Arrow's Theorem for Incomplete Relations
Transcript of Arrow's Theorem for Incomplete Relations
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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
![Page 2: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/2.jpg)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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.
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 4: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/4.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 5: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/5.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 6: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/6.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 7: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/7.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 8: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/8.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 9: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/9.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 10: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/10.jpg)
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.”
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 11: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/11.jpg)
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.”
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 12: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/12.jpg)
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.”
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 13: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/13.jpg)
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.”
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 14: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/14.jpg)
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.”
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 15: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/15.jpg)
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.”
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 16: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/16.jpg)
Goals
Recast Arrow’s Theorem as a contribution to the Calculus ofRelations
What does the proof of Arrow’s Theorem show?
Avoid Axiom I
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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Goals
Recast Arrow’s Theorem as a contribution to the Calculus ofRelations
What does the proof of Arrow’s Theorem show?
Avoid Axiom I
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 18: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/18.jpg)
Goals
Recast Arrow’s Theorem as a contribution to the Calculus ofRelations
What does the proof of Arrow’s Theorem show?
Avoid Axiom I
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 19: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/19.jpg)
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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 22: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/22.jpg)
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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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.
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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.
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 25: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/25.jpg)
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.
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 26: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/26.jpg)
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
![Page 27: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/27.jpg)
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
![Page 28: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/28.jpg)
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
![Page 29: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/29.jpg)
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
![Page 30: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/30.jpg)
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
![Page 31: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/31.jpg)
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
![Page 32: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/32.jpg)
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
![Page 33: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/33.jpg)
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
![Page 34: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/34.jpg)
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
![Page 35: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/35.jpg)
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
![Page 36: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/36.jpg)
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
![Page 37: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/37.jpg)
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
![Page 38: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/38.jpg)
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}
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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}
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 40: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/40.jpg)
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}
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 41: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/41.jpg)
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}
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 42: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/42.jpg)
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}
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 43: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/43.jpg)
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}
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 44: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/44.jpg)
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}
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 45: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/45.jpg)
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}
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 46: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/46.jpg)
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}
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 47: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/47.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 48: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/48.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 49: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/49.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 50: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/50.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 51: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/51.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 52: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/52.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 53: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/53.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 54: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/54.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 55: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/55.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 56: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/56.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 57: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/57.jpg)
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?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 58: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/58.jpg)
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?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 59: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/59.jpg)
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?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 60: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/60.jpg)
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?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 61: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/61.jpg)
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?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 62: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/62.jpg)
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?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 63: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/63.jpg)
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?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 64: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/64.jpg)
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?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 65: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/65.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 66: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/66.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 67: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/67.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 69: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/69.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 70: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/70.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 71: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/71.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 72: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/72.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 73: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/73.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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.”
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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.”
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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.”
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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.”
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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.”
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 79: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/79.jpg)
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.”
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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!
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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!
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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!
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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!
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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!
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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 ′
i y)
⇒ 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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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 ′
i y)
⇒ 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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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 ′
i y)
⇒ 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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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 ′
i y)
⇒ 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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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 ′
i y)
⇒ 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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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 ′
i y)
⇒ 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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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 ′
i y)
⇒ 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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 95: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/95.jpg)
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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 96: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/96.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 97: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/97.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 98: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/98.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 99: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/99.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 100: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/100.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 101: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/101.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 102: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/102.jpg)
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
(xRy ∧ xRz ∧ zRy
))∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
) )∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
) )R is very diverse if R is diverse and also
∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
)∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
)2 (of 8) conditions are missing—the ones that can’t happen fortransitive, co-transitive relations
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 103: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/103.jpg)
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
(xRy ∧ xRz ∧ zRy
))∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
) )∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
) )R is very diverse if R is diverse and also
∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
)∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
)2 (of 8) conditions are missing—the ones that can’t happen fortransitive, co-transitive relations
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 104: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/104.jpg)
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
(xRy ∧ xRz ∧ zRy
))∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
) )∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
) )R is very diverse if R is diverse and also
∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
)∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
)2 (of 8) conditions are missing—the ones that can’t happen fortransitive, co-transitive relations
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 105: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/105.jpg)
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
(xRy ∧ xRz ∧ zRy
))∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
) )∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
) )R is very diverse if R is diverse and also
∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
)∀x ,y ,z∈U
(|{x , y , z}| = 3 ⇒ ∃R∈R
(xRy ∧ xRz ∧ zRy
)2 (of 8) conditions are missing—the ones that can’t happen fortransitive, co-transitive relations
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 106: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/106.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 107: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/107.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 108: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/108.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 109: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/109.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 110: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/110.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 111: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/111.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 112: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/112.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 113: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/113.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 114: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/114.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 115: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/115.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 116: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/116.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 117: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/117.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 118: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/118.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 119: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/119.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 120: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/120.jpg)
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 ′
i y)
⇒(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
Ri
)R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 121: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/121.jpg)
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 ′
i y)
⇒(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
Ri
)R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 122: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/122.jpg)
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 ′
i y)
⇒(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
Ri
)R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 123: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/123.jpg)
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 ′
i y)
⇒(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
Ri
)R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 124: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/124.jpg)
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 ′
i y)
⇒(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
Ri
)R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 125: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/125.jpg)
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 ′
i y)
⇒(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
Ri
)R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 126: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/126.jpg)
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 ′
i y)
⇒(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
Ri
)R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 127: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/127.jpg)
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 ′
i y)
⇒(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
Ri
)R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 128: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/128.jpg)
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 ′
i y)
⇒(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
Ri
)R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 129: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/129.jpg)
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 ′
i y)
⇒(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
Ri
)R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 130: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/130.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 131: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/131.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 132: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/132.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 133: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/133.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 134: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/134.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 135: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/135.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 136: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/136.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 137: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/137.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 138: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/138.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 139: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/139.jpg)
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 140: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/140.jpg)
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
Ri
)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—
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 141: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/141.jpg)
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
Ri
)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—
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 142: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/142.jpg)
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
Ri
)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—
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 143: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/143.jpg)
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
Ri
)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—
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 144: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/144.jpg)
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
Ri
)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—
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 145: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/145.jpg)
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
Ri
)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—
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 146: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/146.jpg)
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
Ri
)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—
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 147: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/147.jpg)
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 = ∅?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 148: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/148.jpg)
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 = ∅?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 149: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/149.jpg)
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 = ∅?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 150: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/150.jpg)
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 = ∅?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 151: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/151.jpg)
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 = ∅?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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 = ∅?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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 = ∅?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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 = ∅?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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 = ∅?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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Proof of Arrow’s theorem, cont.
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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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Proof of Arrow’s theorem, cont.
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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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Proof of Arrow’s theorem, cont.
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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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Proof of Arrow’s theorem, cont.
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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 160: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/160.jpg)
Proof of Arrow’s theorem, cont.
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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 161: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/161.jpg)
Proof of Arrow’s theorem, cont.
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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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Proof of Arrow’s theorem, cont.
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)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 163: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/163.jpg)
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
)then
∃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.
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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
)then
∃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.
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 165: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/165.jpg)
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
)then
∃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.
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 166: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/166.jpg)
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
)then
∃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.
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 167: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/167.jpg)
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
)then
∃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.
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 168: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/168.jpg)
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—
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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—
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 170: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/170.jpg)
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—
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 171: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/171.jpg)
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—
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 172: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/172.jpg)
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—
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 173: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/173.jpg)
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. D. Maddux Arrow’s Theorem for Incomplete Relations
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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. D. Maddux Arrow’s Theorem for Incomplete Relations
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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. D. Maddux Arrow’s Theorem for Incomplete Relations
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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. D. Maddux Arrow’s Theorem for Incomplete Relations
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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. D. Maddux Arrow’s Theorem for Incomplete Relations
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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. D. Maddux Arrow’s Theorem for Incomplete Relations
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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. D. Maddux Arrow’s Theorem for Incomplete Relations
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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. D. Maddux Arrow’s Theorem for Incomplete Relations
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Prospects
Other theorems, Gibbard?, Sen?,. . . ,Other operations, converse?, union?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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Prospects
Other theorems, Gibbard?, Sen?,. . . ,Other operations, converse?, union?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
![Page 183: Arrow's Theorem for Incomplete Relations](https://reader030.fdocuments.in/reader030/viewer/2022020702/61fb10922e268c58cd59bc99/html5/thumbnails/183.jpg)
Prospects
Other theorems, Gibbard?, Sen?,. . . ,Other operations, converse?, union?
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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})
)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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 ′
i w))
⇒ C ({v ,w}) = C ′({v ,w})
)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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Condition 3 for pairs
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 ′
i w))
⇒ ((vRv ∧ vRw) ⇔ (vR ′v ∧ vR ′w))∧
((wRv ∧ wRw) ⇔ (wR ′v ∧ wR ′w))
)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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Condition 3 for pairs
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)
)
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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 ′
i−1
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
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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):
R. D. Maddux Arrow’s Theorem for Incomplete Relations
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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 ′
i w)⇒ (vRw ⇔ vR ′w)
)
This is IIA with R = T .
R. D. Maddux Arrow’s Theorem for Incomplete Relations