arrow's theorem for incomplete...

Arrow’s Theorem for Incomplete Relations

R. D. Maddux

Department of MathematicsIowa State University

Ames, Iowa, USA

Ames, IowaTuesdays, October 9 and 16, 2012

R. D. Maddux Arrow’s Theorem for Incomplete Relations

Abstract

Let U be a set with three or more elements, let W be the set ofweak orderings of U, let T be the set of total orderings of U, andlet f be an n-ary function mapping Wn to W. Arrow’sImpossibility Theorem asserts that if f satisfiesArrow’s Condition P (“Pareto”) and Condition 3 (“independenceof irrelevant alternatives”) then f is a projection function on totalorderings, i. e., there is some k ∈ {1, . . . , n} such thatf (R1, . . . ,Rn) = Rk for all total orderings R1, . . . ,Rn ∈ T .Th. If a transitive-valued multivariate relational operator fsatisfies versions of Arrow’s Conditions P and 3, and maps allprofiles from a diverse set R of binary relations on U to transitiverelations on U, then f must be the unanimous consent function forsome set of input variables, and if R is very diverse then f is aprojection function.Cor. Characterizations of intersection and projection functions;Arrow’s Theorem.

History

Kenneth J. Arrow (Nobel, 1973, Econonics)

learned the Calculus of Relations from Alfred Tarski in 1940

applied it to his problems in economics

obtained Arrow’s Impossibility Theorem (Arrow 1950, 1963)

created Social Choice Theory

History

History, cont.

spring semester of 1940

Tarski was a Visiting Professor of Philosophy at the CityCollege of New York

class attended by Kenneth Arrow:

“It was a great course, Calculus of Relations. His organizationwas beautiful — I could tell that immediately — and he wasthorough. In fact what I learned from him played a role in myown later work — not so much the particular theorems butthe language of relations was immediately applicable toeconomics. I could express my problems in those terms”

Arrow proofread Tarski’s Introduction to Logic for no pay atTarski’s request:

“I also owe many thanks to Mr. K. J. Arrow for his help inreading proofs.”

History, cont.

Recast Arrow’s Theorem as a contribution to the Calculus ofRelations

What does the proof of Arrow’s Theorem show?

Avoid Axiom I

References

Arrow, K.J. (1950), ”A difficulty in the concept of social welfare”,Journal of Political Economy 58:328-246.Arrow, K.J. (1951), Social Choice and Individual Values, 1stedition (Wiley, New York).Arrow, K.J. (1963), Social Choice and Individual Values, 2ndedition (Wiley, New York) (all subsequent quotations from here)

References

Arrow’s Theorem — informally

Th. There is no fair social welfare function.Th. There is no multivariable relational operator on weak orderingsof a set with at least three elements that is non-dictatorial,independent of irrevelevant alternatives, and Pareto.

Arrow’s Theorem — more details

Hypotheses:

U is a set with three or more elements

W is the set of weak orderings of U

T is the set of total orderings of U

2 ≤ n < ω

f is an n-ary function mapping Wn to W(f is a social welfare function)

f satisfies Arrow’s Condition P (“Pareto”)and Arrow’s Condition 3 (“independence of irrelevantalternatives”)

Conclusion:

f is a projection function on total orderings, i. e.,for some “dictator” k ∈ {1, . . . , n},

f (R1, . . . ,Rn) = Rk

for all total orderings R1, . . . ,Rn ∈ TR. D. Maddux Arrow’s Theorem for Incomplete Relations

Hypotheses:

2 ≤ n < ω

Conclusion:

f (R1, . . . ,Rn) = Rk

Hypotheses:

2 ≤ n < ω

Conclusion:

f (R1, . . . ,Rn) = Rk

Hypotheses:

2 ≤ n < ω

Conclusion:

f (R1, . . . ,Rn) = Rk

Hypotheses:

2 ≤ n < ω

Conclusion:

f (R1, . . . ,Rn) = Rk

Hypotheses:

2 ≤ n < ω

Conclusion:

f (R1, . . . ,Rn) = Rk

Hypotheses:

2 ≤ n < ω

Conclusion:

f (R1, . . . ,Rn) = Rk

Hypotheses:

2 ≤ n < ω

Conclusion:

f (R1, . . . ,Rn) = Rk

Hypotheses:

2 ≤ n < ω

Conclusion:

f (R1, . . . ,Rn) = Rk

Hypotheses:

2 ≤ n < ω

Conclusion:

f (R1, . . . ,Rn) = Rk

Hypotheses:

2 ≤ n < ω

Conclusion:

f (R1, . . . ,Rn) = Rk

Hypotheses:

2 ≤ n < ω

Conclusion:

f (R1, . . . ,Rn) = Rk

Definitions from the Calculus of Relations

A binary relation R ⊆ U2 = {〈x , y〉 : x , y ∈ U}The diversity relation 0

,= {〈x , y〉 : x , y ∈ U, x 6= y}

The identity relation 1,= {〈x , x〉 : x ∈ U}

xRy iff 〈x , y〉 ∈ RThe complement of R ⊆ U2 is

R = {〈x , y〉 : x , y ∈ U ∧ 〈x , y〉 /∈ R}

for all x , y ∈ U, xRy iff ¬xRyThe relative product of R and R ′ is

R ;R ′ = {〈x , z〉 : ∃y (xRy ∧ yR ′z)}

The converse of R is

R−1 = {〈y , x〉 : xRy}

,= {〈x , y〉 : x , y ∈ U, x 6= y}

R = {〈x , y〉 : x , y ∈ U ∧ 〈x , y〉 /∈ R}

R ;R ′ = {〈x , z〉 : ∃y (xRy ∧ yR ′z)}

R−1 = {〈y , x〉 : xRy}

,= {〈x , y〉 : x , y ∈ U, x 6= y}

R = {〈x , y〉 : x , y ∈ U ∧ 〈x , y〉 /∈ R}

R ;R ′ = {〈x , z〉 : ∃y (xRy ∧ yR ′z)}

R−1 = {〈y , x〉 : xRy}

,= {〈x , y〉 : x , y ∈ U, x 6= y}

R = {〈x , y〉 : x , y ∈ U ∧ 〈x , y〉 /∈ R}

R ;R ′ = {〈x , z〉 : ∃y (xRy ∧ yR ′z)}

R−1 = {〈y , x〉 : xRy}

,= {〈x , y〉 : x , y ∈ U, x 6= y}

R = {〈x , y〉 : x , y ∈ U ∧ 〈x , y〉 /∈ R}

R ;R ′ = {〈x , z〉 : ∃y (xRy ∧ yR ′z)}

R−1 = {〈y , x〉 : xRy}

,= {〈x , y〉 : x , y ∈ U, x 6= y}

R = {〈x , y〉 : x , y ∈ U ∧ 〈x , y〉 /∈ R}

R ;R ′ = {〈x , z〉 : ∃y (xRy ∧ yR ′z)}

R−1 = {〈y , x〉 : xRy}

,= {〈x , y〉 : x , y ∈ U, x 6= y}

R = {〈x , y〉 : x , y ∈ U ∧ 〈x , y〉 /∈ R}

R ;R ′ = {〈x , z〉 : ∃y (xRy ∧ yR ′z)}

R−1 = {〈y , x〉 : xRy}

,= {〈x , y〉 : x , y ∈ U, x 6= y}

R = {〈x , y〉 : x , y ∈ U ∧ 〈x , y〉 /∈ R}

R ;R ′ = {〈x , z〉 : ∃y (xRy ∧ yR ′z)}

R−1 = {〈y , x〉 : xRy}

,= {〈x , y〉 : x , y ∈ U, x 6= y}

R = {〈x , y〉 : x , y ∈ U ∧ 〈x , y〉 /∈ R}

R ;R ′ = {〈x , z〉 : ∃y (xRy ∧ yR ′z)}

R−1 = {〈y , x〉 : xRy}

More definitions from the Calculus of Relations

Binary relation R ⊆ U2 is

transitive if ∀x ,y ,z∈U (xRy ∧ yRz ⇒ xRz) iff R ;R ⊆ R

co-transitive if R is transitive

reflexive if ∀x∈U (xRx) iff 1, ⊆ R

connected if ∀x ,y∈U (x 6= y ⇒ xRy ∨ yRx) iff 0, ⊆ R ∪ R−1

complete if ∀x ,y∈U (xRy ∨ yRx) iff 1 = R ∪ R−1

symmetric if ∀x ,y∈U (xRy ⇒ yRx) iff R = R−1

anti-symmetric if ∀x ,y∈U (xRy ⇒ (x = y ∨ yRx)) iffR ∩ R−1 ⊆ 1

an equivalence relation if R is transitive, symmetric, andreflexive

a weak ordering if R is transitive, reflexive, and connected

a total ordering if R is transitive, reflexive, connected, andanti-symmetric

Definition of a social welfare function

A social welfare function is a multivariate operator that mapsweak orderings to weak orderings:

f : Wn →W

i.e., for all R1, . . . ,Rn ∈ W,

R = f (R1, . . . ,Rn) ∈ W

Interpretations:

U is a set of alternative social states

input Ri ∈ W is a ranking of alternative social states bysociety member i ∈ {1, . . . , n}the output R is the social ordering

Why use weak orderings?

f : Wn →W

i.e., for all R1, . . . ,Rn ∈ W,

R = f (R1, . . . ,Rn) ∈ W

Interpretations:

f : Wn →W

i.e., for all R1, . . . ,Rn ∈ W,

R = f (R1, . . . ,Rn) ∈ W

Interpretations:

f : Wn →W

i.e., for all R1, . . . ,Rn ∈ W,

R = f (R1, . . . ,Rn) ∈ W

Interpretations:

f : Wn →W

i.e., for all R1, . . . ,Rn ∈ W,

R = f (R1, . . . ,Rn) ∈ W

Interpretations:

f : Wn →W

i.e., for all R1, . . . ,Rn ∈ W,

R = f (R1, . . . ,Rn) ∈ W

Interpretations:

f : Wn →W

i.e., for all R1, . . . ,Rn ∈ W,

R = f (R1, . . . ,Rn) ∈ W

Interpretations:

f : Wn →W

i.e., for all R1, . . . ,Rn ∈ W,

R = f (R1, . . . ,Rn) ∈ W

Interpretations:

Why weak orderings?

(p. 129–15). “It is assumed further that . . . the chooser considers inturn all possible pairs of alternatives, say x and y , and for eachsuch pair he makes one and only one of three decisions: x ispreferred to y [xPy ], x is indifferent to y [xIy ], or y is preferred tox [yPx ]. The decisions made for different pairs are assumed to beconsistent with each other, so, for example, if x is preferred to yand y to z then x is preferred to z ; similarly, if x is indifferent to yand y to z , then x is indifferent to z .”Arrow assumes

P ∪ I ∪ P−1 = 0,

P, I , P−1 are pairwise disjoint

P and I are transitive

Arrow replaces P and I with R = P ∪ I

Axiom I. R is complete, reflexive, and connected: 1 = R ∪R−1

Axiom II. R is transitive: R ;R ⊆ R

Why weak orderings?

P ∪ I ∪ P−1 = 0,

Why weak orderings?

P ∪ I ∪ P−1 = 0,

Why weak orderings?

P ∪ I ∪ P−1 = 0,

Why weak orderings?

P ∪ I ∪ P−1 = 0,

Why weak orderings?

P ∪ I ∪ P−1 = 0,

Why weak orderings?

P ∪ I ∪ P−1 = 0,

Why weak orderings?

P ∪ I ∪ P−1 = 0,

Why weak orderings?

P ∪ I ∪ P−1 = 0,

Why weak orderings? (cont.)

Arrow: a weak ordering is a relation satisfying Axioms I and II.(p.146–7) “. . . we ordinarily feel that not only the relation R butalso the relations of (strict) preference and of indifference aretransitive.”Arrow gets these “usually desired properties . . . by definingpreference and indifference suitably in terms of R”.Definition 1. P = R−1

Definition 2. I = R ∩ R−1

(p.1915–19) “. . . it will be assumed that individuals are rational, bywhich is meant that the ordering relations Ri satisfy Axioms Iand II. The problem will be to construct an ordering relation R forsociety as a whole that will also reflect rational choice-making so Rmay also be assumed to satisfy Axioms I and II.”

Arrow’s Conditions 1′, P, and 5 on a social welfare function

(p. 96) Condition 1′. “All logically possible orderings areadmissible” (the domain of f is Wn)

(p. 96) Condition P. N = {1, . . . , n}, Pi = Ri−1, P = R−1,

∀R1,...,Rn∈W ∀x ,y∈U (∀i∈N (xPiy) ⇒ xPy)

(p. 30) Condition 5. The social welfare function is not dictatorial

¬∃k∈N ∀x ,y∈U (xPky ⇒ xPy)

Conditions P and 5 are about P, not R!

Arrow’s Condition 3 on a social choice function

(p. 15) “In terms of the relation R, we may now define the conceptof choice,. . . ”Definition 3. C (S) = {x : x ∈ S ∧ ∀y∈S (xRy)} if S ⊆ U

Lemma 2. ∀x ,y∈U

(xPy ⇔ {x} = C ({x , y})

)(by Axiom I)

(p. 27) Condition 3. (“independence of irrelevant alternatives”)

∀R1,...,Rn,R′1,...,R

′n∈W ∀S⊆U

(∀i∈N ∀x ,y∈S (xRiy ⇔ xR ′

⇒ C (S) = C ′(S)).

(p.97) “Theorem 2: Conditions 1′, 3, P, and 5 are inconsistent.”To eliminate C , use x ∈ C ({x , y}) ⇔ xRx ∧ xRy (by logic alone)

(xPy ⇔ {x} = C ({x , y})

)(by Axiom I)

∀R1,...,Rn,R′1,...,R

′n∈W ∀S⊆U

⇒ C (S) = C ′(S)).

(xPy ⇔ {x} = C ({x , y})

)(by Axiom I)

∀R1,...,Rn,R′1,...,R

′n∈W ∀S⊆U

⇒ C (S) = C ′(S)).

(xPy ⇔ {x} = C ({x , y})

)(by Axiom I)

∀R1,...,Rn,R′1,...,R

′n∈W ∀S⊆U

⇒ C (S) = C ′(S)).

(xPy ⇔ {x} = C ({x , y})

)(by Axiom I)

∀R1,...,Rn,R′1,...,R

′n∈W ∀S⊆U

⇒ C (S) = C ′(S)).

(xPy ⇔ {x} = C ({x , y})

)(by Axiom I)

∀R1,...,Rn,R′1,...,R

′n∈W ∀S⊆U

⇒ C (S) = C ′(S)).

(xPy ⇔ {x} = C ({x , y})

)(by Axiom I)

∀R1,...,Rn,R′1,...,R

′n∈W ∀S⊆U

⇒ C (S) = C ′(S)).

Arrow’s Theorem restated, “weak” vs. “total”

Arrow’s Th. Conditions 1′, 3, and P imply the negation ofCondition 5:

∃k∈N ∀x ,y∈U (xPky ⇒ xPy)

i. e., by Axiom I, there is some “dictator” k ∈ {1, . . . , n} such that

f (R1, . . . ,Rn) = Rk

for all total orderings R1, . . . ,Rn (f is a projection function on T )

Arrow’s theorem characterizes projection functions on T asfunctions f : T n → T satisfying Conditions P and 3 (however, verymany non-projection functions f : Wn →W satisfy Conditions Pand 3)

f (R1, . . . ,Rn) = Rk

Arrow’s Condition 3 for preference relations

Arrow’s characterization and its proof should only involvepreference relations, but Condition 3 applies to R, not P, so use a“unidirectional” property of R ⊆ P(U2), called IIA:

∀R1,...,Rn,R′1,...,R

′n∈R ∀v ,w∈U

v 6= w ∧ ∀i∈N

(vRiw ⇔ vR ′

i w)⇒ (vRw ⇔ vR ′w)

IIAimplies Cond. 3 if R = W, butCond. 3 implies IIA if R = T

∀R1,...,Rn,R′1,...,R

v 6= w ∧ ∀i∈N

(vRiw ⇔ vR ′

∀R1,...,Rn,R′1,...,R

v 6= w ∧ ∀i∈N

(vRiw ⇔ vR ′

∀R1,...,Rn,R′1,...,R

v 6= w ∧ ∀i∈N

(vRiw ⇔ vR ′

∀R1,...,Rn,R′1,...,R

v 6= w ∧ ∀i∈N

(vRiw ⇔ vR ′

∀R1,...,Rn,R′1,...,R

v 6= w ∧ ∀i∈N

(vRiw ⇔ vR ′

Diverse sets of relations

R ⊆ P(U2) is diverse if

∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

(xRy ∧ xRz ∧ zRy

))∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

))∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

) )∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

) )R is very diverse if R is diverse and also

∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

)∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

)2 (of 8) conditions are missing—the ones that can’t happen fortransitive, co-transitive relations

∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

))∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

))∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

) )∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

)∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

))∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

))∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

) )∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

)∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

))∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

))∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

) )∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

)∀x ,y ,z∈U

(|{x , y , z}| = 3 ⇒ ∃R∈R

Examples of diverse sets of relations

Let R = {{〈x , y〉 , 〈x , z〉 , 〈z , y〉} : x , y , z ∈ U, |{x , z , y}| = 3}R contains only transitive relations and is very diverseSome very diverse sets of relations:

transitive relations

co-transitive relations

reflexive relations

connected relations

symmetric relations

anti-symmetric relations

weak orderings Wtotal orderings T

The set of equivalence relations on U is diverse but not very diverse

reflexive relations

connected relations

symmetric relations

reflexive relations

connected relations

symmetric relations

reflexive relations

connected relations

symmetric relations

reflexive relations

connected relations

symmetric relations

reflexive relations

connected relations

symmetric relations

reflexive relations

connected relations

symmetric relations

reflexive relations

connected relations

symmetric relations

reflexive relations

connected relations

symmetric relations

reflexive relations

connected relations

symmetric relations

reflexive relations

connected relations

symmetric relations

reflexive relations

connected relations

symmetric relations

reflexive relations

connected relations

symmetric relations

reflexive relations

connected relations

symmetric relations

Intersection Theorem

Th. 1. Assume

U is a set with at least three elements

R ⊆ P(U2) is diverse

2 ≤ n < ω, f : P(U2)n → P(U2)

Trans ∀R1,...,Rn∈R(f (R1, . . . ,Rn) is transitive

)Unan ∀x ,y∈U ∀R1,...,Rn∈R

(x 6= y ∧ ∀i∈N (xRiy) ⇒ xf (R1, . . . ,Rn)y

)IIA ∀x ,y∈U ∀R1,...,Rn,R′

1,...,R′n∈R

(x 6= y ∧ ∀i∈N (xRiy ⇔ xR ′

⇒(xf (R1, . . . ,Rn)y ⇔ xf (R ′

1, . . . ,R′n)y))

Then f (on diversity relations) is just intersection over some set ofinput variables

∃D0⊆N ∀R1,...,Rn∈R(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

)R. D. Maddux Arrow’s Theorem for Incomplete Relations

Th. 1. Assume

2 ≤ n < ω, f : P(U2)n → P(U2)

(x 6= y ∧ ∀i∈N (xRiy) ⇒ xf (R1, . . . ,Rn)y

)IIA ∀x ,y∈U ∀R1,...,Rn,R′

1,...,R′n∈R

(x 6= y ∧ ∀i∈N (xRiy ⇔ xR ′

⇒(xf (R1, . . . ,Rn)y ⇔ xf (R ′

1, . . . ,R′n)y))

∃D0⊆N ∀R1,...,Rn∈R(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

Th. 1. Assume

2 ≤ n < ω, f : P(U2)n → P(U2)

(x 6= y ∧ ∀i∈N (xRiy) ⇒ xf (R1, . . . ,Rn)y

)IIA ∀x ,y∈U ∀R1,...,Rn,R′

1,...,R′n∈R

(x 6= y ∧ ∀i∈N (xRiy ⇔ xR ′

⇒(xf (R1, . . . ,Rn)y ⇔ xf (R ′

1, . . . ,R′n)y))

∃D0⊆N ∀R1,...,Rn∈R(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

Th. 1. Assume

2 ≤ n < ω, f : P(U2)n → P(U2)

(x 6= y ∧ ∀i∈N (xRiy) ⇒ xf (R1, . . . ,Rn)y

)IIA ∀x ,y∈U ∀R1,...,Rn,R′

1,...,R′n∈R

(x 6= y ∧ ∀i∈N (xRiy ⇔ xR ′

⇒(xf (R1, . . . ,Rn)y ⇔ xf (R ′

1, . . . ,R′n)y))

∃D0⊆N ∀R1,...,Rn∈R(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

Th. 1. Assume

2 ≤ n < ω, f : P(U2)n → P(U2)

(x 6= y ∧ ∀i∈N (xRiy) ⇒ xf (R1, . . . ,Rn)y

)IIA ∀x ,y∈U ∀R1,...,Rn,R′

1,...,R′n∈R

(x 6= y ∧ ∀i∈N (xRiy ⇔ xR ′

⇒(xf (R1, . . . ,Rn)y ⇔ xf (R ′

1, . . . ,R′n)y))

∃D0⊆N ∀R1,...,Rn∈R(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

Th. 1. Assume

2 ≤ n < ω, f : P(U2)n → P(U2)

(x 6= y ∧ ∀i∈N (xRiy) ⇒ xf (R1, . . . ,Rn)y

)IIA ∀x ,y∈U ∀R1,...,Rn,R′

1,...,R′n∈R

(x 6= y ∧ ∀i∈N (xRiy ⇔ xR ′

⇒(xf (R1, . . . ,Rn)y ⇔ xf (R ′

1, . . . ,R′n)y))

∃D0⊆N ∀R1,...,Rn∈R(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

Th. 1. Assume

2 ≤ n < ω, f : P(U2)n → P(U2)

(x 6= y ∧ ∀i∈N (xRiy) ⇒ xf (R1, . . . ,Rn)y

)IIA ∀x ,y∈U ∀R1,...,Rn,R′

1,...,R′n∈R

(x 6= y ∧ ∀i∈N (xRiy ⇔ xR ′

⇒(xf (R1, . . . ,Rn)y ⇔ xf (R ′

1, . . . ,R′n)y))

∃D0⊆N ∀R1,...,Rn∈R(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

Th. 1. Assume

2 ≤ n < ω, f : P(U2)n → P(U2)

(x 6= y ∧ ∀i∈N (xRiy) ⇒ xf (R1, . . . ,Rn)y

)IIA ∀x ,y∈U ∀R1,...,Rn,R′

1,...,R′n∈R

(x 6= y ∧ ∀i∈N (xRiy ⇔ xR ′

⇒(xf (R1, . . . ,Rn)y ⇔ xf (R ′

1, . . . ,R′n)y))

∃D0⊆N ∀R1,...,Rn∈R(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

Th. 1. Assume

2 ≤ n < ω, f : P(U2)n → P(U2)

(x 6= y ∧ ∀i∈N (xRiy) ⇒ xf (R1, . . . ,Rn)y

)IIA ∀x ,y∈U ∀R1,...,Rn,R′

1,...,R′n∈R

(x 6= y ∧ ∀i∈N (xRiy ⇔ xR ′

⇒(xf (R1, . . . ,Rn)y ⇔ xf (R ′

1, . . . ,R′n)y))

∃D0⊆N ∀R1,...,Rn∈R(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

Th. 1. Assume

2 ≤ n < ω, f : P(U2)n → P(U2)

(x 6= y ∧ ∀i∈N (xRiy) ⇒ xf (R1, . . . ,Rn)y

)IIA ∀x ,y∈U ∀R1,...,Rn,R′

1,...,R′n∈R

(x 6= y ∧ ∀i∈N (xRiy ⇔ xR ′

⇒(xf (R1, . . . ,Rn)y ⇔ xf (R ′

1, . . . ,R′n)y))

∃D0⊆N ∀R1,...,Rn∈R(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

Limits to the theorem

D0 may not be a proper subset of N (perhaps D0 = N), e.g.,if R be the set of equivalence relations on U

and ∀R1,...,Rn∈P(U2)

(f (R1, . . . ,Rn) =

⋂i∈N Ri

)then the hypotheses of the Intersection Theorem all hold—

R is diverse

Unan holds trivially

IIA obviously also holds

f preserves transitivity, reflexivity, and symmetry, so

applied to equivalence relations in R, f produces a (transitive)equivalence relation, so

Trans holds

(f (R1, . . . ,Rn) =

⋂i∈N Ri

R is diverse

Trans holds

(f (R1, . . . ,Rn) =

⋂i∈N Ri

R is diverse

Trans holds

(f (R1, . . . ,Rn) =

⋂i∈N Ri

R is diverse

Trans holds

(f (R1, . . . ,Rn) =

⋂i∈N Ri

R is diverse

Trans holds

(f (R1, . . . ,Rn) =

⋂i∈N Ri

R is diverse

Trans holds

(f (R1, . . . ,Rn) =

⋂i∈N Ri

R is diverse

Trans holds

(f (R1, . . . ,Rn) =

⋂i∈N Ri

R is diverse

Trans holds

(f (R1, . . . ,Rn) =

⋂i∈N Ri

R is diverse

Trans holds

(f (R1, . . . ,Rn) =

⋂i∈N Ri

R is diverse

Trans holds

Proof of Arrow’s theorem

Use the Intersection Theorem to prove Arrow’s: assumingf : Wn →W satisfies Conds. P, 3, let with R = T and check—

R is diverse because T is very diverse

f : Wn →W, so f produces only transitive output relations

IIA follows from Condition 3 because R = T (not W!)

Unan follows from Condition P by Axiom I and R = T (pf?)

therefore, by Th. 1, for some subset D0 ⊆ N,

∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

)D0 cannot have two or more elements, because we can feed twototal orderings to f that disagree on a pair of alternatives, and getan output relation from f that is not connected—

∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩⋂

i∈D0

Proof of Arrow’s theorem, cont.

Suppose i 6= j and i , j ∈ D0

Pick distinct x , y ∈ U, pick Ri ,Rj ∈ T so that xRiy , yRix , yRjx ,and xRjy(this is easy with total orderings, which can be anti-symmetric, butit cannot be done with equivalence relations, because they aresymmetric)Let R = f (. . . , Ri , . . . ,Rj , . . . ) = Ri ∩ Rj

with missing input relations chosen arbitrarily from {Ri ,Rj}Then xRy and yRx , i. e., R is not connected, contradicting theassumption that f produces weak (hence connected) orderingsThis shows |D0| < 2Is D0 = ∅?

If D0 = ∅ then ∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

, )but f (R0, . . . ,R0) = R0 by Condition P and the reflexivity ofrelations in W, hence D0 6= ∅, the only possibility left is D0 = {k}for some k ∈ N, hence

∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩ Rk

)However, since all (output and input) relations in W (and T ) arereflexive, f is actually a projection function (onto coordinate k):

∀R1,...,Rn∈T(

f (R1, . . . ,Rn) = Rk

)applying converse-complement and notational conventions gives

∀R1,...,Rn∈T(

P = Pk

)so Condition 5 fails (because there is a “dictator” k)

If D0 = ∅ then ∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩ Rk

∀R1,...,Rn∈T(

f (R1, . . . ,Rn) = Rk

∀R1,...,Rn∈T(

P = Pk

If D0 = ∅ then ∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩ Rk

∀R1,...,Rn∈T(

f (R1, . . . ,Rn) = Rk

∀R1,...,Rn∈T(

P = Pk

If D0 = ∅ then ∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩ Rk

∀R1,...,Rn∈T(

f (R1, . . . ,Rn) = Rk

∀R1,...,Rn∈T(

P = Pk

If D0 = ∅ then ∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩ Rk

∀R1,...,Rn∈T(

f (R1, . . . ,Rn) = Rk

∀R1,...,Rn∈T(

P = Pk

If D0 = ∅ then ∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩ Rk

∀R1,...,Rn∈T(

f (R1, . . . ,Rn) = Rk

∀R1,...,Rn∈T(

P = Pk

If D0 = ∅ then ∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

∀R1,...,Rn∈T(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩ Rk

∀R1,...,Rn∈T(

f (R1, . . . ,Rn) = Rk

∀R1,...,Rn∈T(

P = Pk

Projection Theorem

Is connectedness really the key property that produces a dictator?No, because—Th. 2. If R ⊆ P(U2) is very diverse, |U| > 2, 2 ≤ n < ω,f : Rn → P(U2), IIA, Unan, Trans, and f produces co-transitiveoutputs from inputs in R,

Co-trans ∀R1,...,Rn∈R(f (R1, . . . ,Rn) is co-transitive

∃k∈N ∀R1,...,Rn∈R(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩ Rk

)Pf (of Arrow’s Th.) Given Arrow’s hypotheses, let R = T . ThenTrans, Unan, IIA hold (shown above), T is very diverse, and everytotal ordering is co-transitive (but not conversely). By Th.2, f is aprojection function when restricted to T , hence there is a dictator.

Projection Theorem

∃k∈N ∀R1,...,Rn∈R(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩ Rk

Projection Theorem

∃k∈N ∀R1,...,Rn∈R(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩ Rk

Projection Theorem

∃k∈N ∀R1,...,Rn∈R(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩ Rk

Projection Theorem

∃k∈N ∀R1,...,Rn∈R(

0, ∩ f (R1, . . . ,Rn) = 0

, ∩ Rk

Characterizations

Arrow’s Theorem says that a multivariate operator on T is aprojection function iff Conditions 3 and PLet R be the reflexive, transitive, co-transitive relations on UThe Projection Theorem says that a multivariate operator on R isa projection function iff IIA and UnanT ⊂ W ⊂ R but W and R are “almost” the same—

Characterizations

Transitive, co-transitive relations

Assume R is transitive and co-transitive, e. g., R = ∅ (not refl.)Let I = 1

, ∪ (R ∩ R−1), J = 1, ∪ (R ∩ R−1), E = I ∪ J

Then I , J, E are equivalence relations, I ∩ J = 1,

Let X , Y be disjoint equivalence classes of E , x ∈ X , y ∈ YEither xRy , yRx , X × Y ⊆ R, and (Y × X ) ∩ R = ∅or else yRx , xRy , Y × X ⊆ R, and (X × Y ) ∩ R = ∅The E -classes are totally ordered by ≤ where

X ≤ Y ⇔ ∃x∈X∃y∈Y (xRy)

If any J-class has two or more elements, then R is incomplete(hence not in W) since there are x , y ∈ U with xRy and yRx (Rcomplete ⇔ J = 1

, ∪ (R ∩ R−1), J = 1, ∪ (R ∩ R−1), E = I ∪ J

Prospects

Other theorems, Gibbard?, Sen?,. . . ,Other operations, converse?, union?

Prospects

Condition 3

Arrow’s Condition 3 is

∀R1,...,Rn,R′1,...,R

′n∈W ∀S⊆U(

∀i∈N ∀x ,y∈S (xRiy ⇔ xR ′i y) ⇒ C (S) = C ′(S)

)Instantiate to S = {v ,w} with v 6= w to get Cond. 3 for 2-elementsets (pairs):

∀R1,...,Rn,R′1,...,R

′n∈W ∀v ,w∈U(

v 6= w ∧ ∀i∈N ∀x ,y∈{v ,w} (xRiy ⇔ xR ′i y) ⇒ C ({v ,w}) = C ′({v ,w})

Condition 3 for pairs

Eliminate “ ∀x ,y∈{v ,w} ” to get this equivalent form of Cond. 3 forpairs:

∀R1,...,Rn,R′1,...,R

′n∈W ∀v ,w∈U

(v 6= w ∧ ∀i∈N

((vRiv ⇔ vR ′

i v) ∧ (vRiw ⇔ vR ′i w)∧

(wRiv ⇔ wR ′i v) ∧ (wRiw ⇔ wR ′

⇒ C ({v ,w}) = C ′({v ,w})

By the definition of C (S), x ∈ C ({x , y}) ⇔ xRx ∧ xRy , so Cond. 3for pairs is equivalent to

∀R1,...,Rn,R′1,...,R

′n∈W ∀v ,w∈U(

v 6= w ∧ ∀i∈N

((vRiv ⇔ vR ′

i v) ∧ (vRiw ⇔ vR ′i w)∧

(wRiv ⇔ wR ′i v) ∧ (wRiw ⇔ wR ′

⇒ ((vRv ∧ vRw) ⇔ (vR ′v ∧ vR ′w))∧

((wRv ∧ wRw) ⇔ (wR ′v ∧ wR ′w))

All the relations in W and all the output relations are reflexive, soCond. 3 for pairs is equivalent to

∀R1,...,Rn,R′1,...,R

′n∈W ∀v ,w∈U(

v 6= w ∧ ∀i∈N

((vRiw ⇔ vR ′

i w) ∧ (wRiv ⇔ wR ′i v))

⇒ (vRw ⇔ vR ′w) ∧ (wRv ⇔ wR ′v)

restricted Condition 3 for pairs

Restrict Cond. 3 for pairs from W to T ⊆ W (from weak to totalorderings)Note that, since 0

, ∩ R = 0, ∩ R−1 whenever R ∈ T , if v 6= w then

(vRiw ⇔ vR ′i w) ⇔(vRiw ⇔ vR ′

i w) by logic

⇔(vRi−1w ⇔ vR ′

w) Ri ,R′i ∈ T , v 6= w

⇔(wRiv ⇔ wR ′i v) def. of −1

so the Cond 3. for pairs, restricted to total orderings, is equivalentto

Condition 3 restricted to pairs and total orderings

∀R1,...,Rn,R′1,...,R

′n∈T ∀v ,w∈U

v 6= w ∧ ∀i∈N

(vRiw ⇔ vR ′

i w)⇒ (vRw ⇔ vR ′w) ∧ (wRv ⇔ wR ′v)

We only need the consequence obtained by deleting the finalconjunct (equivalent if R and R ′ are also total orderings):

A consequence of Cond. 3 restricted to pairs and T

∀R1,...,Rn,R′1,...,R

′n∈T ∀v ,w∈U

v 6= w ∧ ∀i∈N

(vRiw ⇔ vR ′

This is IIA with R = T .

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