Group Coordination: A History of Paradox and Impossibility

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Group Coordination A History of Paradox and
Impossibility

• David M. Pennock

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Voting Paradox I(Condorcet 1785)
C gt A (2 1)
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Voting Paradox II
Plurality vote A gt B gt C (432)
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How bad can it get?

• Plurality vote A gt B gt C gt D gt E gt gt Z
• Remove Z Y gt X gt W gt V gt U gt gt A or any
  other pattern! Saari 95

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Other voting schemes

• Borda count
• Dodgson (Lewis Carroll) winner
• adjacent swap AgtBgtCgtD ? AgtCgtBgtD
• alternative that requires fewest adjacent swaps
  to become a Condorcet winner

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Other voting schemes

• Kemeny winner
• d(A,B,gti,gtj) 0 if gti and gtj agree on A,B 1
  if one is indiff, the other not 2 if gti and
  gtj are opposite
• dist(gti,gtj) ?all pairs A,B d(A,B,gti,gtj)
• Winner ordering gt with min ?i dist(gt,gti)
• 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.

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How good can it get?

• General case
• gt f(gt1,gt2,...,gtn)
• where gt, gti weak order preference relations
• Q What aggregation function f (e.g., voting
  scheme) is independent of irrelevant
  alternatives?

A Essentially none!
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Arrows Conditions

• Individual collective rationality gt, gti are
  weak orders (transitive)
• Universal domain (U)
• Pareto (P) If A gti B for all i, then A gt B
• Indep. of irrelevant alternatives (IIA) gt on
  A,B depends only on the gti on A,B
• Non-dictatorship (ND) no i s.t. A gti B ? A gt B,
  for all A,B

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Arrows Impossibility Theorem

• If persons finite, alternatives gt 2 then
• There is no aggregation function fthat can
  simultaneously satisfyU, P, IIA, ND.

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Proof Sketch

• A subgroup G is decisive over A,B if A gti B ,
  for all i in G ? A gt 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.

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Another Explanation

• IIA ? procedure cannot distinguish btw transitive
  intransitive inputs Saari
• For example, pairwise vote cannot distinguish
  between

A gt B gt C
AgtB, BgtC, CgtA

B gt C gt A
AgtB, BgtC, CgtA
C gt A gt B
AltB, BltC, CltA
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The Impossibility of aParetian Liberal (Sen 1970)

• Liberalism (L) For each i, there is at least one
  pair A,B such that A gti B ? A gt 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.

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Back Doors?

• Fishburn If persons infinite Arrows axioms
  are mutually consistent.
• But Kirman Sondermann Infinite society
  controlled by an arbitrarily small group. An
  invisible dictator.
• Mihara Determining whether A gt B is uncomputable

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Back Doors?

• Blacks Single-peakedness
• If all voters preferences are single-peaked, then
  pairwise (majority) vote satisfiesP, IIA, ND

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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)
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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 gt Clinton gt Bush
• Gibbard and Satterthwaite (independently)
• If of alternatives gt 2,
• Any deterministic, strategy-proof voting scheme
  is dictatorial.

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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,TT)

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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

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Other ImpossibilitiesBelief 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


EF
EF
EF
EF
/


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Belief Aggregation

• Impossibilities
• IPP, PDS are inconsistent
• MP, EB, UNAM ND are inconsistent

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Other ImpossibilitiesGroup Decision Making

• Setup
• individual probabilities Pri(E), i1,...,n
• individual utilities ui(A?E), i1,...,n
• set of events E
• set of collective actions A

Pr, u
Pr2, u2
Pr3, u3
Pr1, u1
A
E
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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)gtEUi(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

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Other ImpossibilitiesIncentive-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

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Other ImpossibilitiesDistributed 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)

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Other ImpossibilitiesApportionment

• 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

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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 gt M2 gt M3
• Your default theory M2 gt M3 gt M1

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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

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Collaborative Filtering

• Goal predict preferences of one user based on
  other users preferences (e.g., movie
  recommendations)

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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 applicableP
  H 99

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Ensemble Learning

• censemble f(c1, c2, ... , cn)
• Variants of Arrows thm applies to multiclass
  case
• Mays 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

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Combining Bayesian networks

• Structural unanimity
• Proportional dependence on states
• Pr0(?) ? f(Pr1(?), Pr2(?), , Prn(?))
• Unanimity
• Nondictatorship P W 99

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Combining Bayesian networks

• Structural unanimity
• Proportional dependence on states
• Pr0(?) ? f(Pr1(?), Pr2(?), , Prn(?))
• Unanimity
• Nondictatorship P W 99

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Combining Bayesian networks

• Family aggregation
• Pr0(Epa(E)) fPr1(Epa(E)), , Prn(Epa(E))
• Unanimity
• Nondictatorship P W 99

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Combining Bayesian networks

• Family aggregation
• Pr0(Epa(E)) fPr1(Epa(E)), , Prn(Epa(E))
• Unanimity
• Nondictatorship P W 99

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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

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Conclusion II

• Arrows Theorem initiated social choice theory
  remains powerful, compelling
• May provide a valuable perspective for computer
  scientists interested in multi-agent or
  distributed systems

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Simpsons 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

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Magic Dice Paradox
gt
gt