Path Independent Choice and the Ranking of Opportunity Sets
Path Independent Choice and the Ranking of Opportunity Sets Matthew Ryan University of Auckland
3rd CMSS Workshop
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
1 / 23
Theory of Choice
Finite “consumption set” X .
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
2 / 23
Theory of Choice
Finite “consumption set” X . A choice function is a mapping c : 2X ! 2X that satis…es:
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
2 / 23
Theory of Choice
Finite “consumption set” X . A choice function is a mapping c : 2X ! 2X that satis…es:
(CF0) c (A)
A for each A
Ryan (University of Auckland)
X , and
Plott Consistent Rankings
3rd CMSS Workshop
2 / 23
Theory of Choice
Finite “consumption set” X . A choice function is a mapping c : 2X ! 2X that satis…es:
(CF0) c (A) A for each A (CF1) c (A) = ∅ i¤ A = ∅.
Ryan (University of Auckland)
X , and
Plott Consistent Rankings
3rd CMSS Workshop
2 / 23
Theory of Choice
Finite “consumption set” X . A choice function is a mapping c : 2X ! 2X that satis…es:
(CF0) c (A) A for each A (CF1) c (A) = ∅ i¤ A = ∅.
X , and
A choice function is path independent if it also satis…es
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
2 / 23
Theory of Choice
Finite “consumption set” X . A choice function is a mapping c : 2X ! 2X that satis…es:
(CF0) c (A) A for each A (CF1) c (A) = ∅ i¤ A = ∅.
X , and
A choice function is path independent if it also satis…es (CF2) For any A, B 2 2X : c (A [ B ) = c (c (A) [ B )
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
2 / 23
Theory of Choice
Finite “consumption set” X . A choice function is a mapping c : 2X ! 2X that satis…es:
(CF0) c (A) A for each A (CF1) c (A) = ∅ i¤ A = ∅.
X , and
A choice function is path independent if it also satis…es (CF2) For any A, B 2 2X : c (A [ B ) = c (c (A) [ B )
A path independent choice function is also called a Plott function (after Plott, 1973).
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
2 / 23
Non-Binary Choice
Path independence is not su¢ cient to ensure that the choice function is binary.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
3 / 23
Non-Binary Choice
Path independence is not su¢ cient to ensure that the choice function is binary. A choice function c : 2X ! 2X is binary if there exists a binary relation % X X such that c (A) = max A %
where max A %
Ryan (University of Auckland)
fx 2 A j there is no y 2 A
Plott Consistent Rankings
fx g with y
xg .
3rd CMSS Workshop
3 / 23
Non-Binary Choice
Example (Plott, 1973) Let X = fx, y , z g and de…ne c : 2X ! 2X as follows: c (A) =
fx, y g if A = X A if A 6= X
It is easily veri…ed that c is a Plott function. Since c (X ) = fx, y g, we must have x z or y z if c is binary. But these contradict c (fx, z g) = fx, z g and c (fy , z g) = fy , z g, respectively, so c is non-binary.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
4 / 23
Ranking Opportunity Sets
De…nition An opportunity set ranking is a binary relation % re‡exive, complete and satis…es A
∅
2X
2X which is
for all A 6= ∅.
Think of ranking restaurants (i.e., menus).
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
5 / 23
Ranking Opportunity Sets
If (meal) choice is governed by the binary relation % X X , then opportunity sets (i.e., restaurants) are naturally ranked according to the following indirect utility (IU) principle: A% B
Ryan (University of Auckland)
,
max (A [ B ) \ A 6= ∅ %
Plott Consistent Rankings
3rd CMSS Workshop
(IU)
6 / 23
Ranking Opportunity Sets
If (meal) choice is governed by the binary relation % X X , then opportunity sets (i.e., restaurants) are naturally ranked according to the following indirect utility (IU) principle: A% B
,
max (A [ B ) \ A 6= ∅ %
(IU)
What are the hallmarks of opportunity set rankings that obey the IU principle?
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
6 / 23
Ranking Opportunity Sets
Theorem (Kreps, 1979) The opportunity set ranking % satis…es (IU) for some complete, re‡exive and transitive % X X i¤ % is transitive and satis…es A% B for every A, B
)
A
A[B
(K)
X.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
7 / 23
Ranking Opportunity Sets
De…nition (Lahiri, 2003) An opportunity set ranking % is justi…able if it satis…es (IU) for some complete and re‡exive (but not necessarily transitive) % X X
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
8 / 23
Ranking Opportunity Sets
The IU principle embodies two fundamental ideas:
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
9 / 23
Ranking Opportunity Sets
The IU principle embodies two fundamental ideas: 1
Consequentialism.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
9 / 23
Ranking Opportunity Sets
The IU principle embodies two fundamental ideas: 1 2
Consequentialism. Binariness.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
9 / 23
Ranking Opportunity Sets
The IU principle embodies two fundamental ideas: 1 2
Consequentialism. Binariness.
Many papers relax (1). We maintain (1) but relax (2).
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
9 / 23
Ranking Opportunity Sets By analogy with the IU condition A% B
,
max (A [ B ) \ A 6= ∅, %
we propose:
De…nition An opportunity set ranking % is Plott consistent if there exists a Plott function c : 2X ! 2X such that A% B for any A, B
,
c (A [ B ) \ A 6 = ∅
X.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
10 / 23
Ranking Opportunity Sets
We will... 1
Characterise the Plott consistent rankings.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
11 / 23
Ranking Opportunity Sets
We will... 1
Characterise the Plott consistent rankings.
2
Compare Plott consistency and justi…ability.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
11 / 23
Ranking Opportunity Sets
We will... 1
Characterise the Plott consistent rankings.
2
Compare Plott consistency and justi…ability.
3
Raise a question of interpretation and pose an open problem.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
11 / 23
Plott Consistency
1. What are the necessary and su¢ cient conditions (on % ) for Plott consistency?
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
12 / 23
Plott Consistency
1. What are the necessary and su¢ cient conditions (on % ) for Plott consistency? It is easy to verify that the Kreps condition A% B
)
A
A[B
(K)
is necessary.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
12 / 23
Plott Consistency
1. What are the necessary and su¢ cient conditions (on % ) for Plott consistency? It is easy to verify that the Kreps condition A% B
)
A
A[B
(K)
is necessary. However, transitivity of % is not:
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
12 / 23
Plott Consistency
Example (continued) Recall that X = fx, y , z g and c (A) =
fx, y g if A = X A if A 6= X
is a Plott function. Applying Plott consistency, we have: fx, y g and fy g fz g, but fx, y g fz g.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
fy g
13 / 23
Plott Consistency Theorem Given an opportunity set ranking % , the following are equivalent: (i) % is Plott consistent.
(ii) % satis…es the following conditions for any A, B, C condition (K), plus B
A
)
B [C
A and B
X : the Kreps
A C
(SM)
and
[B
A and B
C] ) B
A[C
(U)
This result is “tight” in that none of (K), (SM) or (U) can be dropped without violating the equivalence.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
14 / 23
Plott Consistency
The proof of the theorem draws liberally on results from abstract convexity theory, and especially the papers by Aizerman and Malishevski (1981) and Danilov and Koshevoy (2006).
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
15 / 23
Plott Consistency
The proof of the theorem draws liberally on results from abstract convexity theory, and especially the papers by Aizerman and Malishevski (1981) and Danilov and Koshevoy (2006). Given an abstract convex geometry on X , consider the complete and re‡exive binary relation % 2X 2X de…ned as follows: A B i¤ all the extreme points of A [ B are contained in A B.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
15 / 23
Plott Consistency and Justi…ability
2. What is the relationship between Plott consistency and Justi…ability?
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
16 / 23
Plott Consistency and Justi…ability
2. What is the relationship between Plott consistency and Justi…ability? Not all Plott consistent rankings are justi…able – this can be proved using Plott’s example.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
16 / 23
Plott Consistency and Justi…ability
2. What is the relationship between Plott consistency and Justi…ability? Not all Plott consistent rankings are justi…able – this can be proved using Plott’s example. Plott consistency permits fundamental non-binariness.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
16 / 23
Plott Consistency and Justi…ability
2. What is the relationship between Plott consistency and Justi…ability? Not all Plott consistent rankings are justi…able – this can be proved using Plott’s example. Plott consistency permits fundamental non-binariness.
Neither are all justi…able rankings Plott consistent.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
16 / 23
Plott Consistency and Justi…ability
2. What is the relationship between Plott consistency and Justi…ability? Not all Plott consistent rankings are justi…able – this can be proved using Plott’s example. Plott consistency permits fundamental non-binariness.
Neither are all justi…able rankings Plott consistent. Plott consistency imposes quasi-transitivity of % .
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
16 / 23
Plott Consistency and Justi…ability
Theorem A justi…able opportunity set ranking % is Plott consistent i¤ it is quasi-transitive.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
17 / 23
Plott Consistency and Justi…ability
An opportunity set ranking % 2X for any A, B 2 B and any x 2 X ,
fx g % A and fx g % B
2X satis…es Weak Expansion if:
)
fx g % A [ B
(WE)
Theorem A Plott consistent opportunity set ranking % is justi…able i¤ it satis…es Weak Expansion.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
18 / 23
A question of interpretation
3. Can the principles of consequentialism and binariness really be separated?
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
19 / 23
A question of interpretation
3. Can the principles of consequentialism and binariness really be separated? Non-binary choice implies context-dependence.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
19 / 23
A question of interpretation
3. Can the principles of consequentialism and binariness really be separated? Non-binary choice implies context-dependence. Is Plott consistency appropriate for context-dependent choice?
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
19 / 23
A question of interpretation Example (continued) X = fx, y , z g and c (A) =
fx, y g if A = X A if A 6= X
Consider the following two-player game:
α β
x , 3 , 0
y , 0 , 3
z , 1 , 1
Then c corresponds to choosing undominated strategies. Can we rule out fx, y g Ryan (University of Auckland)
fx g? Plott Consistent Rankings
3rd CMSS Workshop
20 / 23
A question of interpretation
Given an opportunity set ranking % , de…ne c (A) = for each A
X.
\
fB
A jB
A Bg
De…nition Say that % is strongly consequentialist if c (A) 6= ∅ for every non-empty A
X , and
A% B
i¤ c (A [ B ) \ A 6= ∅.
In this case, we call c : 2X ! 2X the revealed choice function for % .
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
21 / 23
A question of interpretation
Theorem An opportunity set ranking %
2X
2X is strongly consequentialist i¤
...??? Plott consistency is su¢ cient but not necessary.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
22 / 23
A question of interpretation
Theorem A strongly consequentialist opportunity set ranking % consistent i¤ ...???
Ryan (University of Auckland)
Plott Consistent Rankings
2X
2X is Plott
3rd CMSS Workshop
23 / 23
3rd CMSS Workshop
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
1 / 23
Theory of Choice
Finite “consumption set” X .
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
2 / 23
Theory of Choice
Finite “consumption set” X . A choice function is a mapping c : 2X ! 2X that satis…es:
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
2 / 23
Theory of Choice
Finite “consumption set” X . A choice function is a mapping c : 2X ! 2X that satis…es:
(CF0) c (A)
A for each A
Ryan (University of Auckland)
X , and
Plott Consistent Rankings
3rd CMSS Workshop
2 / 23
Theory of Choice
Finite “consumption set” X . A choice function is a mapping c : 2X ! 2X that satis…es:
(CF0) c (A) A for each A (CF1) c (A) = ∅ i¤ A = ∅.
Ryan (University of Auckland)
X , and
Plott Consistent Rankings
3rd CMSS Workshop
2 / 23
Theory of Choice
Finite “consumption set” X . A choice function is a mapping c : 2X ! 2X that satis…es:
(CF0) c (A) A for each A (CF1) c (A) = ∅ i¤ A = ∅.
X , and
A choice function is path independent if it also satis…es
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
2 / 23
Theory of Choice
Finite “consumption set” X . A choice function is a mapping c : 2X ! 2X that satis…es:
(CF0) c (A) A for each A (CF1) c (A) = ∅ i¤ A = ∅.
X , and
A choice function is path independent if it also satis…es (CF2) For any A, B 2 2X : c (A [ B ) = c (c (A) [ B )
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
2 / 23
Theory of Choice
Finite “consumption set” X . A choice function is a mapping c : 2X ! 2X that satis…es:
(CF0) c (A) A for each A (CF1) c (A) = ∅ i¤ A = ∅.
X , and
A choice function is path independent if it also satis…es (CF2) For any A, B 2 2X : c (A [ B ) = c (c (A) [ B )
A path independent choice function is also called a Plott function (after Plott, 1973).
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
2 / 23
Non-Binary Choice
Path independence is not su¢ cient to ensure that the choice function is binary.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
3 / 23
Non-Binary Choice
Path independence is not su¢ cient to ensure that the choice function is binary. A choice function c : 2X ! 2X is binary if there exists a binary relation % X X such that c (A) = max A %
where max A %
Ryan (University of Auckland)
fx 2 A j there is no y 2 A
Plott Consistent Rankings
fx g with y
xg .
3rd CMSS Workshop
3 / 23
Non-Binary Choice
Example (Plott, 1973) Let X = fx, y , z g and de…ne c : 2X ! 2X as follows: c (A) =
fx, y g if A = X A if A 6= X
It is easily veri…ed that c is a Plott function. Since c (X ) = fx, y g, we must have x z or y z if c is binary. But these contradict c (fx, z g) = fx, z g and c (fy , z g) = fy , z g, respectively, so c is non-binary.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
4 / 23
Ranking Opportunity Sets
De…nition An opportunity set ranking is a binary relation % re‡exive, complete and satis…es A
∅
2X
2X which is
for all A 6= ∅.
Think of ranking restaurants (i.e., menus).
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
5 / 23
Ranking Opportunity Sets
If (meal) choice is governed by the binary relation % X X , then opportunity sets (i.e., restaurants) are naturally ranked according to the following indirect utility (IU) principle: A% B
Ryan (University of Auckland)
,
max (A [ B ) \ A 6= ∅ %
Plott Consistent Rankings
3rd CMSS Workshop
(IU)
6 / 23
Ranking Opportunity Sets
If (meal) choice is governed by the binary relation % X X , then opportunity sets (i.e., restaurants) are naturally ranked according to the following indirect utility (IU) principle: A% B
,
max (A [ B ) \ A 6= ∅ %
(IU)
What are the hallmarks of opportunity set rankings that obey the IU principle?
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
6 / 23
Ranking Opportunity Sets
Theorem (Kreps, 1979) The opportunity set ranking % satis…es (IU) for some complete, re‡exive and transitive % X X i¤ % is transitive and satis…es A% B for every A, B
)
A
A[B
(K)
X.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
7 / 23
Ranking Opportunity Sets
De…nition (Lahiri, 2003) An opportunity set ranking % is justi…able if it satis…es (IU) for some complete and re‡exive (but not necessarily transitive) % X X
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
8 / 23
Ranking Opportunity Sets
The IU principle embodies two fundamental ideas:
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
9 / 23
Ranking Opportunity Sets
The IU principle embodies two fundamental ideas: 1
Consequentialism.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
9 / 23
Ranking Opportunity Sets
The IU principle embodies two fundamental ideas: 1 2
Consequentialism. Binariness.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
9 / 23
Ranking Opportunity Sets
The IU principle embodies two fundamental ideas: 1 2
Consequentialism. Binariness.
Many papers relax (1). We maintain (1) but relax (2).
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
9 / 23
Ranking Opportunity Sets By analogy with the IU condition A% B
,
max (A [ B ) \ A 6= ∅, %
we propose:
De…nition An opportunity set ranking % is Plott consistent if there exists a Plott function c : 2X ! 2X such that A% B for any A, B
,
c (A [ B ) \ A 6 = ∅
X.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
10 / 23
Ranking Opportunity Sets
We will... 1
Characterise the Plott consistent rankings.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
11 / 23
Ranking Opportunity Sets
We will... 1
Characterise the Plott consistent rankings.
2
Compare Plott consistency and justi…ability.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
11 / 23
Ranking Opportunity Sets
We will... 1
Characterise the Plott consistent rankings.
2
Compare Plott consistency and justi…ability.
3
Raise a question of interpretation and pose an open problem.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
11 / 23
Plott Consistency
1. What are the necessary and su¢ cient conditions (on % ) for Plott consistency?
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
12 / 23
Plott Consistency
1. What are the necessary and su¢ cient conditions (on % ) for Plott consistency? It is easy to verify that the Kreps condition A% B
)
A
A[B
(K)
is necessary.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
12 / 23
Plott Consistency
1. What are the necessary and su¢ cient conditions (on % ) for Plott consistency? It is easy to verify that the Kreps condition A% B
)
A
A[B
(K)
is necessary. However, transitivity of % is not:
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
12 / 23
Plott Consistency
Example (continued) Recall that X = fx, y , z g and c (A) =
fx, y g if A = X A if A 6= X
is a Plott function. Applying Plott consistency, we have: fx, y g and fy g fz g, but fx, y g fz g.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
fy g
13 / 23
Plott Consistency Theorem Given an opportunity set ranking % , the following are equivalent: (i) % is Plott consistent.
(ii) % satis…es the following conditions for any A, B, C condition (K), plus B
A
)
B [C
A and B
X : the Kreps
A C
(SM)
and
[B
A and B
C] ) B
A[C
(U)
This result is “tight” in that none of (K), (SM) or (U) can be dropped without violating the equivalence.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
14 / 23
Plott Consistency
The proof of the theorem draws liberally on results from abstract convexity theory, and especially the papers by Aizerman and Malishevski (1981) and Danilov and Koshevoy (2006).
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
15 / 23
Plott Consistency
The proof of the theorem draws liberally on results from abstract convexity theory, and especially the papers by Aizerman and Malishevski (1981) and Danilov and Koshevoy (2006). Given an abstract convex geometry on X , consider the complete and re‡exive binary relation % 2X 2X de…ned as follows: A B i¤ all the extreme points of A [ B are contained in A B.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
15 / 23
Plott Consistency and Justi…ability
2. What is the relationship between Plott consistency and Justi…ability?
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
16 / 23
Plott Consistency and Justi…ability
2. What is the relationship between Plott consistency and Justi…ability? Not all Plott consistent rankings are justi…able – this can be proved using Plott’s example.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
16 / 23
Plott Consistency and Justi…ability
2. What is the relationship between Plott consistency and Justi…ability? Not all Plott consistent rankings are justi…able – this can be proved using Plott’s example. Plott consistency permits fundamental non-binariness.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
16 / 23
Plott Consistency and Justi…ability
2. What is the relationship between Plott consistency and Justi…ability? Not all Plott consistent rankings are justi…able – this can be proved using Plott’s example. Plott consistency permits fundamental non-binariness.
Neither are all justi…able rankings Plott consistent.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
16 / 23
Plott Consistency and Justi…ability
2. What is the relationship between Plott consistency and Justi…ability? Not all Plott consistent rankings are justi…able – this can be proved using Plott’s example. Plott consistency permits fundamental non-binariness.
Neither are all justi…able rankings Plott consistent. Plott consistency imposes quasi-transitivity of % .
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
16 / 23
Plott Consistency and Justi…ability
Theorem A justi…able opportunity set ranking % is Plott consistent i¤ it is quasi-transitive.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
17 / 23
Plott Consistency and Justi…ability
An opportunity set ranking % 2X for any A, B 2 B and any x 2 X ,
fx g % A and fx g % B
2X satis…es Weak Expansion if:
)
fx g % A [ B
(WE)
Theorem A Plott consistent opportunity set ranking % is justi…able i¤ it satis…es Weak Expansion.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
18 / 23
A question of interpretation
3. Can the principles of consequentialism and binariness really be separated?
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
19 / 23
A question of interpretation
3. Can the principles of consequentialism and binariness really be separated? Non-binary choice implies context-dependence.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
19 / 23
A question of interpretation
3. Can the principles of consequentialism and binariness really be separated? Non-binary choice implies context-dependence. Is Plott consistency appropriate for context-dependent choice?
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
19 / 23
A question of interpretation Example (continued) X = fx, y , z g and c (A) =
fx, y g if A = X A if A 6= X
Consider the following two-player game:
α β
x , 3 , 0
y , 0 , 3
z , 1 , 1
Then c corresponds to choosing undominated strategies. Can we rule out fx, y g Ryan (University of Auckland)
fx g? Plott Consistent Rankings
3rd CMSS Workshop
20 / 23
A question of interpretation
Given an opportunity set ranking % , de…ne c (A) = for each A
X.
\
fB
A jB
A Bg
De…nition Say that % is strongly consequentialist if c (A) 6= ∅ for every non-empty A
X , and
A% B
i¤ c (A [ B ) \ A 6= ∅.
In this case, we call c : 2X ! 2X the revealed choice function for % .
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
21 / 23
A question of interpretation
Theorem An opportunity set ranking %
2X
2X is strongly consequentialist i¤
...??? Plott consistency is su¢ cient but not necessary.
Ryan (University of Auckland)
Plott Consistent Rankings
3rd CMSS Workshop
22 / 23
A question of interpretation
Theorem A strongly consequentialist opportunity set ranking % consistent i¤ ...???
Ryan (University of Auckland)
Plott Consistent Rankings
2X
2X is Plott
3rd CMSS Workshop
23 / 23
