Democratic Rules in Context

Transcription

1 Democratic Rules in Context Hannu Nurmi Public Choice Research Centre and Department of Political Science University of Turku Institutions in Context 2012 (PCRC, Turku) Democratic Rules in Context 4 June, / 48

2 The main points of the presentation The main points many non-equivalent procedures are used for a seemingly same purpose, each one of them has at least one major flaw these flaws occur in specific contexts not much is known about the determinants of those contexts knowledge of the determinants would be helpful in choosing adequate choice methods new approaches are essentially expanding the domain of applicability of social choice results (PCRC, Turku) Democratic Rules in Context 4 June, / 48

3 Dramatis personae (Very) brief history Jean-Charles de Borda, (PCRC, Turku) Democratic Rules in Context 4 June, / 48

4 (Very) brief history Marie Jean Antoine Nicolas de Caritat, Marquis de Condorcet, (PCRC, Turku) Democratic Rules in Context 4 June, / 48

5 (Very) brief history Charles Lutwidge Dodgson (a.k.a. Lewis Carroll), (PCRC, Turku) Democratic Rules in Context 4 June, / 48

6 (Very) brief history Edward John Nanson, (PCRC, Turku) Democratic Rules in Context 4 June, / 48

7 Many systems, many flaws Paretian bliss: We can all become (unique) winners Example Five alternatives, five winners 4 voters 3 voters 2 voters A B C E C D D E E C D B B A A (PCRC, Turku) Democratic Rules in Context 4 June, / 48

8 Two winner intuitions Highest average ranking Borda Count Example 2 voters 2 voters 2 voters 1 voter D A B D C D A C B C D B A B C A This yields the ranking DABC. Now remove D. This gives: CBA, i.e. reversal of collective preference over A, B and C. Fishburn: it is possible that the Borda winner wins in only one of the proper subsets of the alternative set. Obviously, fiddling with the alternative set opens promising vistas for outcome control. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

9 Two winner intuitions 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? (PCRC, Turku) Democratic Rules in Context 4 June, / 48

10 Surprise? Two winner intuitions 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. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

11 Improving old systems Borda s paradox Example 4 voters 3 voters 2 voters A B C B C B C A A Borda s points: plurality voting results in a bad outcome a superior system exists (Borda Count) (PCRC, Turku) Democratic Rules in Context 4 June, / 48

12 Improving old systems Improving Borda Count: Nanson s rule How does it work? Compute Borda scores and eliminate all candidates with no more than average score. Repeat until the winner is found. Properties: Guarantees Condorcet consistency Is nonmonotonic (PCRC, Turku) Democratic Rules in Context 4 June, / 48

13 Improving old systems Nanson s rule is nonmonotonic Example C B A B A A A D B A C C D C D C B D B A C D D B The Borda ranking: A C B D with D s score 97 being the only one that does not exceed the average of 150. Recomputing the scores for A, B and C, results in both B and C failing to reach the average of 100. Thus, A wins. Suppose now that those 12 voters who had the ranking B A C D improve A s position, i.e. rank it first, ceteris paribus. Now, both B and D are deleted and the winner is C. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

14 Improving old systems Improving plurality rule: plurality runoff Properties: Does not elect Condorcet losers Is nonmonotonic Example 6 voters 5 voters 4 voters 2 voters A C B B B A C A C B A C (PCRC, Turku) Democratic Rules in Context 4 June, / 48

15 Improving old systems Black s system: a synthesis of two ideas How does it work? Pick the Condorcet winner. If none exists, choose the Borda winner. Properties: Satisfies Cordorcet criteria Is monotonic Is inconsistent Example 4 voters 3 voters 3 voters 2 voters 2 voters A B A B C B C B C A C A C A B (PCRC, Turku) Democratic Rules in Context 4 June, / 48

16 Varieties of goodness Some systems and performance criteria Criterion Voting system a b c d e f g h i Amendment Copeland Dodgson Maximin Kemeny Plurality Borda Approval Black Pl. runoff Nanson Hare (PCRC, Turku) Democratic Rules in Context 4 June, / 48

17 Varieties of goodness Criteria a: the Condorcet winner criterion b: the Condorcet loser criterion c: the strong Condorcet criterion d: monotonicity e: Pareto f: consistency g: Chernoff property h: independence of irrelevant alternatives i: invulnerability to the no-show paradox (PCRC, Turku) Democratic Rules in Context 4 June, / 48

18 Dramatis personae II Modern classics Kenneth Arrow (PCRC, Turku) Democratic Rules in Context 4 June, / 48

19 Modern classics Alan Gibbard (PCRC, Turku) Democratic Rules in Context 4 June, / 48

20 Modern classics Arrow s theorem Theorem Arrow (1963): No social welfare function satisfies the following conditions: 1 unrestricted domain (U) 2 independence of irrelevant alternatives (IIA) 3 Pareto (P) 4 non-dictatorship (D) Remark: social welfare functions assigns to each n-tuple of connected and transitive individual preference relations a (collective) connected and transitive preference relation. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

21 Modern classics Gibbard-Satterthwaite theorem Definition A social choice function is manipulable (by individuals) iff there is a situation and an individual so that the latter can bring about a preferable outcome by preference misrepresentation than by truthful revelation of his/her preference ranking, ceteris paribus. Definition A social choice function is non-trivial (non-degenerate) iff for each alternative x, there is a preference profile so that x is chosen. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

22 Modern classics Theorem (Gibbard-Satterthwaite ). Every universal and non-trivial resolute social choice function is either manipulable or dictatorial. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

23 How often are the criteria violated? The role of culture impartial culture: each ranking is drawn from uniform probability distribution over all rankings impartial anonymous culture: all profiles (i.e. distributions of voters over preference rankings) equally likely unipolar cultures bipolar cultures (PCRC, Turku) Democratic Rules in Context 4 June, / 48

24 How often are the criteria violated? Lessons from probability and simulation studies cultures make a difference (Condorcet cycles, Condorcet efficiencies, discrepancies of choices) none of the cultures mimics reality IC is useful in studying the proximity of intuitions underlying various procedures (PCRC, Turku) Democratic Rules in Context 4 June, / 48

25 The no-show paradox What makes some incompatibilities particularly dramatic? The fact that they involve intuitively plausible, natural or obvious desiderata. The more plausible etc. the more dramatic is the incompatibility. Theorem Moulin, Pérez: all Condorcet extensions are vulnerable to the no-show paradox. Example 26% 47% 2% 25% A B B C B C C A C A A B (PCRC, Turku) Democratic Rules in Context 4 June, / 48

26 The no-show paradox Some difficult counterexamples: Black Black procedure is vulnerable to the no-show paradox, indeed, to the strong version thereof. Example 1 voter 1 voter 1 voter 1 voter 1 voter D E C D E E A D E B A C E B A B B A C D C D B A C Here D is the Condorcet winner and, hence, is elected by Black. Suppose now that the right-most voter abstains. Then the Condorcet winner disappears and E emerges as the Borda winner. It is thus elected by Black. E is the first-ranked alternative of the abstainer. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

27 The no-show paradox Another difficult one: Nanson 5 voters 5 voters 6 voters 1 voter 2 voters A B C C C B C A B B D D D A D C A B D A Here Nanson s method results in B. If one of the right-most two voters abstain, C their favorite wins. Again the strong version of no-show paradox appears. The twin paradox occurs whenever a voter is better off if one or several individuals, with identical preferences to those of the voter, abstain. Here we have an instance of the twin paradox as well: if there is only one CBDA voter, C wins. If he is joined by another, B wins. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

28 Justifying systems by their goals What do we aim at? Possible consensus states: consensus about everything, i.e. first, second, etc. consensus about the winner majority consensus about first rank majority consensus about Condorcet winner... (PCRC, Turku) Democratic Rules in Context 4 June, / 48

29 Justifying systems by their goals How far are we? Possible distance measures: inversion metric (Kemeny) discrete metric (PCRC, Turku) Democratic Rules in Context 4 June, / 48

30 Upshot Upshot We have (hopefully) seen that: system-criterion pairs give asymmetric information only important criteria ought to be focused upon the likelihood of encountering problems varies with the culture some counterexamples are much harder to find than others What is called for is (much) more work on structural properties of problematic profiles. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

31 Expert choice Condorcet s jury theorem Consider a group N consisting of n(> 2) individuals, each with identical probability p of being right on a dichotomous issue. Assume that the individuals are voting independently of each other. Theorem The probability P of the majority being right depends on the individuals probability p of being right in the following way: 1 If 0.5 < p < 1, then P > p, P increases with n and, when n, P converges to 1. 2 If 0 < p < 0.5, then P < p, P decreases when n increases and P 0 when n. 3 If p = 0.5, then P = 0.5, for all values of n. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

32 Expert choice Asymptotic part does not always hold It is clear that adding persons with competence values just barely exceeding 1/2 does not increase the competence of the majority. In other words, the asymptotic part of the generalized CJT is conditional on competence distribution. The following theorem gives a restriction on the validity of the asymptotic part. Theorem (Nitzan and Paroush; Shapley and Grofman). Assume that the individuals vote independently of each other and that p i > 1/2 for all i N. Let the individuals be labeled in a non-increasing order of competence, that is, p i p j if i < j. The asymptotic part of CJT does not hold if p 1 1 p 1 > n i=2 p i 1 p i. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

33 Optimal weights Expert choice We might ask if there is an optimal assignment of voting weights to individuals such that the probability of the correct majority decision is maximized. Theorem (Nitzan and Paroush; Shapley and Grofman) Assume that p i > 1/2 for all i N. Then the decision method that maximizes the probability of the group decision being right is the weighted majority rule with weights w i being assigned as follows: p i w i = log( ). (1) 1 p i (PCRC, Turku) Democratic Rules in Context 4 June, / 48

34 Dramatis personae III Judgment aggregation Philip Pettit (PCRC, Turku) Democratic Rules in Context 4 June, / 48

35 Judgment aggregation Christian List (PCRC, Turku) Democratic Rules in Context 4 June, / 48

36 Judgment aggregation Doctrinal paradox Example (Vacca 1921, Kornhauser and Sager 1986, List 2011) judge prop. p prop. q prop. r judge 1 true true true judge 2 true false false judge 3 false true false majority true true false p: the defendant was contractually obliged not to do A q: the defendant did A r: the defendant breached the contract (PCRC, Turku) Democratic Rules in Context 4 June, / 48

37 Judgment aggregation Propositional logic Example Modus ponens: voter p p q q voter 1 true true true voter 2 true false false voter 3 false true false majority true true false The majority outcome is: p, p q, q, i.e. a logically inconsistent set of propositions. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

38 Judgment aggregation The flavor of some results List & Pettit: Let {p, q, p q} X. Then there exists no aggregation rule satisfying universal domain, collective rationality, systematicity and anonymity. Dietrich: If X contains at least two propositions, then an aggregation rule F is strongly independent and weakly responsive (and satisfies universal domain and collective rationality) if and only if it is dictatorial. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

39 Judgment aggregation Some definitions Definition Collective rationality. Aggregation rule is to provide a fully rational collective judgment set for every combination of individual judgments. Definition Systematicity. Collective judgment on each proposition depends only on individual judgments on that proposition and the criterion determining the collective judgment on each proposition is the same. Definition Weak responsiveness. A rule is weakly responsive there are at least two judgment profiles that lead to different collective judgments under it. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

40 Computational social choice Dramatis personae IV Edith Elkind (PCRC, Turku) Democratic Rules in Context 4 June, / 48

41 Computational social choice Arkadii Slinko (PCRC, Turku) Democratic Rules in Context 4 June, / 48

42 Computational social choice Computational social choice Background: the theory of (effective) computability Problems in P: there is an polynomial (in terms of the size of the problem) algorithm for solving the problem. Problems in N P: no such algorithm is known to exist. Question: P = N P? N P complete problems: if one of them is shown to be in P, then P = N P. Questions addressed: how difficult it is to determine the winner(s) how difficult it is to manipulate how hard it is to control the election (PCRC, Turku) Democratic Rules in Context 4 June, / 48

43 Computational social choice Sample of results Bartholdi et. al: determining the winner may be computationally intractable in some systems (e.g Dodgson, Kemeny) Bartholdi and Orlin: STV is computationally resistant to manipulation Brandt et al.: when restricted to single-peaked electorates, determining the winner for Kemeny, Young and Dodgson elections are in P. Hazon and Elkind: For the Bucklin rule, finding out whether a safe manipulation exists, is in P. Hazon and Elkind: For the Borda rule, finding out whether a safe manipulation exists, is N P -hard. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

44 Computational social choice Safe manipulation Definition Let i be a voter and the set F a set of like-minded other voters. Assume that i announces his manipulative (non-sincere) vote L. If the following two conditions are met, then the i s manipulation is safe: 1 there is a subset U F such that if all voters in U switch to L, the outcome is better (for them ) than under sincere voting, and 2 for any W F, if the voters in W switch to L, the outcome is no worse than under sincere voting. (PCRC, Turku) Democratic Rules in Context 4 June, / 48

45 Concluding remarks By way of conclusion Arrow s impossibility theorem extends far beyond social welfare functions while no system of aggregation is perfect, there are variations in their properties contextual variables (frequency and difficulty of finding counterexamples) should play a role in the choice of procedures majority (simple or qualified) differs essentially from individuals (PCRC, Turku) Democratic Rules in Context 4 June, / 48

46 Some References I References V. Conitzer and J. Rothe, eds. Proceedings of the Third International Workshop on Computational Social Choice (COMSOC 2010). Düsseldorf: Düsseldorf University Press D. Felsenthal and M. Machover, eds. Electoral Systems: Paradoxes, Assumptions, and Procedures. Berlin: Springer W. Gehrlein and D. Lepelley. Voting Paradoxes and Group Coherence. Berlin: Springer H. Nurmi. Voting Procedures under Uncertainty. Berlin: Springer, (PCRC, Turku) Democratic Rules in Context 4 June, / 48

47 Some References II References D. Saari. Basic Geometry of Voting. Berlin: Springer P. Fishburn. Condorcet social choice functions. SIAM Journal of Applied Mathematics, 33, 1977, H. Moulin. Condorcet s principle implies the no show paradox. Journal of Economic Theory 45, 1988, S. Nitzan and J. Paroush. Optimal decision rules in uncertain dichotomous choice situation. International Economic Review, 23,1982, (PCRC, Turku) Democratic Rules in Context 4 June, / 48

48 Some References III References J. Pérez. The strong no show paradoxes are common flaw in Condorcet voting correspondences. Social Choice and Welfare 18,2001, L. Shapley and B. Grofman. Optimizing group judgmental accuracy in the presence of uncertainties. Public Choice 43, 1984, (PCRC, Turku) Democratic Rules in Context 4 June, / 48