Inequality of Representation

Transcription

1 Inequality of Representation Hannu Nurmi Public Choice Research Centre University of Turku Institutions in Context: Inequality (HN/PCRC) Inequality of Representation June 3 9, / 31

2 The main points The main points proportionality is a matter of degree proportionality is often implicitly based on a choice rule what is (a reasonably) proportional outcome under one rule, may not be that under another proportionality is not only vague, but also ambiguous: what is it that we want to distribute proportionally? if it is a priori voting power (rather than seats) then we might have a major problem in our hands there are methods for composing optimal committees requiring no more than pairwise comparisons of candidates by voters (HN/PCRC) Inequality of Representation June 3 9, / 31

3 The vagueness of proportionality The recent Finnish parliamentary election Results is mainland Finland: parties votes % seats: current seats: proposed KOK SDP PS KESK Vas Vihr SFP KD (HN/PCRC) Inequality of Representation June 3 9, / 31

4 The vagueness of proportionality Three P.R. systems in action parties votes Sainte-Laguë Hamilton d Hondt A B C D E (HN/PCRC) Inequality of Representation June 3 9, / 31

5 The vagueness of proportionality Many methods of apportionment largest remainder s method (Hamilton) d Hondt s method (Jefferson) Webster s method Huntington-Hill or equal proportions (currently in use in the US) (HN/PCRC) Inequality of Representation June 3 9, / 31

6 The vagueness of proportionality Referendum paradox Example voter states opinion state 1... state 45 state state 50 total Obama Romney O-electors R-electors (HN/PCRC) Inequality of Representation June 3 9, / 31

7 The ambiguity of proportionality Pairwise victories Condorcet extensions Example Condorcet s paradox 4 voters 4 voters 4 voters A B C C A B B C A Surely, there is no winner here, or what? If so, then removing this kind of component from any larger profile or adding it to some profile should not change the winners, right? (HN/PCRC) Inequality of Representation June 3 9, / 31

8 Surprise? The ambiguity of proportionality Example A profile with a strong Condorcet winner 7 voters 4 voters A B B C C A Adding the Condorcet paradox profile to this one results in a new Condorcet winner. N.B. the Borda winner remains the same in the 11- and 23-voter profiles. (HN/PCRC) Inequality of Representation June 3 9, / 31

9 The ambiguity of proportionality Proportionality and social choice rule Problem: elect proportionally 2 candidates out of 4. Consider the following profile: 4 voters 3 voters 2 voters 1 voter A B C A C D D D D C B C B A A B Plurality choice set: {A, B} Borda choice set: {C, D} (HN/PCRC) Inequality of Representation June 3 9, / 31

10 The ambiguity of proportionality Plurality vs. Condorcet Plurality-based proportionality does not guarantee the election of an eventual Condorcet winner. 1 voter 2 voters 2 voters A B C B A A C C B The Condorcet (and Borda) winner A is not elected even if all but one candidate is elected proportionally. (HN/PCRC) Inequality of Representation June 3 9, / 31

11 The ambiguity of proportionality Borda vs. Condorcet Borda-based proportionality does not guarantee the election of (even strong) Condorcet winner. 8 voters 7 voters A B B C C D D E E A If two candidates are chosen based on Borda-proportionality, the strong Condorcet winner A is not elected, B and C are. (HN/PCRC) Inequality of Representation June 3 9, / 31

12 Optimal committees Composing optimal committees A person, A, is represented in some matter by another person, B, to the extent that B s actions in the matter reflect what might be called A s ideal preferences the choices that A would make if A were ideally informed, ideally expert, and ideally clear about his own interests. (Rogowski) A committee member M represents a voter V to the extent that 1 M makes present V s opinions in the deliberations that take place within the committee, 2 M is similarly responsive to various kinds of arguments presented in those deliberations as V, and 3 M votes in the same way as V, should V be present in the committee. (Chamberlin and Courant) (HN/PCRC) Inequality of Representation June 3 9, / 31

13 Optimal committees Optimal fuzzy representation Consider now voter i and a committee c t consisting of k candidates as required. We are interested in finding the members of c t that best represent i. Denote the set of these representatives by B(i, c t ). Several plausible ways of finding the best representatives can be envisioned: 1 B i sum (c t) = {j c t l r jl l r ql, q c t }, 2 B i min (c t) = {j c t min l r jl min l r ql, l K, q c t }, 3 B i h (c t) = {j c t h(j) h(q), q c t } where h(j) = p (max l r jl ) + (1 p)(min l r jl ), 4 B i cop (c t) = {j c t cop(j) cop(q), q c t } where cop(j) = {l c t r jl > r lj, l K } (HN/PCRC) Inequality of Representation June 3 9, / 31

14 Optimal committees 1 determines the best representatives on the basis of the sums of the preference degrees obtained by candidates in all pairwise comparisons (cf. Borda). 2 looks at the minimum preference degree of each candidate when compared with all others and picks the candidate with the largest minimum n(min-max). 3 is a version of Hurwicz s rule which maximizes the weighted sum of the smallest and largest preference degrees. 4 is motivated by Copeland s rule in social choice theory. (HN/PCRC) Inequality of Representation June 3 9, / 31

15 Optimal committees Ranking committees The most straightforward way to accomplish this is to define the score of committee c t as follows: S t = i N raj i. a c t j K Thus, the score of a committee is the sum of values given by voters to each of its members. The most representative committee RC B would then be: RC B = {c i C k S i S j, c j C k }. (HN/PCRC) Inequality of Representation June 3 9, / 31

16 A priori voting power Seats or power The Shapley-Shubik index value of player i is: (s 1)!(n s)! φ i = Σ S N [v(s) v(s \ {i})]. n! The standardized Banzhaf index value of i is defined as: β i = Σ S N[v(S) v(s \ {i})] Σ j N Σ S N [v(s) v(s \ {j})]. The absolute Penrose-Banzhaf index is defined as: β i = Σ S N[v(S) v(s \ {i})] 2 n 1. (HN/PCRC) Inequality of Representation June 3 9, / 31

17 More votes, less power Non-monotonicity of runoff 22 voters 21 voters 20 voters A B C B C A C A B Table: Additional support paradox (HN/PCRC) Inequality of Representation June 3 9, / 31

18 More votes, less power No-show paradox and runoff 5 voters 5 voters 4 voters A B C B C A C A B Table: No-Show Paradox (HN/PCRC) Inequality of Representation June 3 9, / 31

19 More votes, less power Schwartz paradox and amendment party A party B party C 23 seats 28 seats 49 seats a b c b c a c a b Table: Schwartz Paradox Motion b has been presented and that also an amendment to it c is on the table. Hence the amendment agenda: motion b vs. amendment c, the winner of the preceding vs. a (HN/PCRC) Inequality of Representation June 3 9, / 31

20 Power and proximity of outcomes Baigent s result Theorem Anonymity and respect for unanimity cannot be reconciled with proximity preservation: choices made in profiles more close to each other ought to be closer to each other than those made in profiles less close to each other (Baigent 1987). I.e. if a small group of voters changes its mind about preference ranking, the change in outcomes can be larger than had a large group of voters changed its mind. That is, smaller groups can, under any reasonable voting rule, have larger impact on outcomes than larger groups. (HN/PCRC) Inequality of Representation June 3 9, / 31

21 A toy example Power and proximity of outcomes Let us assume that there are only two voters in NATO (1 and 2) and two alternatives: impose a no-fly zone in Libya (NFZ) and refrain from military interference (R) in Libya. To simplify things even further, assume that only strict preferences are possible, i.e both decision makers have a strictly preferred policy. Four profiles are now possible: P 1 P 2 P 3 P NFZ NFZ R R R NFZ NFZ R R R NFZ NFZ NFZ R R NFZ (HN/PCRC) Inequality of Representation June 3 9, / 31

22 Power and proximity of outcomes Toy example, cont d We denote the voters rankings in various profiles by P mi where m is the number of the profile and i the voter. We consider two types of metrics: one that is defined on pairs of rankings and one defined on profiles. The former is denoted by d r an the latter by d P. They are related as follows: d P (P m, P j ) = i N d r (P mi, P ji ). In other words, the distance between two profiles is the sum of distances between the pairs of rankings of the first, second, etc. voters. (HN/PCRC) Inequality of Representation June 3 9, / 31

23 Power and proximity of outcomes Toy example, cont d Take now two profiles, P 1 and P 3, from the above list and express their distance using metric d P as follows: d P (P 1, P 3 ) = d r (P 11, P 31 ) + d r (P 12, P 32 ). Since, P 12 = P 32 = NFZ R, and hence the latter summand equals zero, this reduces to: d P (P 1, P 3 ) = d r (P 11, P 31 ) = d r ((NFZ R), (R NFZ )). Taking now the distance between P 3 and P 4, we get: d P (P 3, P 4 ) = d r (P 31, P 41 ) + d r (P 32, P 42 ). (HN/PCRC) Inequality of Representation June 3 9, / 31

24 Power and proximity of outcomes Toy example, cont d Both summands are equal since by definition: Thus, d r ((R NFZ ), (NFZ R)) = d r ((NFZ R), (R NFZ )). d P (P 3, P 4 ) = 2 d r ((NFZ R), (R NFZ )). In terms of d P, then, P 3 is closer to P 1 than to P 4. This makes sense intuitively. (HN/PCRC) Inequality of Representation June 3 9, / 31

25 Power and proximity of outcomes Toy example, cont d The proximity of the social choices emerging out of various profiles depends on the choice procedures, denoted by g, being applied. Let us make two very mild restrictions on choice procedures, viz. that they are anonymous and respect unanimity. In our example, anonymity requires that whatever is the choice in P 3 is also the choice in P 4 since these two profiles can be reduced to each other by relabelling the voters. Unanimity, in turn, requires that g(p 1 ) = NFZ, while g(p 2 ) = R. Therefore, either g(p 3 ) g(p 1 ) or g(p 3 ) g(p 2 ). Assume the former. It then follows that d r (g(p 3 ), g(p 1 )) > 0. Recalling the implication of anonymity, we now have: d r (g(p 3 ), g(p 1 )) > 0 = d r (g(p 3 ), g(p 4 )). (HN/PCRC) Inequality of Representation June 3 9, / 31

26 Power and proximity of outcomes Toy example, cont d In other words, even though P 3 is closer to P 1 than to P 4, the choice made in P 3 is closer to - indeed identical with - that made in P 4. This argument rests on the assumption that g(p 3 ) g(p 1 ). Similar argument can, however, easily be made for the alternative assumption, viz. that g(p 3 ) g(p 2 ). (HN/PCRC) Inequality of Representation June 3 9, / 31

27 Power and proximity of outcomes Toy example, cont d The example shows that small mistakes or errors made by voters are not necessarily accompanied with small changes in voting outcomes. Indeed, if the true preferences of voters are those of P 3, then voter 1 s mistaken report of his preferences leads to profile P 1, while both voters making a mistake leads to P 4. Yet, the outcome ensuing from P 1 is further away from the outcome resulting from P 3 than the outcome that would have resulted had more - indeed both - voters made a mistake whereupon P 4 would have emerged. (HN/PCRC) Inequality of Representation June 3 9, / 31

28 Power and proximity of outcomes Toy example, cont d N.B.: the violation of proximity preservation occurs in a wide variety of voting systems, viz. those that satisfy anonymity and unanimity. This result is not dependent on any particular metric with respect to which the distances between profiles and outcomes are measured. I.e. in nearly all reasonable voting systems it is possible that a small group of voters has greater impact on voting outcomes than a big group. (Thus, we have yet another form of violating local monotonicity.) (HN/PCRC) Inequality of Representation June 3 9, / 31

29 Conclusion By way of concluding inequality of representation is typically a result of the election system the current representation systems are implicitly based on plurality principle, i.e. the voters are expected to reveal their unique favorite even in terms of this plurality-based PR, systems vary in how faithfully they reflect the distribution of voter opinions the distribution of influence over outcomes should this be our focus often deviates from the seat distribution there are several non-equivalent ways of measuring the influence none of them seems quite adequate (HN/PCRC) Inequality of Representation June 3 9, / 31

30 Some References I Conclusion M. Balinski and H. P. Young. Fair Representation. 2nd ed. Washington, D.C.: Brookings Institution Press, 2001 M. J. Holler and H. Nurmi, eds. Power, Voting, and Voting Power: 30 Years After. Berlin: Springer-Verlag, H. Nurmi. Voting Procedures under Uncertainty. Berlin: Springer-Verlag, D. Saari. Basic Geometry of Voting. Cambridge: Cambridge University Press, (HN/PCRC) Inequality of Representation June 3 9, / 31

31 Some References II Conclusion N. Baigent. Preference proximity and anonymous social choice. The Quarterly Journal of Economics 102, 1987, J. Chamberlin and P. Courant. Representative deliberations and representative decisions. American Political Science Review 77, 1983, D. Saari and K. Sieberg. The sum of the parts can violate the whole. American Political Science Review, 95, 2001, T. Schwartz. The paradox of representation. The Journal of Politics 57, 1995, (HN/PCRC) Inequality of Representation June 3 9, / 31