Group Coordination: A History of Paradox and Impossibility

1. Group Coordination: A History of Paradox and Impossibility David M. Pennock

2. anana arrot pple Pairwise (majority) votes: A > B > C A > B (2 : 1) B > C > A B > C (2 : 1) C > A > B Voting Paradox I(Condorcet 1785) C > A (2 : 1)

3. anana arrot pple A > C > B B > C > A Pairwise votes: C > B > A B > A (5 : 4) C > A (5 : 4) C > B (6 : 3) Voting Paradox II Plurality vote: A > B > C (4:3:2)

4. How bad can it get? • Plurality vote: A > B > C > D > E > ••• > Z • Remove Z: Y > X > W > V > U > ••• > A or any other pattern! [Saari 95]

5. A > B > C  B > A > C (5:4:3) B > C > A3 2 1 Other voting schemes • Borda count: • Dodgson (Lewis Carroll) winner: • adjacent swap: A>B>C>D  A>C>B>D • alternative that requires fewest adjacent swaps to become a Condorcet winner

6. Other voting schemes • Kemeny winner: • d(A,B,>i,>j) = 0 if >i and >j agree on A,B = 1 if one is indiff, the other not = 2 if >i and >j are opposite • dist(>i,>j) = all pairs {A,B} d(A,B,>i,>j) • Winner: ordering > with min i dist(>,>i) • Dodgson and Kemeny winner are NP-hard! [Bartholdi, Tovey, & Trick 89] • Plurality, Borda, Dodgson, Kemeny all depend on “irrelevant alternatives”; pairwise can lead to intransitivities.

7. How good can it get? • General case: > = f(>1,>2,...,>n) where >, >i weak order preference relations • Q: What aggregation function f (e.g., voting scheme) is independent of irrelevant alternatives? A: Essentially none!

8. Arrow’s Conditions • Individual & collective rationality: >, >i are weak orders (transitive) • Universal domain (U) • Pareto (P): If A >i B for all i, then A > B • Indep. of irrelevant alternatives (IIA): > on A,B depends only on the >i on A,B • Non-dictatorship (ND): no i s.t. A >i B  A > B, for all A,B

9. Arrow’s Impossibility Theorem • If # persons finite, # alternatives > 2 then There is no aggregation function fthat can simultaneously satisfyU, P, IIA, ND.

10. Proof Sketch • A subgroup G is decisive over {A,B} if A >i B , for all i in G  A > B • Field Expansion: If G is almost decisive over {A,B}, then G is decisive over all pairs. • Group Contraction: If any group G is decisive, then so is some proper subset of G.

11. Another Explanation • IIA  procedure cannot distinguish btw transitive & intransitive inputs [Saari] • For example, pairwise vote cannot distinguish between: A > B > C A>B, B>C, C>A & B > C > A A>B, B>C, C>A C > A > B A<B, B<C, C<A

12. The Impossibility of aParetian Liberal (Sen 1970) • Liberalism (L): For each i, there is at least one pair A,B such that A >i B  A > B • Minimal Liberalism (L*): There are at least two such “free” individuals. There is no aggregation function fthat can simultaneously satisfyU, P and L*. • Does not require IIA.

13. Back Doors? • Fishburn: If # persons infinite: Arrow’s axioms are mutually consistent. • But Kirman & Sondermann: Infinite society controlled by an arbitrarily small group. An “invisible dictator”. • Mihara: Determining whether A > B is uncomputable

14. E E E B B B A A A C C C D D D Back Doors? • Black’s Single-peakedness: • If all voters preferences are single-peaked, then pairwise (majority) vote satisfiesP, IIA, ND

15. u1(A)=10, u1(B)=5, u1(C)=1 u2(A)=-4, u2(B)=3, u2(C)=10 Back Doors? • Cardinal preferences / no interpersonal comparability  impossibility remains • Cardinal preferences / interpersonal comparability  utilitarianism u(A) ui(A) u1(A) not comparable to u2(C)

16. Strategy-proofness(Non-manipulability) • A voting scheme is manipulable if, in some situation, it can be advantageous to lie; otherwise it is strategy-proof. • Example: Perot > Clinton > Bush • Gibbard and Satterthwaite (independently): If # of alternatives > 2, Any deterministic, strategy-proof voting scheme is dictatorial.

17. Probabilistic Voting • Hat of Ballots (HOB): place all ballots in a hat and choose one top choice at random. • Hat of Alternatives (HOA): Collect ballots. Choose two alternatives at random. Use any standard vote to pick one of these two. • HOB & HOA are strategy-proof andnon-dictatorial, but not very appealing. Gibbard: Any strategy-proof voting scheme is a probability mixture of HOB & HOA (Computing strategy may be intractable [B,T&T])

18. Arrow  Gibbard-Satterthwaite • One-to-one correspondence • Suppose we find a preference aggregation function f that satisfies U, P, IIA, and ND. • Then the associated vote is strategy-proof • Suppose we find a strategy-proof vote • Then an associated f satisfies P, IIA, ND, and U • Contrapositive: another justification for IIA

19. Other Impossibilities:Belief Aggregation • Combining probabilities: Pr = f(Pr1,Pr2,...,Prn) • Properties / axioms: • Marginalization property (MP) • Externally Bayesian (EB) • Proportional Dependence on States (PDS) • Unanimity (UNAM) • Independence Preservation Property (IPP) • Non-dictatorship (ND) EF EF E + = E|F EF EF EF = / +

20. Belief Aggregation • Impossibilities: • IPP, PDS are inconsistent • MP, EB, UNAM & ND are inconsistent

21. Other Impossibilities:Group Decision Making • Setup: • individual probabilities Pri(E), i=1,...,n • individual utilities ui(AE), i=1,...,n • set of events E • set of collective actions A Pr, u Pr2, u2 Pr1, u1 Pr3, u3 A E

22. Group Decision Making • Desirable properties / axioms: (1) Universal domain (2) Pr = f(Pr1,Pr2,...,Prn) ; u = g(u1,u2,...,un) (3) Choice aA maximizes EU: EPr(E)u(a,E) (4) Pareto Optimal: if for all i EUi(a1)>EUi(a2), then a2 not chosen (5) Unanimous beliefs prevail: f(Pr,Pr,...,Pr) = Pr (6) no prob dictator i such that f(Pr1,...,Prn) = Pri • (1)(6) mutually inconsistent [H & Z 1979] • does not require IIA

23. Other Impossibilities:Incentive-compatible trade • Setup: 1 good, 1 buyer w/ value [a1,b1],seller w/ value [a2,b2], nonempty intersect. • Desirable properties / axioms: (1) incentive compatible (2) individually rational (3) efficient (4) no outside subsidy • (1)(4) are inconsistent [M & S 83]

24. Other Impossibilities:Distributed Computation • Consensus: a fundamental building block • all processors agree on a value from {0,1} • if all agents choose 0 (1), then output is 0 (1) • Impossibilities: • unbounded msg delay & 1 proc fail by stopping (common knowledge problem) • no shared mem & 1/3 procs fail maliciously (Byzantine generals problem)

25. Other Impossibilities:Apportionment • Setup: n congressional seats, pop. of all states; how do we apportion seats to states? • Alabama Paradox • Desirable properties / axioms: (1) monotone (2) consistent (3) satisfying quota • (1)(3) are inconsistent [B & Y 77]

26. Default Logic • In default logic, we must sometime choose among conflicting models: • Republicans are by default not pacifists • Quakers are by default pacifists • Nixon is both a Republican and a Quaker • Many conflict resolution strategies: • specificity, chronological, skepticism, credulity • My default theory: M1 > M2 > M3 • Your default theory: M2 > M3 > M1

27. Default Logic • Q: Is it possible to construct a universal default theory, which combines current & future theories? • A: No, assuming we want the universal theory to obey U, P, IIA, & ND. • Aside: applicability to societies of minds [Doyle and Wellman 91]

28. Collaborative Filtering Goal: predict preferences of one user based on other users’ preferences (e.g., movie recommendations)

29. CF and Social Choice Usociety = f(u1, u2, …, un) ra = f(r1, r2, ... , rn) • Same functional form • Similar semantics • Some of the same constraints on f are desirable, and have been advocated • Modified limitative theorems are applicable[P & H 99]

30. Ensemble Learning censemble = f(c1, c2, ... , cn) • Variants of Arrow’s thm applies to multiclass case • May’s axiomatization of majority rule applies to binary classification case • Common ensemble methods destroy unanimous independencies • Voting paradoxes can and do occur [P, M-R, & G 2000]

31. &  & &  Combining Bayesian networks • Structural unanimity • Proportional dependence on states Pr0()  f(Pr1(), Pr2(), … , Prn()) • Unanimity • Nondictatorship [P & W 99]

32. &  & &  Combining Bayesian networks • Structural unanimity • Proportional dependence on states Pr0()  f(Pr1(), Pr2(), … , Prn()) • Unanimity • Nondictatorship [P & W 99]

33. , ,…, = Combining Bayesian networks • Family aggregation Pr0(E|pa(E)) = f[Pr1(E|pa(E)), … , Prn(E|pa(E))] • Unanimity • Nondictatorship [P & W 99]

34. , ,…, = Combining Bayesian networks • Family aggregation Pr0(E|pa(E)) = f[Pr1(E|pa(E)), … , Prn(E|pa(E))] • Unanimity • Nondictatorship [P & W 99]

35. Conclusion I • Group coordination is fraught w/ paradox and impossibilities: • voting • preference aggregation • belief aggregation • group decision making • trading • distributed computing • Non-ideal tradeoffs are inevitable • Standard acceptable solutions seem unlikely

36. Conclusion II • Arrow’s Theorem initiated social choice theory & remains powerful, compelling • May provide a valuable perspective for computer scientists interested in multi-agent or distributed systems

37. Simpson’s Paradox • New York • experiment: 54 / 144 (0.375) subjects are cured • control: 12 / 36 (0.333) cured • California • experiment: 18 / 36 (0.5) cured • control: 66 / 144 (0.458) cured • Totals • experiment: 70 / 180 cured • control: 78 / 180 cured

38. 1 3 2 1 3 2 6 5 4 6 5 4 8 7 9 8 7 9 “Magic” Dice Paradox > >