EC9A4 Social Choice and Voting Lecture 1

1. EC9A4Social Choice and VotingLecture 1 Prof. Francesco Squintani f.squintani@warwick.ac.uk

2. Syllabus 1. Social Preference Orders May's Theorem Arrow Impossibility Theorem (Pref. Orders) 2. Social Choice Functions Arrow Impossibility Theorem (Choice Funct.) Rawlsian Theory of Justice Arrow Theory of Justice

3. 3. Single Peaked Preferences Black's Theorem Downsian Electoral Competition 4. Probabilistic Voting and Ideological Parties 5. Private Polling and Elections Citizen Candidate Models

4. References • Jehle and Reny Ch. 6 • MasColell et al. Ch 21 • Lecture Notes • D. Bernhardt, J. Duggan and F. Squintani (2009): “The Case for Responsible Parties”, American Political Science Review, 103(4): 570-587.

5. D. Bernhardt, J. Duggan and F. Squintani (2009): “Private Polling in Elections and Voters’ Welfare” Journal of Economic Theory, 144(5): 2021-2056. • M. Osborne and A. Slivinski (1996): “A Model of Political Competition with Citizen-Candidates,” Quarterly Journal of Economics, 111(1): 65-96. • T. Besley and S. Coate (1997), “An Economic Model of Representative Democracy,” Quarterly Journal of Economics, 112: 85-114.

6. What is Social Choice? • Normative economics. What is right or wrong, fair or unfair for an economic environment. • Axiomatic approach. Appropriate axioms describing efficiency, and fairness are introduced. Allocations and rules are derived from axioms. • Beyond economics, social choice applies to politics, and sociology.

7. Social Preferences Consider a set of social alternatives X, in a society of N individuals. Each individual i, has preferences over X, described by the binary relation R(i), a subset of X2. The notation `x R(i) y’ means that individual i weakly prefers x to y. Strict preferences P(i) are derived from R(i): `x P(i) y’ corresponds to `x R(i) y but not y R(i) x.’

8. Indifference relations I(i) are derived from R(i): `x I(i) y’ corresponds to `x R(i) y and y R(i) x.’ The relation R(i) is complete: for any x, y in X, either `x R(i) y’ or `y R(i) x’, or both. The relation R(i) is transitive: for any x, y, z in X, if `x R(i) y’ and `y R(i) z’, then `x R(i) z’. Social preference relation: a complete and transitive relation R = f (R (1), …, R(N)) over the set of alternative X, that aggregates the preferences R(i) for all i = 1,…, N, and satisfies appropriate efficiency and fairness axioms.

9. Example: Exchange Economy x12 x21 I1 I2 w x11 x22 In the exchange economy with 2 consumers, and 2 goods, x are such that + = + , where w is the initial endowment, and j is the good. xj1 xj2 wj1 wj2

10. x12 x21 I1 I2 x y w x11 x22 For individual i, `x I(i) y’ if ( , ) and ( , ) are on the same indifference curve Ii The relations R(i) and P(i) are described by the contour sets of the utilities ui x2i x2i y2i y2i

11. x12 x21 I1 I2 x w x11 x22 The line of contracts describes all Pareto optimal allocations. One possible social preference is R such that `x P y’ if and only if x is on the line of contracts, and y is not.

12. The case of two alternatives Suppose that there are only two alternatives: x is the status quo, and y is the alternative. Each individual preference R(i) is indexed as q in {-1, 0, 1}, where 1 is a strict preference for x. The social welfare rule is a functional F(q(1), …, q(N)) in {-1, 0, 1}.

13. May’s Axioms AN: The social rule F is anonymous if for every permutation p, F(q(1), …, q(N))= F(q(p(1)),…q(p(N))) NE: The social rule F is neutral if F(q) = - F (- q ). PR: The rule F is positively responsive if q > q’, q = q’ and F(q’) > 0 imply that F(q) = 1.

14. May’s Theorem A social welfare rule is majoritarian (i.e. F(q) =1 if and only if n+(q) =#{i: q(i) = 1} > n-(q) =#{i: q(i) = - 1} , F(q) =-1 if and only if n+(q) < n-(q) , F(q) =0 if and only if n+(q) = n-(q) ), if and only if it is neutral, anonymous, and positively responsive.

15. Proof. Clearly, majority rule satisfies the 3 axioms. By anonimity, F(q) = G(n+(q), n-(q)). If n+(q) = n-(q), then n+(-q) = n-(-q), and so: F(q)= G(n+(q), n-(q))= G(n+(-q), n-(-q))= =F(-q)=-F(q), by NE. This implies that F(q)=0. If n+(q) > n-(q), pick q’ with q’<q and n+(q’) = n-(q’), Because F(q’) = 0, by PR, it follows that F(q) = 1. When n+(q) < n-(q), it follows that n+(-q) > n-(-q), hence F(-q) = 1 and by NE, F(q) = -1.

16. Transitivity Transitivity is apparently a sound axiom. But it fails for the majority voting rule. Suppose that X={x, y, z}, and `x P(1) y P(1) z’, `y P(2) z P(2) x’ and `z P(3) x P(3) y’. Aggregating preferences by the majority voting rule yields `x P y’, `y P z’, and `z P x’. This is called a Condorcet cycle.

17. Arrow’s Axioms U. Unrestricted Domain. The domain of f must include all possible (R(1), …, R(n)) over X. WP. Weak Pareto Principle. For any x, y in X, if `x P(i) y’ for all i, then `x P y’. ND. Non-Dictatorship. There is no individual i such that for all x,y, if `x P(i) y’ i, then `x P y’, regardless of the relations R(j), for j other than i.

18. IIA. Independence of Irrelevant Alternatives. Let R = f(R(1), …, R(N)), and R’ = f(R’(1),…, R’(N)). For any x,y, if every individual i ranks x and y in the same way under R(i) and R’(i), then the ranking of x and y must be the same under R and R’. This axiom requires some comments. In some sense, it requires that each comparison can be taken without considering the other alternatives at play. The axiom fails in some very reasonable voting rules.

19. Borda Count Suppose that X is a finite set. Let Bi(x) = #{y : x P(i) y}. The Borda rule is: x R y if and only if B1(x) + …+ BN(x) > B1(y) + …+ BN(y). This rule does not satisfy IIA. Suppose 2 agents and {x,y,z} alternatives. x P(1) z P(1) y, y P(2) x P(2) z yields x P y x P*(1) y P*(1) z, y P*(2) z P*(2) x yields y P x

20. Arrow Impossibility Theorem Theorem If there are at least 3 allocations in X, then the axioms of Unrestricted Domain, Weak Pareto and Independence of Irrelevant Alternatives imply the existence of a dictator. Corollary There is no social welfare function f that satisfies all the Arrow axioms for the aggregation of individual preferences.

21. Proof (Geanakoplos 1996). Step 1. Consider any social state c. Suppose that `x P(i) c’ for any x other than c, and for any i. By Weak Pareto, it must be that `x P c’ for all such x. Step 2. In order, move c to the top of the ranking of 1, than of 2, all the way to n. Index these orders as (P1(1),…, P1(N)), … (PN(1),…, PN(N)). By WP, as `c PN(i) x’ for all i and x other than c, it must be that `c PN x’ for all x other than c. The allocation c is at the top of the ranking.

22. Because c is at the top of the ranking P after raising it to the top in all individual i’ s ranking P(i), there must be an individual n such that c raises in P, after raising c to the top in all rankings P(i) for i smaller or equal to n. We let this ranking be (Pn(1),… , Pn(N)) We now show that c is raised to the top of Pn for the ranking (Pn(1), …, Pn(N)), i.e. when raising c to the top of P(i), for all i < n.

23. By contradiction, say that `a Pn c’ and `c Pn b.’ Because c is at the top of Pn(i) for i < n, and at the bottom of Pn(i) for i > n, we can change all i‘s preferences to P*(i) so that `b P*(i) a’. By WP, `b P* a’. By IIA, `a P* c’ and `c P* b.’ By transitivity `a P* b’, which is a contradiction. This concludes that c is at the top of Pn, i.e. when raising c to the top of P(i) for all individuals i < n.

24. Step 3. Consider any a,b different from c. Change the preferences Pn to P* such that `a P*(n) c P*(n) b’, and for any other i, a and b are ranked in any way, as long as the ranking of c (either bottom or top), did not change. Compare Pn+1 to P*: by IIA, `a P* c’. Compare Pn-1 to P*: by IIA, `c P* b’. By transitivity, `a P* b’, for all a, b other than c. Because a,b are arbitrary, we have: if `a P*(n) b’, then `a P* b’. n is a dictator for all a, b other than c.

25. Step 4. Repeat all previous steps with state a, playing the role of c. Because, again, it is the ranking Pn of n which determines whether d is at the top or at the bottom of the social ranking, we can reapply step 3. Again, we have: if `c P*(n) b’, then `c P* b’. n is a dictator for all c, b other than a. Because n is a dictator for all a, b other than c, and n is a dictator for all c, b other than a, we obtain that n is a dictator for all states.

26. A diagrammatic proof We assume that preferences are continuous. Continuity. For any i, x, the sets {y: y R(i) x} and {y: x R(i) y} are closed. Complete, transitive and continuous preferences R(i) can be represented as continuous utility functions ui

27. We aggregate the utility functions u into a social welfare function V (x) = f (u1(x), …, uN(x)). PI. Pareto indifference. If ui(x)= ui(y), for all i, then V(x)=V(y). If V satisfies U, IIA, and P, then 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.

28. The axioms of Arrow’s theorem require that the utility is an ordinal concept (OS) and that utility is interpersonally noncomparable (NC). If ui represents R(i), then so does any increasing transformation vi (ui). All increasing transformations vi must be allowed, independently across i. The function W aggregates the preferences (ui)i=1 if and only if the function g(W) aggregates the preferences (vi(ui))i=1, where g is an increasing transformation.

29. Suppose that N =2 (when N>2 the analysis is a simple extension) Pick an arbitrary utility vector u. u2 II I u III IV u1

30. By weak Pareto, W(u) > W(w) for all utility indexes w in III, and W(w)>W(u) for all utility indexes w in I. u2 II I u III IV u1

31. Suppose that W (u) > W (w) for some w in II. Applying the transformation v1 (w1) = w*1 and v2 (w2) = w*2, the OS/NC principle implies that W (u) > W (w*). u2 w* II I w u III IV u1

32. This concludes that for all w in II, either W (u)<W (w), W (u) = W (w), or W (u) > W (w). It cannot be that W (u) = W (w) because transitivity would imply W (w*) = W (w). u2 w* II I w u III IV u1

33. Suppose that W (u) > W (w). In particular, W (u) > W (u1-1, u2+1). Consider the transform v1(u1) = u1+1, v2(u2) = u1-1. By the OS/NC principle, W (v(u)) > W (v1 (u1-1), v2(u2+1)) = W (u). u2 w II I u v(w) v(u) III IV u1

34. By repeating the step of quadrant II, the transform v1 (w1) = w*1 and v2 (w2) = w*2, the OS/NC principle implies W (u) < W (w*) for all w* in IV. u2 II I u w* v(u) III IV u1

35. The indifference curves are either horizontal or vertical. Hence there must be a dictator. In the figure, agent 1 is the dictator. u2 II I u III IV u1

36. Conclusion 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.

37. Preview next lecture We will extend Arrow’ theorem to social choice functions. We will introduce the possibility of interpersonal utility comparisons. We will axiomatize the Rawlsian Theory of Justice and the Arrowian Theory of Justice