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

“Magic” Dice Paradox

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