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UTMD Working Papers can be downloaded without charge from:
https://www.mdc.e.u-tokyo.ac.jp/category/wp/
Working Papers are a series of manuscripts in their draft form. They are not intended for
circulation or distribution except as indicated by the author. For that reason Working Papers
may not be reproduced or distributed without the written consent of the author.
UTMD-007
Assignments with Ethical Concerns
Hitoshi Matsushima
University of Tokyo
May 9, 2021
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Assignments with Ethical Concerns1
Hitoshi Matsushima2
University of Tokyo
May 9, 2021
Abstract
This study investigates an axiomatic approach to in-kind assignment problems with single-unit demand. We consider multiple ethical criteria regarding which agents should be assigned that conflict with each other. To make compromises between criteria, we introduce two methods for configuring social choice rules that map from various problems to agents who are assigned slots: the method of procedure and the method of aggregation. From inter-problem regularities, we demonstrate characterization results, implying that the method of procedure emphasizes consistent respect for individual criteria across problems, while the method of aggregation emphasizes consistent respect for individual agents across problems. These methods are incompatible because only ethical dictatorships are induced by both methods at the same time. We show that the method of aggregation is superior when we can utilize detailed information about ethical concerns such as cardinality and comparability, while the method of procedure is superior when there are severe informational limitations.
JEL Classification Codes: D30, D45, D63, D71
Keywords: Multiple Criteria, Ethical Social Choice Theory, Procedure and Aggregation, Ethical Dictatorship, Fair Justification, Informational Basis.
1 This study was supported by a grant-in-aid for scientific research (KAKENHI 20H00070) from the Japan Society for the Promotion of Science (JSPS) and the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) of the Japanese government. 2 Department of Economics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-0033, Japan. E-mail: hitoshi @ e.u-tokyo.ac.jp
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1. Introduction
This study investigates assignment problems, where there exist multiple
homogeneous slots and multiple agents (participants), and the number of available slots
is less than that of the participants. We assume a single-unit demand in that each agent
prefers a single slot to nothing, but she (or he) does not need more slots. We demonstrate
an axiomatic approach to characterize social choice rules (SCRs) that determine which
agents are assigned slots in various assignment problems that are associated with different
numbers of slots and participants, as a manifestation of distributive justice in the
community.
Centrally, we consider multiple conflicting ethical concerns about who should be
prioritized in slot assignment and reflect them in configuring an SCR. In particular, we
even investigate situations in which there may be some need to interfere with consumer
sovereignty. The quality of assignments that are involved in life, dignity, health,
Section 5 introduces the state space and investigates the informational basis of state-
dependent social choice rules (SSCRs). We show that there exist nontrivial SSCRs that
are induced by the method of aggregation and satisfy comparability, while any SSCR that
is induced by the method of procedure fails to satisfy comparability. Section 6 concludes.
2. Assignment Problem
Let {1, ..., }N n denote a finite set of agents, where 3n . Let a nonempty
subset of agents I N denote the set of participants. Let a positive integer q denote
the number of available slots. The assignment problem is defined as ( , )I q , where
q I . Let X denote the set of all the assignment problems.
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The SCR is defined as : 2NC X , where ( , )C I q I and ( , )C I q q . An
SCR C determines which participants are assigned slots in various assignment
problems: any agent in ( , )C I q obtains a single slot, while any agent who is not in
( , )C I q obtains nothing. We introduce two basic axioms for an SCR C .
Axiom 1: For every ( , )I q X and i I ,
[ ( , )i C I q ] [ ( , 1)i C I q ].
Axiom 2: For every ( , )I q X , i I , and \ { }j I i ,
[ ( , )i C I q ] [ ( \ { }, )i C I j q ].
Axiom 1 implies that the same agents can obtain slots when the number of available
slots increases. Axiom 2 implies that the same agents can obtain slots when the set of
participants becomes smaller. Since both axioms are quite reasonable, this study will
focus on SCRs that satisfy Axioms 1 and 2.3
We denote a set of criteria as {1,2, ..., }D d . We denote a priority order over
agents at each criterion d D using a one-to-one mapping :{1, ..., }d n N . Some
examples are orders of willingness to pay multiplied by welfare weights, income order,
age order, orders of degree of diseases, and hybrids of these orders. For each {1, ..., }h n ,
agent ( )d h N has the h th rank of the criterion d D . For simplicity, we assume
strict ordering over agents to eliminate tie-breaking cases.4 We denote the profile of these
priority orders as ( )d d D .
3. Method of Procedure
3 Axiom 2 excludes SCRs such as the Borda rule (Borda, 1781; Maskin, 2020). The Borda rule uses each participant’s priority order over potential agents, instead of over actual participants, which causes the contradiction with Axiom 2. 4 Because of this assumption, we do not handle some categories such as gender differences without spectrum considerations. However, this study does not depend on it; we can simply add arbitrary strict orderings (age order, for example), which eliminates tie-breaking cases.
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We first demonstrate a basic characterization of SCRs that satisfy Axioms 1 and 2.
We then introduce the method of procedure to configure SCRs C that are associated
with a pre-existing combination of a set of criteria and a profile of priority orders over
agents ( , )D .
3.1. Basic Characterization
We denote a priority order over the criteria by : {1, 2, ..., }z D , where z n . In
all parts of this study, except for Subsection 3. 3, we consider the case in which z n .
We define a procedure as a combination of a set of criteria, a profile of priority orders
over agents, and a priority order over criteria, which is denoted by ( , , )D .
A procedure uniquely determines an SCR, which is denoted by C , according
to the following steps. Consider an arbitrary assignment problem ( , )I q X . In step 1,
the top-ranked agent at the criterion (1) D among I is selected. This agent is
denoted by (1)i I . At each step {2, ..., }k q , the top-ranked agent at the criterion
( )k D among the set of remaining participants \ { (1), ..., ( 1)}I i i k is selected. This
agent is denoted by ( ) \ { (1), ..., ( 1)}i k I i i k . We then define C by:
( , ) { (1), ..., ( )}C I q i i q for all ( , )I q X .
Note that for each {1, ..., }k n , the corresponding agent ( )i k N is selected and
assigned a slot based on the corresponding criterion ( )k D in any assignment
problem ( , )I q X , provided that i I and k q . We interpret a priority order over
criteria as a device to minimize biases regarding which criteria are used to justify
assigned agents across various problems.5
5 What the specification of C and the serial dictatorship (Luce and Raiffa, 1957; Satterthwaite and Sonnenschein, 1981; Abdulkadiroglu and Sonmez, 1998; Piccione and Rubinstein, 2007) have in common is that priority is given in order, but these are essentially different. Unlike the serial dictatorship, the steps for specifying C are not avaricious: these steps give each criterion chances of priority many times, but in each chance only the priority to select one agent is allowed. We have the same point of difference from the reserve system in Pathak et al. (2020). See Subsection 3. 3.
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The following theorem states that an SCR satisfies Axioms 1 and 2 if and only if it
can be induced by an artificially created procedure.
Theorem 1: An SCR C satisfies Axioms 1 and 2 if and only if there exists ( , , )D
such that C C .
Proof: Consider an arbitrary procedure ( , , )D . Clearly, C satisfies Axiom 1:
for each I N and {1, ..., }h n , the same agent ( )i h I is assigned the h th slot in
an assignment problem ( , )I q whenever q h . The SCR C also satisfies Axiom 2:
any agent who has a better rank than agent ( )i h at the criterion ( )h is either absent
or assigned a slot before agent ( )i h . Hence, the fact that agent ( )i h has the highest
rank at the criterion ( )h at the h th step is unchanged after eliminating agent j ,
irrespective of whether agent j is assigned a slot before agent ( )i h or not.
Next, consider an arbitrary SCR C that satisfies Axioms 1 and 2. Let d n , i.e.,
{1, ..., }D n ,
and specify 1 as follows:
1{ (1)} ( ,1)C N ,
and recursively, for each {2, ..., }k n ,
1 1 1{ ( )} ( \ { (1), ..., ( 1)},1)k C N k .
From Axiom 1, for each {2, ..., }d n , we can recursively specify d as follows:
1(1) (1)d , 2(2) (2)d , …, 1( 1) ( 1)d dd d ,
1 1{ ( )} ( , ) \ { (1), ..., ( 1)}d dd C N d d ,
and for each { 1, ..., }k d n ,
1 1{ ( )} ( \ { ( ), ..., ( 1)}, ) \ { (1), ..., ( 1)}d d d dk C N d k d d .
Specify to satisfy
( )d d for all {1, ..., }d n .
Hence, we have specified ( , , )D .
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We show C C as follows. Suppose C C . From Axiom 1, there exist
( , )I q X and 2( , ) { , ..., 1}k k q n such that k k , ( )q k I , ( ) ( , )q k C I q ,
and ( ) ( , )q k C I q . From Axiom 2, we have
( ( , ) { ( )}, ) ( , )qC C I q k q C I q ,
that is,
( ) ( ( , ) { ( )}, )q qk C C I q k q .
However, from the specification of and Axiom 2, we have
( ) ( \ { ( ), ..., ( 1)}, )q q qk C N q k q .
Since
( , ) { ( )} \ { ( ), ..., ( 1)}q q qC I q k N q k ,
The procedure in the proof of Theorem 1 was artificially created to explain how an
SCR can be configured. Hereafter, we shall fix ( , )D and then investigate the SCRs
C that are associated with the pre-existing ( , )D . Since ( , )D pre-exists, we shall
regard a priority order over criteria as a procedure, instead of ( , , )D . We also
denote the SCR induced by C , instead of C . We term the method to configure
an SCR by specifying the procedure as the method of procedure.
To associate an SCR C with a pre-existing ( , )D , we introduce a justification as
( , )( ( , )) I q XI q , where ( , ) : ( , )I q C I q D for each ( , )I q X . Each agent
( , )i C I q uses a criterion ( , )( )d I q i D to explain why she is assigned a slot in
an assignment problem ( , )I q X . We introduce an axiom ( , , )C D on fairness in
justification from three points of view: respecting priorities, diversity in justification, and
invariance across problems.
Axiom 3 (Fair Justification): There exists that satisfies the following properties:
(i) (Priority): For every ( , )I q X , ( , )i C I q , and \ ( , )j I C I q ,
1 1( , )( ) ( , )( )( ) ( )I q i I q ii j .
(ii) (Diversity): For every ( , )I q X and d D ,
( , ) | ( , )( ) ( , ) | ( , )( )i C I q I q i d i C N q N q i d .
(iii) (Invariance): For every ( , )I q X and ( , )i C I q ,
( , )( ) ( , 1)( )I q i I q i .
To justify why an agent is assigned to push others away, Axiom 3 recommends
coherently using a single criterion that is appropriate for this agent, rather than using a
makeshift mixture of conflicting criteria. To be precise, property (i) implies respecting
priorities in a way that any assigned agent ( , )i C I q can explain why agent i was
assigned to push any unassigned agent away by using criterion ( , )( )I q i . Hence, any
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unassigned agent \ ( , )j I C I q has a worse rank than the assigned agent i in this
criterion ( , )( )I q i .6
Properties (ii) and (iii) are the main components of Axiom 3, which concern
regularities across different assignment problems. Property (ii) implies diversity in
justification, that is, a consistent respect for individual criteria across various problems,
in that for each criterion d D , the number of agents who are assigned slots and justified
by d is unaffected by who actually participate in the problem. This reflects that the
procedure avoids biases due to a change in who participates and maintains the same
diversity concerning which criteria are used for justification.7
Property (iii) implies invariance across problems, that is, another aspect of consistent
respect for individual criteria across problems, in that the criterion ( , )( )I q i that
justifies an assigned agent ( , )i C I q is unchanged as the number of slots q increases.
(Note from Axiom 2 that this agent i is still assigned a slot in this case.)
The following theorem states that any SCR C that satisfies Axioms 1, 2, and 3 can
be induced by a procedure associated with the pre-existing ( , )D .
Theorem 2: An SCR C satisfies Axioms 1, 2, and 3 if and only if exists such that
C C .
Proof: For every procedure , the corresponding SCR C ( C ) satisfies Axiom 3.
According to the steps explained in Subsection 3.1, we can specify a justification so
that for each ( , )I q X ,
( , )( ( )) ( )I q i k k for all {1, ..., }k q .
Clearly, C satisfies Axiom 3, where we set . From Theorem 1, C satisfies
Axioms 1 and 2.
6 Property (i) corresponds to the third definition of Pathak et al. (2020). 7 Property (ii) permits who are assigned and justified by criterion d to be affected by who participate. See Section 4.
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Suppose that C satisfies Axioms 1, 2, and 3. We set an arbitrary justification
to satisfy Axiom 3. We then specify as follows. From Axiom 1, we can define
( )i k N for each {1, ..., }k n in a recursive manner:
{ (1)} ( ,1)i C N ,
and for each {2, ..., }k n ,
{ ( )} ( , ) \ { (1), ..., ( 1)}i k C N k i i k .
From Axiom 3, for each {1, ..., }k n , we select
( ) ( , )( ( ))k N k i k D .
Hence, we have specified .
We show C C as follows. Note
( , ) ( , )C N k C N k for all {1, ..., }k n .
Consider any arbitrary ( , )I q X , where I N . Suppose that
( , ) ( , )C I q C I q .
From properties (i) and (ii) in Axiom 3, 2q must hold. Without loss of generality, we
assume that
( , 1) ( , 1)C I q C I q .
From Axiom 2, we have
( , 1) ( , )C I q C I q and ( , 1) ( , )C I q C I q .
From properties (ii) and (iii) in Axiom 3, the added agent must be justified by thecriterion
( )q . From property (i) in Axiom 3, she must be the top-ranked agent among the set of
the remaining participants at the criterion ( )q . Hence, both ( , )C I q and ( , )C I q
must include the same agent in addition to ( , 1) ( , 1)C I q C I q . This is a
contradiction.
Q.E.D.
Remark 1: An example is the reserve procedure * , defined as follows. We regard
criterion 1 as a baseline, for example, the order of willingness to pay. Any other criterion
{2, ..., }d d has a reserve {1, ..., }dv n . The reserve procedure * secures these
reserves in an equal manner irrespective of the number of available slots:
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* (1) 2 , * (2) 3 , …, * ( 1)d d ,
* ( ) 2d , *( 1) 3d , …. .
Once the reserve is filled for a criterion {2, ..., }d d , this criterion is excluded, and the
steps continue without it. Once the reserves are filled for all criteria except the baseline
(criterion 1), the steps continue to select the baseline until all slots are assigned.
Remark 2: We have multiplicities of procedure and justification for an SCR to satisfy
Axiom 3. Consider ( , )D and C addressed in Example 1. We specify the justification