Group Coordination: A History of Paradox and Impossibility David M. Pennock
Voting Paradox I(Condorcet 1785)
B > C > A
C > A > B
A > B > Cpple arrot
ananaPairwise (majority) votes:
A > B (2 : 1)
B > C (2 : 1)
C > A (2 : 1)
Voting Paradox II
Plurality vote:A > B > C (4:3:2)A > C > B
B > C > A
C > B > A
pple arrotanana
Pairwise votes:
B > A (5 : 4)
C > A (5 : 4)
C > B (6 : 3)
How bad can it get?
• Plurality vote:A > B > C > D > E > ••• > Z
• Remove Z:Y > X > W > V > U > ••• > Aor any other pattern! [Saari 95]
• 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
Other voting schemes
B > C > A3 2 1
A > B > C B > A > C (5:4:3)
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.
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!
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
Arrow’s Impossibility Theorem
• If # persons finite, # alternatives > 2 then
There is no aggregation function fthat can simultaneously satisfy
U, P, IIA, ND.
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 issome proper subset of G.
Another Explanation
• IIA procedure cannot distinguish btw transitive & intransitive inputs [Saari]
• For example, pairwise vote cannot distinguish between:
B > C > A
C > A > B
A > B > C
A>B, B>C, C>A
A<B, B<C, C<A
A>B, B>C, C>A
&
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 satisfy
U, P and L*.• Does not require IIA.
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
• Black’s Single-peakedness:
• If all voters preferences are single-peaked, then pairwise (majority) vote satisfiesP, IIA, ND
Back Doors?
E B A C D
E B A C D E B A C D
Back Doors?
• Cardinal preferences / no interpersonal comparability impossibility remains
• Cardinal preferences / interpersonal comparability utilitarianism
u(A) ui(A)
u1(A)=10, u1(B)=5, u1(C)=1
u2(A)=-4, u2(B)=3, u2(C)=10
u1(A) notcomparable
to u2(C)
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.
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])
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
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+
Belief Aggregation
• Impossibilities:– IPP, PDS are inconsistent– MP, EB, UNAM & ND are inconsistent
Other Impossibilities:Group Decision Making
• Setup:– individual probabilities Pri(E), i=1,...,n
– individual utilities ui(AE), i=1,...,n
– set of events E– set of collective actions A
Pr3, u3Pr2, u2Pr1, u1
E
APr, u
Group Decision Making• Desirable properties / axioms:
(1) Universal domain
(2) Pr = f(Pr1,Pr2,...,Prn) ; u = g(u1,u2,...,un)
(3) Choice aA 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
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]
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)
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]
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
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]
Collaborative Filtering
Goal: predict preferences of one user based on other users’ preferences
(e.g., movie recommendations)
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]
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]
• Structural unanimity
• Proportional dependence on statesPr0() f(Pr1(), Pr2(), … , Prn())
• Unanimity
• Nondictatorship
[P & W 99]
Combining Bayesian networks
& & &
Combining Bayesian networks
& & &
• Structural unanimity
• Proportional dependence on statesPr0() f(Pr1(), Pr2(), … , Prn())
• Unanimity
• Nondictatorship
[P & W 99]
• Family aggregation
Pr0(E|pa(E)) = f[Pr1(E|pa(E)), … , Prn(E|pa(E))]
• Unanimity
• Nondictatorship
[P & W 99]
Combining Bayesian networks
, ,…, =
, ,…, =
• Family aggregation
Pr0(E|pa(E)) = f[Pr1(E|pa(E)), … , Prn(E|pa(E))]
• Unanimity
• Nondictatorship
[P & W 99]
Combining Bayesian networks
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
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
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