1 EC9A4 Social Choice and Voting Lecture 2 EC9A4 Social Choice and Voting Lecture 2 Prof. Francesco Squintani

2 Summary from previous lecture We have defined the general set up of the social choice problem. We have shown that majority voting is particularly valuable to choose between two alternatives. We have proved Arrow’s theorem: the only transitive complete social rule satisfying weak Pareto, IIA and unrestricted domain is a dictatorial rule.

3 Preview of the lecture We will extend Arrow’ theorem to social choice functions. We will introduce the possibility of interpersonal comparisons of utility. We describe different concept of social welfare: the utilitarian Arrowian representation and the maximin Rawlsian representation.

4 Social Choice Functions The Arrow theorem concerns the impossibility to find a social preference order that satisfies minimal requirements. But the object of social choice may be restricted to find simply a socially optimal alternative rather than to rank all possible alternatives. We will show that a similar impossibility result holds when considering social choice functions.

5 Let R be a subset of the set of all profile of preferences R = R(1),…R(N). A social choice function f defined on R assigns a single element to every profile R in R. The function f is monotonic if, for any R and R’ s.t. the alternative x = f (R) maintains its position from R to R’ we have that x = f (R’). The alternative x maintains its position from R to R’, if x R(i) y implies x R’(i) y for all i and y.

6 The social choice function f is weakly Paretian if y = f (R) whenever there is x s.t. x R(i) y for all i. An individual n is dictatorial for f if, for every R f (R)  {x : x R(n) y for all y in X}. Theorem. Suppose that there are at least three distinct alternatives. Then any weakly Paretian and monotonic social function defined on the whole domain of preferences is dictatorial.

7 Proof. The proof shows the result as a corollary of Arrow impossibility theorem. We derive a social rule F(R) from f, using the properties of f on all R. We show that F satisfies all Arrow axioms but ND. Step 0. Given a subset X’ of X and R, we say that R’ takes X’ to the top from R, if for every i, x P’(i) y for all x in X’ and y not in X’; x P’(i) y for all x in X’ and y not in X’; x R(i) y if and only if x R’(i) y for all x, y in X’. x R(i) y if and only if x R’(i) y for all x, y in X’.

8 Step 1. If both R’ and R’’ take X’ to the top from R, then f(R’) = f(R’’). In fact, by WP, f(R’) is in X’. Because f(R’) maintains its position from R’ to R’’, f(R’) = f(R’’) by monotonicity. Step 2. For every R, we let `x F(R) y’ if x = f(R’) when R’ is any profile that takes {x,y} to the top of profile R. By Step 1, F is well defined.

9 Step 3. Because f is weakly Paretian, if R’ takes {x,y} to the top from R, then f(R) in {x,y}. By step 1 and 2, either x F(R) y or y F(R) x (but not both). Hence F is complete. Step 4. Suppose that x F(R) y and y F(R) z. If R’’ takes {x,y,z} to the top of R, then F(R’’) in {x,y,z}, because f is weakly Paretian. Say by contradiction that y = f(R’’). Consider R’ that takes {x,y} to the top from R’’. By monotonicity, f(R’) = y.

10 But R’ also takes {x,y} to the top from R. This contradicts x F(R) y. In sum, y = f (R’’). Similarly z = f (R’’), by contradiction with y F(R) z. Thus x = f (R’’). Let R’ take {x,z} to the top from R’’. By monotonicity, x = f (R’). But R’ also takes {x,z} to the top from R, and hence x F(R) z. Hence F(R) is transitive.

11 Step 5. The social rule F(.) satisfies Weak Pareto. In fact, if x R(i) y for all i, then x = f (R’) whenever R’ takes {x,y} to the top from R. Thus x F(R) y, by step 1. Step 6. The social rule F(.) satisfies IIA. In fact, if R and R’ have the same ordering for x,y, and R’’ takes {x,y} to the top of R, then it also takes {x,y} to the top of R’. Hence, x = f (R’’) implies: x F(R) y and x F(R’) y.

12 Step 7. The social choice function f is dictatorial. By Arrow theorem, there is an agent i such that for every profile R, we have x F(R) y when x R(i)y. Thus if x R(i) y for all y, then x = f(R). Thus if x R(i) y for all y, then x = f(R). This final step concludes the proof. We have shown how to construct the social rule F(R) starting from the social choice function f. Applying Arrow’s theorem, we have shown that there is a dictator for F(R), and so f is dictatorial.

13 Interpersonal comparisons We will assume that all social choice functions f satisfy welfarism, i.e. U, P and IIA, and continuity. Hence there is a continuous function W such that: V(x) > V(y) if and only if W(u(x)) > W(u(y)). The welfare function depends only on the utility ranking, not on how the ranking comes about.

14 Under Arrow axiom, utilities are measured along an ordinal scale, and are non-comparable across individuals. Specifically, the function W aggregates the preferences (u i ) i=1,…,n if and only if W aggregates the preferences (v(u i )) i=1,…,n, for all increasing transformation v i (u i ), for any i, independently across i. We modify the framework to allow for cardinal comparisons of utility, and comparability across individuals.

15 Suppose that preferences are fully comparable but measured on the ordinal scale. The social ranking V must be invariant to arbitrary, but common, increasing transformations v i applied to every individual’s utility function u i. Specifically, the function W aggregates the preferences (u i ) i=1,…,n if and only if W aggregates the preferences (v(u i )) i=1,…,n, for all increasing transformation v i (.), such that v’ i is constant across i.

16 Suppose that preferences are fully comparable and measured on the cardinal scale. The social ranking V must be invariant to increasing, linear transformations v(u i )= a i +bu i, where b is common to every individual. Specifically, the function W aggregates the preferences (u i ) i=1,…,n if and only if W aggregates the preferences (v(u i )) i=1,…,n, for all transformation v i (.), such that v i (u i )= a i +bu i, with b>0.

17 HE: Let u and u’ be two distinct utility vectors. Suppose that u k = u’ k for all k other than i and j. If u i > u’ i > u’ j > u j, then W(u’) > W(u). Condition HE states that the society has a preference towards decreasing the dispersion of utilities across individuals. Rawlsian Form

18 AN: The social rule W is anonymous if for every permutation p, W(u 1, …, u N )=F(u p(1),…u p(N) ). Theorem: Suppose that preferences are fully comparable and measured on the ordinal scale. The social welfare function W satisfies Weak Pareto, Anonymity and Hammond equality if and only if it takes the Rawlsian form W(u)=min{u 1, …, u N }.

19 Proof. It is easy to see that the function W(u)=min{u 1, …, u N } satisfies Weak Pareto, Anonymity and Hammond equality. To show the converse, we will see only the case for N=2. Consider a utility index u, with u 1 > u 2 u1u1 u2u2 u u* I II III IV

20 By anonymity, the utility profile By anonymity, the utility profile u* must be ranked in the same way as u: hence W(u) = W(u*). By Weak Pareto, all u’ such that u’ > u or u’ > u* must be such that W(u’) > W(u). Hence the whole area in blue is s.t. W(u’) > W(u). u1u1 u2u2 u u* I II III IV

21 By Weak Pareto, all u’ such that u’ < u or u’ < u* must be such that W(u’) < W(u). Hence the whole area in green is such that W(u’) < W(u). u1u1 u2u2 u u* I II III IV

22 Pick a point u’ in zone III. To be in III, it must be that u 2 < u’ 2 < u’ 1 < u 1 Every linear transform v such that v i (u i )=u i yields: u 2 < v 2 (u’ 2 ) < v 1 (u’ 1 ) < u 1 This concludes that all points in III are ranked the same way wrt to u. u1u1 u2u2 u u* I II III IV u’

23 To be in III, it must be that u 2 < u’ 2 < u’ 1 < u 1 Hammond equality implies that W(u’) > W(u). u1u1 u2u2 u u* I II III IV u’

24 By anonymity, the ranking of each u’ in III relative to u must be the same as the ranking of any utility vector u’’ in II: W(u’’)>W(u). u1u1 u2u2 u u* I II III IV u’

25 Any linear transform v such that v 1 (u’ 1 ) = u 1, v 2 (u’ 2 ) = u 2, yields: W (v(u)) < W (v 1 (u’ 1 ), v 2 (u’ 2 )) = W (u). Hence all the utility vectors u’’ in IV are ranked opposite to all utility vectors u’ in III, relative to u. u1u1 u2u2 u u* I II III IV u’ u’’

26 Hence W(u) > W(u’’) for all u’’ in IV, and, by anonymity, W(u) > W(u’’) for all u’’ in I. u1u1 u2u2 u I II III IV u’’

27 We conclude that V(II) and V(III) > W(u) > V(I) and V(IV). We are left to consider the boundaries of these sets. u1u1 u2u2 u u* I II III IV u’

28 Because W is continuous, the boundaries opposite to each other, relative to u must be indifferent to u’. Thus, the boundaries between II and III and the blue set must be better than u in W terms. The boundaries between I and IV and the green set must be indifferent to u. u1u1 u2u2 u u* I II III IV u’

29 We have obtained the Rawlsian indifference curves, where W (u) = min {u 1, u 2 }. u1u1 u2u2 u u* II III IV u

30 Utilitarian Form Theorem: Suppose that preferences are fully comparable and measured on the cardinal scale. The social welfare function W satisfies Weak Pareto and Anonymity if and only if it takes the utilitarian form: W(u)= u 1 +… + u N. W(u)= u 1 +… + u N.

31 Proof. It is easy to see that the function W(u)= u 1 + …+ u N, satisfies Weak Pareto and Anonymity. To show the converse, we will see only the case for N=2. Consider a utility index u, with u 1 = u 2 u1u1 u2u2 u’ u* u

32 Define the constant k = u 1 + u 2. Consider the locus k(u) = {(u 1, u 2 ) : k = u 1 + u 2 }. For any vector u’ on k(u), the vector u* such that u* = (u’ 2, u’ 1 ) is also on k(u). By Anonymity, W(u’)=W(u*). u1u1 u2u2 u’ u* u

33 Suppose now that W(u) > W(u’). Under CS/IC, this ranking must be invariant to transformation v i (u i ) = a i + bu i u1u1 u2u2 u’ u* u

34 Let v i (u i ) = (u i - u’ i )+ u i for i=1,2. Hence, (v 1 (u’ 1 ), v i (u’ i )) = u and (v 1 (u 1 ), v i (u i )) = u*. If W(u) > W(u’), then W(u*) > W(u), which contradicts W(u’) = W(u*). u1u1 u2u2 u’ u* u

35 If W(u) < W(u’), then W(u*) < W(u), which contradicts W(u’) = W(u*). Hence we conclude that W(u)=W(u’) for all vectors u’ on k(u). u1u1 u2u2 u’ u* u

36 By Weak Pareto, each vector u’’ to the north-east of a vector u’ on k(u) is strictly preferred to u. Thus W(u’’) > W(u) for u’’ such that u’’ 1 + u’’ 2 > u 1 + u 2. u1u1 u2u2 u

37 Similarly, W(u’’) < W(u) for u’’ such that u’’ 1 + u’’ 2 < u 1 + u 2 u1u1 u2u2 u

38 We concluded that the indifference curve of any vector u is k(u) = {(u 1, u 2 ) : k = u 1 + u 2 }. Hence W(u) = u 1 + u 2 Indifference curves are straight lines of slope -1. u1u1 u2u2 u u

39 If we drop the axiom of Anonymity, the full range of generalized utilitarian orderings is allowed. These are the linear social welfare functions W(u)= a 1 u 1 +…+ a N u N, with a i > 0 for all i, and a i > 0 for some i. Indifference curves are lines of negative slope. u1u1 u2u2 u

40 The maximin Rawlsian form and the utilitarian form both belong to constant elasticity class with the formula: W = ( u 1  +… + u N  )  where 0 =  and  is the constant elasticity of social substitution between any pair of individuals. As  approaches  W approaches the utilitarian form As  approaches –infinity,  W approaches the Rawlsian form.

41 A Theory of Justice Behind a ``veil of ignorance’’, an individual does not know which position she will take in a society. Will she be rich or poor, successful or unsuccessful? If she assigns equal probability to any of the possible economic and social identities that exist in the society, a rational evaluation would

42 evaluate welfare according to the expected utility [u 1 (x) + … + u N (x) ]/N This is equivalent to adopt the utilitarian criterion: W(u(x)) = u 1 (x) + … + u N (x). But the approach is also consistent with every CES form, embodying different degrees of risk aversion. Consider the positive transformation v i (x) = - u i (x) -a with a>0.

43 Suppose that u i (x) represents utility over social states “with certainty,” whereas v i (x) represents utility over social states “with uncertainty.” In the form v i (x) = - u i (x) -a, a>0 represents the degree of risk aversion. Suppose that the social welfare function is given by the expected utility: W = [v 1 (x) +…+ v N (x)]/N = [-u 1 (x) -a - …-u N (x) -a ]/N = [-u 1 (x) -a - …-u N (x) -a ]/N

44 Because the monotonic transformation W = ( -u 1 (x) -a - …-u N (x) -a ) -1/a equivalently represents welfare, we obtain that any CES form is compatible with the expected utility formulation behind a veil of ignorance. The extreme risk aversion case of CES is W = min {u 1 (x), …, u N (x)}, the Rawlsian form that describes a social concern for the agent with the lowest utility.

45 Conclusion We have extended Arrow’ theorem to social choice functions. We have introduced the possibility of interpersonal comparisons of utility. We have described different concept of social welfare: the utilitarian Arrowian representation and the maximin Rawlsian representation.

46 Preview of the next lecture We will introduce single-peaked utilities. We will prove Black’s theorem: Majority voting is socially fair when utilities are single-peaked. We will prove Median Voter Convergence in the Downsian model of elections. We will introduce the probabilistic voting model.