 x ˆ 2 z X 1 2 Society x y z Now, x ˆ y ˆ z So, the social preference between x and y has reversed yet x 1 y and x ˆ 1 y y 2 x and y ˆ 2 x IDEA - Microeconomics II

8 Nevertheless, the [PI] condition (besides being intuitively reasonable) has the additional appealing that if it were true then we could proceed by steps to determine the social preference by mean of iterative pairwise comparisons between alternatives. This could be helpful because of the good properties of the Majority voting rule. Unfortunately, this is not possible Example: The Condorcet Paradox Let X = {x, y, z}, preferences I = {1, 2, 3} and consider the x 1 y 1 z y 2 z 2 x z 3 x 3 y By majority voting between pairs of alternatives we would have {x, y} x y {y, x} y z {x, z} z x Therefore, the iterated majority voting between pairs of alternatives does not always result in a transitive social preference relation (as in this example) IDEA - Microeconomics II

9 Arrow s Impossibility Theorem Definition. Satisfactory Social Welfare Aggregators F A = {F : A R F satisfies [P], [PI], and [ND]} Theorem Arrow s Impossibility Theorem Let #X > 2. Then, A = R or A = P F A = IDEA - Microeconomics II

10 Definitions Given F : A R, we say that a coalition S I is (i) Decisive for x over y if i S i / S x i y y i x } x y (ii) Decisive if for any {x, y} X, S is decisive for x over y (iii) Completely decisive for x over y if i S x i y x y IDEA - Microeconomics II

11 Proof of Arrow s Theorem Proof. (1) If for some {x, y} X, S I is decisive for x over y, then z x and z y S is decisive for x over z and S is decisive for z over y Assume z y. Consider the profile ( 1,..., I ) A such that x i y i z i S y i z i x i / S Then, S decisive for x over y x y F satisfies [P] y z } x z Hence, by [PI] we have that S is decisive for x over z (Similarly, if z y, we can prove that S is decisive for z over y) IDEA - Microeconomics II

12 (2) If for some {x, y} X, S I is decisive for x over y and z X is a third alternative (z x, z y), then S is decisive for z over w and S is decisive for w over z, where w z From (1) we know that S is decisive for x over z and for z over y Apply (1) to the pair {x, z} and the alternative w to conclude that S is decisive for w over z Apply (1) to the pair {z, y} and the alternative w to conclude that S is decisive for z over w (3) If for some {x, y} X, S I is decisive for x over y, then S is decisive Consider any pair {v, w} X and recall that, besides x and y, there is at least one more alternative z in X (i) If v = z or w = z, then step (2) can be applied directly to conclude that S is decisive for v over w (ii) If v z and w z then, by (2), S is decisive for z over w. We can then apply (1) to the pair {z, w} and the alternative v to conclude that S is decisive for v over w IDEA - Microeconomics II

13 (4) If S I and T I are decisive, then S T is decisive Let {x, y, z} X. Consider the profile ( 1,..., I ) A such that z i y i x i S \ (S T ) x i z i y i S T y i x i z i T \(S T ) y i z i x i I\(S T ) Then, S = (S\(S T )) (S T ) is decisive T = (T \(S T )) (S T ) is decisive z y x z Hence, by Transitivity, x y. Finally, by [PI], we conclude that S T is decisive for x over y. Therefore, by (3), we have that (S T ) is decisive IDEA - Microeconomics II

14 (5) For any S I, either S or I\S is decisive Let {x, y, z} X. Consider the profile ( 1,..., I ) A such that x i z i y y i x i z i S i / S (i) If x y then, by [PI], we have that S is decisive for x over y and, because of (3) we conclude that S is decisive. (ii) If y x then, since by [P] x z, we have that y z (by transitivity). Then, by [PI], we have that I\S is decisive for y over z and, because of (3) we conclude that I\S is decisive. (6) If S is decisive and S T then T is decisive Suppose T is not decisive. Then, because of (5), I\T is decisive and, therefore, S (I\T ) is decisive because of (4). But if S T, S (I\T ) =, and can not be decisive because of [P]. Hence, T must be decisive. IDEA - Microeconomics II

15 (7) If S is decisive and #S > 1 then S S, (S S) such that S is decisive Take any h S (i) If S\{h} is decisive we are done. (ii) If S\{h} is not decisive then, because of (5), I\(S\{h}) = (I\S) {h} is decisive. Hence, because of (4) {h} = S ((I\S) {h}) is decisive. (8) h I such that S = {h} is decisive Iterate (7) starting with S = I taking into account that, by [P], I is decisive and that I is finite IDEA - Microeconomics II