Social choice theory = preference aggregation = truthful voting

1. Social choice theory= preference aggregation= truthful voting Tuomas Sandholm Professor Computer Science Department Carnegie Mellon University

2. Social choice • Collectively choosing among outcomes • E.g. presidents • Outcome can also be a vector • E.g. allocation of money, goods, tasks, and resources • Agents have preferences over outcomes • Center knows each agent’s preferences • Or agents reveal them truthfully by assumption • Social choice function aggregates those preferences & picks outcome • Outcome is enforced on all agents • CS applications: Multiagent planning [Ephrati&Rosenschein], computerized elections [Cranor&Cytron], accepting a joint project, collaborative filtering, rating Webarticles [Avery,Resnick&Zeckhauser], rating CDs...

3. Condorcet paradox [year 1785] • x  z  y • y  x z • z  y  x • Majority rule • Three voters: x  z  y  x

4. x x y z y z x y z z x y z y y x z x Agenda paradox • x  z  y (35%) • y  x  z (33%) • z  y  x (32%) • Binary protocol (majority rule) = cup • Three types of agents: • Power of agenda setter (e.g. chairman) • Vulnerable to irrelevant alternatives (z) • Plurality protocol • For each agent, most preferred outcome gets 1 vote • Would result in x

5. Pareto dominated winner paradox Voters: • x  y  b  a • a  x  y  b • b  a  x  y

6. Inverted-order paradox • Borda rule with 4 alternatives • Each agent gives 4 points to best option, 3 to second best... • Agents: • x=22, a=17, b=16, c=15 • Remove x: c=15, b=14, a=13 • x  c  b  a • a  x  c  b • b  a  x  c • x  c  b  a • a  x  c  b • b  a  x  c • x  c  b  a

7. Borda rule also vulnerable to irrelevant alternatives • Three types of agents: • Borda winner is x • Remove z: Borda winner is y • x  z  y (35%) • y  x  z (33%) • z  y  x (32%)

8. Majority-winner paradox • a  b  c • a  b  c • a  b  c • b  c  a • b  c  a • b  a  c • c  a  b • Agents: • Majority rule with any binary protocol: a • Borda protocol: b=16, a=15, c=11

9. Is there a desirable way to aggregate agents’ preferences? • Set of alternatives A • Each agent i {1,..,n} has a ranking iof A • Social welfare function F: LnL • To avoid unilluminating technicalities in proof, assume iand  are strict total orders • Some possible (weak) desiderata of F • 1. Unanimity: If all voters have the same ranking, then the aggregate ranking equals that. Formally,    L, F( ,…,) =. • 2. Nondictatorship: No voter is a dictator. Voter i is a dictator if for all 1 ,…,n , F(1 ,…,n) =i • 3. Independence of irrelevant alternatives: The social preference between any alternatives a and b only depends on the voters’ preferences between a and b. Formally, for every a, b  A and every 1 ,…,n ,  ’1 ,…, ’n  L , if we denote  = F(1 ,…,n) and  ’ = F( ’1 ,…, ’n), then a i b  a  ’i b for all i implies that a  b  a  ’ b. • Arrow’s impossibility theorem [1951]: If |A| ≥ 3, then no F satisfies desiderata 1-3.

10. Proof • Assume F satisfies unanimity and independence of irrelevant alternatives • Lemma.Any function F: LnL that satisfies unanimity and independence of irrelevant alternatives also satisfies pairwise unanimity. That is, if for all i, a i b, then a  b. • Proof.Let* L be such that a * b. By unanimity, a  b in F(* ,..,*). If1,...,nare all such that a i b, then we have a i b  a * b, and so by independence of irrelevant alternatives, for '  F(1,...,n), we have a *' b. ■ • Lemma (pairwise neutrality). Let 1 ,…,nand ’1 ,…, ’n be two player profiles such that a i b  c ’id. Then a b  c ’ d. • Proof. Assume wlog that a b and c  d. We merge each i and ’i into a single preference i by putting c just above a (unless c = a) and d just below b (unless d = b) and preserving the internal order within the pairs (a,b) and (c,d). • By pairwise unanimity, c a and b  d. Thus, by transitivity, c d. ■ • Take any a  b A, and for every i {0,…,n} define a preference profile pi in which exactly the first i players rank b above a. By pairwise unanimity, in F(p0) we have a  b, while in F(pn) we have b a. Thus, for some i* the ranking of a and b flips: in F(pi*-1) we have a  b, while in F(pi*) we have b a. • We conclude the proof by showing that i* is a dictator: • Lemma. Take any c  d A. If c i*d then c  d. • Proof. Take some alternative e that is different from c and d. For i < i*movee to the top in i, for i > i*movee to the bottom in i, and for i* movee so that c i*e i*d. By independence of irrelevant alternatives, we have not changed the social ranking between c and d. • Notice that the players’ preferences for the ordered pair (d,e) are identical to their preferences for (a,b) in pi*, but the preferences for (c,e) are identical to the preferences for (a,b) in pi*-1, and thus using the pairwise neutrality claim, socially e d and c e, and thus by transitivity c d in the preferences where e was moved. By independence of irrelevant alternatives, moving e does not affect the relative ranking of c and d; thus c d also under the original preferences. ■ ■

11. Stronger version of Arrow’s theorem • In Arrow’s theorem, social welfare function F outputs a ranking of the outcomes • The impossibility holds even if only the highest ranked outcome is sought: • Thm. Let |A| ≥ 3. If a social choice function f:LnA is monotonic and Paretian, then f is dictatorial • f is monotonic if [ x = f() and x maintains its position in ’ ] => f(’) = x • x maintains its position whenever x i y => x i’ y • Proof. From f we construct a social welfare function F that satisfies the conditions of Arrow’s theorem • For each pair x, y of outcomes in turn, to determine whether x > y in F, move x and y to the top of each voter’s preference order • don’t change their relative order • order of other alternatives can change • Lemma 1. If ’ and ’’ are constructed from  by moving a set X of outcomes to the top in this way, then f(’) = f(’’) • Proof. Because f is Paretian, f(’)  X. Thus f(’) maintains its position in going from ’ to ’’. Then, by monotonicity of f, we have f(’) = f(’’) • Note: Because f is Paretian, we have f = x or f = y (and, by lemma 1, not both) • F is transitive (total order) (we omit proving this part) • F is Paretian (if everyone prefers x over y, then x gets chosen and vice versa) • F satisfies independence of irrelevant alternatives (immediate from lemma 1) • By earlier version of the impossibility, F (and thus f) must be dictatorial. ■

12. Voting rules that avoid Arrow’s impossibility (by changing what the voters can express) • Approval voting • Each voter gets to specify which alternatives he/she approves • The alternative with the largest number of approvals wins • Avoids Arrow’s impossibility • Unanimity • Nondictatorial • Independent of irrelevant alternatives • Range voting • Instead of submitting a ranking of the alternatives, each voter gets to assign a value (from a given range) to each alternative • The alternative with the highest sum of values wins • Avoids Arrow’s impossibility • Unanimity • Nondictatorial • Independent of irrelevant alternatives (one intuition: one can assign a value to an alternative without changing the value of other alternatives) • More information about range voting available at www.rangevoting.org • These still fall prey to strategic voting (e.g., Gibbard-Satterthwaite impossibility, discussed in the next lecture)