Transcript Group Coordination: A History of Paradox and Impossibility

Group Coordination: A History of
Paradox and Impossibility
David M. Pennock
Voting Paradox I
(Condorcet 1785)
A>B>C
B > C>A
C>A>B
Pairwise (majority) votes:
A > B (2 : 1)
B > C (2 : 1)
C > A (2 : 1)
Voting Paradox II
A>C>B
B > C>A
C > B >A
Plurality vote:
A > B > C (4:3:2)
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 > ••• > A
or any other pattern! [Saari 95]
Other voting schemes
• Borda count:
A>B>C
B > C>A
3 2 1

B > A > C (5:4:3)
• 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
• 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 f
that 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 is
some proper subset of G.
Another Explanation
• IIA  procedure cannot distinguish btw
transitive & intransitive inputs [Saari]
• For example, pairwise vote cannot
distinguish between:
A>B>C
B > C >A
C>A>B
A>B, B>C, C>A
&
A>B, B>C, C>A
A<B, B<C, C<A
The Impossibility of a
Paretian 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 f
that 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
Back Doors?
• Black’s Single-peakedness:
E B A C D
E B A C D
E B A C D
• If all voters preferences are single-peaked,
then pairwise (majority) vote satisfies
P, IIA, ND
Back Doors?
• Cardinal preferences / no interpersonal
comparability  impossibility remains
u1(A)=10, u1(B)=5, u1(C)=1
u2(A)=-4, u2(B)=3, u2(C)=10
u1(A) not
comparable
to u2(C)
• Cardinal preferences / interpersonal
comparability  utilitarianism
u(A)  ui(A)
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 and
non-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) EF + EF = E
Externally Bayesian (EB) E|F = EF / EF + EF
Proportional Dependence on States (PDS)
Unanimity (UNAM)
Independence Preservation Property (IPP)
Non-dictatorship (ND)
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(AE), i=1,...,n
set of events E
set of collective actions A
Pr1, u1 Pr2, u2 Pr3, u3
A
E
Pr, u
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
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]
Combining Bayesian networks
• Structural unanimity
&
&  &

• Proportional dependence on states
Pr0()  f(Pr1(), Pr2(), … , Prn())
• Unanimity
• Nondictatorship
[P & W 99]
Combining Bayesian networks
• Structural unanimity
&
&  &

• Proportional dependence on states
Pr0()  f(Pr1(), Pr2(), … , Prn())
• 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
• Family aggregation
,
,…,
=
Pr0(E|pa(E)) = f[Pr1(E|pa(E)), … , Prn(E|pa(E))]
• Unanimity
• Nondictatorship
[P & W 99]
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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