# Obvious Manipulations in Matching with and without Contracts††thanks: We
acknowledge financial support from UNSL through grants 032016, 030120, from
Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) through
grant PIP 112-200801-00655, and from Agencia Nacional de Promoción Científica
y Tecnológica through grant PICT 2017-2355.

R. Pablo Arribillaga Instituto de Matemática Aplicada San Luis (CONICET-UNSL)
and Departamento de Matemática, Universidad Nacional de San Luis, San Luis,
Argentina, and RedNIE. Emails<EMAIL_ADDRESS>and
ebpepa<EMAIL_ADDRESS>Eliana Pepa Risma22footnotemark: 2

###### Abstract

In a many-to-one matching model, with or without contracts, where doctors’
preferences are private information and hospitals’ preferences are
substitutable and public information, any stable matching rule could be
manipulated for doctors. Since manipulations can not be completely avoided, we
consider the concept of obvious manipulations and look for stable matching
rules that prevent at least such manipulations (for doctors). For the model
with contracts, we prove that: (i) the doctor-optimal matching rule is non-
obviously manipulable and (ii) the hospital-optimal matching rule is obviously
manipulable, even in the one-to-one model. In contrast to (ii), for a many-to-
one model without contracts, we prove that the hospital-optimal matching rule
is not obviously manipulable.Furthermore, if we focus on quantile stable
rules, then we prove that the doctor-optimal matching rule is the only non-
obviously manipulable quantile stable rule.

_JEL classification:_ D71, D72.

_Keywords:_ obvious manipulations, matching, contracts

##  1 Introduction

In the two-sided many-to-one matching model with contracts, there is a
bilateral market whose disjoint sides are typically referred to as doctors and
hospitals. Each contract refers to a doctor-hospital pair, although there may
exist two or more contracts involving the same agents under different
contractual conditions. The classical matching models (without contracts) can
be addressed as special cases where there exists at most one contract
involving each pair of agents. The problem consists of assigning agents from
one side of the market to agents on the opposite side, through some contracts.
In the many-to-one model, each doctor is allowed to sign one contract at most,
whereas hospitals can sign multiple contracts. A set of contracts that
contains at most one contract involving a doctor is called  allocation and is
a possible outcome of the matching problem. All the agents have preferences
defined over the set of contracts involving them in some allocation. A
matching rule is a function that, for each preference profile declared by the
agents, selects an allocation. Since two agents wishing to sign an existing
contract are free to do it, and also the agents can unilaterally terminate
previous contracts if they find it convenient, there are outcomes that are
unstable. We will consider stable allocations, i.e., outcomes that are
sustainable over time, supposing the market remains unchanged. Hatfield and
Milgrom (2005) proved that if all hospitals have substitutable
preferences,111Substitutability is a basic condition, widely used in matching
literature, and means that the hospitals do not consider the contracts as
complementary among themselves. then the set of stable allocations is a
nonempty lattice whose maximum and minimum elements can be obtained through a
generalization of the deferred acceptance algorithm introduced by Gale and
Shapley (1962). As in models without contracts, these extreme points
correspond to the unanimously most preferred stable allocation for the doctors
(doctor-optimal stable allocation), and the unanimously most preferred stable
allocation for the hospitals (hospital-optimal stable allocation)

In addition to stability, the non-manipulability of a matching rule also plays
a central role in two-sided matching literature. An agent manipulates a
matching rule if there exists a situation in which it obtains a better result
for itself by declaring an alternative preference to his true one. In this
paper, we assume that hospitals’ preferences are public and only focus on the
manipulations that can be performed by doctors, whose preferences are private
information. It is well known that the doctor-optimal matching rule, i.e., the
matching rule that for each preference profile for doctors, returns the
doctor-optimal stable allocation, is the only candidate to be stable and non-
manipulable. Roth (1982) shows that, in the classical one-to-one matching
model without contracts, the only strategy-proof and stable matching rule is
the doctor-optimal one. In many-to-one matching models, such kind of result
depends on the characteristics of the (fixed and public) hospitals’
preferences. In the context without contracts, if hospitals’ preferences are
$q$-responsive (this model is known as college admission problem) the doctor-
optimal matching rule is also the only strategy-proof and stable matching rule
(see Roth, 1985). However, if hospitals’ preferences are substitutable, the
doctor-optimal matching rule could fail to be strategy-proof and it might not
exist a strategy-proof stable rule. This is shown by Hatfield and Milgrom
(2005) and Martínez et al. (2004) for the matching models with and without
contracts, respectively. Those papers add different conditions to
substitutability to recover the strategy-proofness of the doctor-optimal
matching rule in each context.222 Martínez et al. (2004) add $q$-separability
and Hatfield and Milgrom (2005) add the law of aggregate demand.

Therefore, in the many-to-one matching model (with and without contracts) with
substitutable preferences, any stable matching could be susceptible to
manipulations. Given that manipulations can not be completely avoided in this
context, we look for stable matching rules that at least prevent obvious
manipulations (for doctors), as defined by Troyan and Morrill (2020). A
manipulation is "obvious" if it is much easier for agents to recognize and
execute successfully than others in a specific and formal sense. To formalize
the word "obvious", it is necessary to specify how much information each agent
has about other agents’ preferences. Troyan and Morrill (2020) assume that
each agent has complete ignorance in this respect and, therefore, each agent
focuses on the set of all outcomes that can be chosen by the rule given its
own report. Now, a manipulation is obvious if the best possible outcome under
the manipulation is strictly better than the best possible outcome under
truth-telling or the worst possible outcome under the manipulation is strictly
better than the worst possible outcome under truth-telling. Complementary, the
term "obvious" reflects that an agent could deduce that a mechanism is
manipulable even if it does not fully know how the mechanism is defined (see
Theorem 1 in Troyan and Morrill, 2020, for this interpretation). In the
context of college admission, Troyan and Morrill (2020) prove that any stable
matching rule is no-obviously manipulable. Our results show that such a
statement can be partially extended to the context of matching with contracts
and substitutable preferences, arising some significant differences between
the models with and without contracts.

First, we prove that the doctor-optimal matching rule is non-obviously
manipulable in the general context of a many-to-one matching model with
contracts and substitutable preferences for hospitals. Hence, although there
are no matching rules that are non-manipulable, at least there is a matching
rule that is non-obviously manipulable in such context. This result can be
seen as an extension to the model with contracts of the one obtained by Troyan
and Morrill (2020) but its proof requires completely different arguments and
techniques. Surprisingly, we show that the opposite result holds for the
hospital-optimal matching rule which turns out to be obviously manipulable
(for doctors) even in the particular context of a one-to-one matching model
with contracts. This result is remarkable because it reveals a substantial
difference between the models with and without contracts from the point of
view of the strategic behavior of agents. In the context of the many-to-one
classical matching model without contracts and with substitutable hospitals’
preferences we prove that the hospital-optimal matching rule is non-obviously
manipulable.

Finally, we focus on the class of quantile-stable matching rules (or
mechanisms) defined in Chen et al. (2016) and studied in Chen et al. (2021)
and Fernandez (2020), among others. It is a new and relevant class of matching
rules that generate stable matchings that can be seen as a compromise between
the two sides of the market. Two of the (extremal) quantile-stable rules are
the optimal matching rules that assign the best stable outcome for one side
and the worst for the other. Assume that given a profile, we have a market
with $k$ stable matching, Chen et al. (2016) show that if the hospitals’
preferences are substitutes and satisfy the law of aggregate demand333Law of
aggregate demand estates that the number of contracts chosen by the agent
either rises or stays the same if the set of available contracts increases.,
then the set of contracts that assigns each doctor the $i$-th ($1\leq i\leq
k$) best stable matching outcome is a stable matching. Now given $q\in[0,1]$,
the $q-$quantile-stable matching rule is a rule that for each profile selects
the $\lceil kq\rceil$-th best stable allocation for each doctor in such
profile.444 $\lceil x\rceil$ denotes the lowest positive integer equal to or
larger than $x$. When $q=0$, the $q-$ quantile-stable matching rule is the
doctor-optimal matching rule. Fernandez (2020) proposes the notion of regret-
free truth-telling as a weakness of strategy-proofness. This is a notion which
assumes that the agents obtain limited information about the preferences of
the other agents by observing the outcome of the rule. That paper shows that,
when both doctors and hospitals are strategic, the doctor-optimal and
hospital-optimal are the only regret-free truth-telling matching rules in the
class of the quantile-stable. In line with that result, our last theorem
states that the doctor-optimal is the only quantile-stable matching rule which
is non-obvious manipulable when doctors are strategic.

Troyan et al. (2020) apply the notion of obvious manipulation in the context
of one-sided matching markets. They present the notion of essential stable
matching and show that no essentially stable mechanism is obviously
manipulable. Other recent papers that study the notion of obvious
manipulation, in contexts other than two-side matching markets are Aziz and
Lam (2021) and Arribillaga and Bonifacio (2022) in the context of voting;
Ortega and Segal-Halevi (2022) in cake-cutting and Psomas and Verma (2022) in
allocation problems.

The rest of the paper is organized as follows. The model and the concepts of
stability and obvious manipulations are introduced in Section 2. Section 3
presents the main results of our paper in three subsections. Subsection 3.1
contains the formal definition of the doctor-optimal matching rule and the
proof that such matching rule is non-obvious manipulable in the general
context with contracts and substitutable preferences. Subsection 3.2 contains
the formal definition of the hospital-optimal matching rule and the proof that
such matching rule is: (i) obvious manipulable in the context with contracts,
even in the one-to-one model and (ii) non-obvious manipulable in the context
without contracts and substitutable preferences. In Subsection 3.3 the
quantile stable rules are introduced and it is shown that the doctor-optimal
matching rule is the only non-obvious manipulable quantile stable rule. To
conclude, some final remarks are gathered in Section 4.

##  2 Preliminaries

###  2.1 Matching model with contracts and strategic doctors

We consider a many-to-one matching model with contracts where particular
markets have two disjoint sides: a set of doctors $D$ and a set of hospitals
$H$, both finite. The problem consists of assigning agents from one side of
the market to agents on the opposite side but, unlike in the classical
matching model without contracts, the contractual conditions (salary,
schedules, work tasks, etc.) characterizing the relationship between two
agents are not fixed beforehand. In every particular market, there is a finite
universal set of contracts $\mathbf{X}.$ Each contract $x\in\mathbf{X}$ is
bilateral, involving exactly one doctor $x_{D}\in D$ and one hospital
$x_{H}\in H.$ The set $\mathbf{X}$ could contain two or more contracts
relating the same pair of agents $(d,h)\in\mathbf{D\times H,}$ under different
conditions. The classical matching model without contracts can be considered
as a special case of this setting, where $\mathbf{X}$ contains one and only
one contract involving each pair $(d,h)\in\mathbf{D\times H}.$

In the many-to-one matching model that we study here, each hospital can sign
many contracts and each doctor can sign one contract at most. An allocation is
a subset of contracts meeting such requirements.

###### Definition 1

A set of contracts $Z\subseteq\mathbf{X}$ is an allocation if $x\neq y$
implies $x_{D}\neq y_{D}$ for all $x,y\in Z$.

Given a set of contracts $Y\subseteq\mathbf{X}$ and $i\in D\cup H\mathbf{,}$
we will denote by $A(Y)$ the set of all allocations which are subsets of $Y;$
and by $Y_{i}$ the set of all contracts in $Y$ involving $i.$ Note that the
empty set (referring to a situation where no contract is signed) is an
allocation and $\varnothing\in A(Y)$ for all $Y\subseteq\mathbf{X}$.

Given a set of contracts $\mathbf{X}$, a particular market is determined by a
preference relation, over the set of allocations
$A(\mathbf{X}_{i})$,555Observe that every allocation in $A(\mathbf{X}_{i})$
contains only contracts involving $i.$ for each agent $i\in D\cup H$. Such
preferences are antisymmetric, transitive, and complete. Observe that
$\left|Z\right|\leq 1$ for all $Z\in A(\mathbf{X}_{d})$ and $d\in D.$
Therefore, we might visualize a doctor’s preference relation as an order on
the contracts involving itself and the empty set. In our analysis, as is usual
in the literature, we will assume that only one side of the market is
strategic: the doctors; while hospitals’ preferences are fixed and common
knowledge.

An arbitrary preference for doctor $d$ is denoted by $P_{d}$. The weak
preference relation over $A(\mathbf{X}_{i})$ associated to $P_{i}$ will be
denoted by $R_{i}$.666This is, for all $x,y\in A(\mathbf{X}_{i})$, $xR_{d}y$
if and only if either $x=y$ or $xP_{d}y.$ By $\mathcal{P}_{d}$ we will denote
the set of all feasible preference relations that a doctor $d$ has in a given
market. A preference profile $P=(P_{d})_{d\in D}$ will identify a preference
relation for each doctor. By $\mathcal{P}=\prod\limits_{i\in
D}\mathcal{P}_{i}$ we will denote the set of all profiles of preferences that
could take place in the given market. Finally, for each profile $P$ and doctor
$d\in D,$ we will denote by $P_{-d}$ the subprofile in
$\mathcal{P}_{-d}=\prod\limits_{i\in D\backslash\\{d\\}}\mathcal{P}_{i}$
obtained by deleting $P_{d}$ from $P.$

###### Definition 2

A (matching) rule is a function
$\phi:\mathcal{P}\mathbf{\rightarrow}A(\mathbf{X})$ that returns an allocation
in $A(\mathbf{X})$ for each profile of preferences $P\in\mathcal{P}.$

Given $d\in D,$ we will denote by $\phi_{d}(P)$ the (only) contract in
$\phi(P)$ involving $d$.

###  2.2 Stability

An essential property to be considered in matching models is the stability of
allocations. In order to introduce the notion of stability, it is necessary to
consider the preferences of the hospitals over the subsets of contracts. As we
mentioned before, each hospital $h\in H$ has an antisymmetric, transitive and
complete preference relation over $A(\mathbf{X}_{h})$ that we will denote by
$\succ_{h}.$ A preference profile for hospitals is denoted by
$\succ:=(\succ_{h})_{h\in H}$. Along the paper $\succ$ is assumed to be
arbitrary but fixed and known by doctors.

The choice set of $h\in H$ given a preference $\succ_{h}$ and
$Y\subseteq\mathbf{X,}$ is the subset of $Y_{h}$ that $h$ likes best according
to $\succ_{h}$

$C_{h}(\succ_{h},Y)=\max_{\succ_{h}}A(Y_{h})$

Similarly, the choice set of $d\in D$ given a preference
$P_{d}\in\mathcal{P}_{d}$ and $Y\subseteq\mathbf{X,}$ is the subset of $Y_{d}$
that $d$ likes best according to $P_{d}$ 777Observe that for each $d\in D,$
the choice set $C_{d}(Y,P_{d})$ contains only the best element in $Y_{d}$
according to $P_{d}.$ However, we introduce this definition here to have a
symmetric notation for hospitals and doctors which makes the exposition
simpler.

$C_{d}(P_{d},Y)=\max_{P_{d}}A(Y_{d})$

###### Remark 1

To keep the notation simpler, we will omit describing the preferences used to
obtain the choice sets unless this information were relevant and not obvious
from the context. We will write $C_{h}(Y)$ and $C_{d}(Y)$ instead of
$C_{h}(Y,\succ_{h})$ and $C_{d}(Y,P_{d}),$ respectively.

In addition, given $Y\subseteq\mathbf{X}$ we define $C_{H}(Y)=\cup_{h\in
H}C_{h}(Y)$ and $C_{D}(Y)=\cup_{d\in D}C_{d}(Y).$

Now, we are ready to introduce the concept of individually rational and stable
allocation. Given $P\in\mathcal{P}$, an allocation $Y\in A(\mathbf{X})$ is
individually rational at $P$ if it does not contain unwanted contracts, i.e.,

$C_{D}(Y)=C_{H}(Y)=Y.$

Whenever two agents wish to sign a contract, they are free to do it, and they
are also free to terminate previous contracts. Consequently, an allocation can
be blocked by a contract that it does not include. Given $P\in\mathcal{P}$ and
$Y\in A(\mathbf{X})$ , the contract $x\in\mathbf{X}\setminus Y$ is a blocking
contract at $P$ for $Y$ if

$x\in C_{x_{D}}(Y\cup\\{x\\})\cap C_{x_{H}}(Y\cup\\{x\\}).$

###### Definition 3

Let $P\in\mathcal{P}$. A set $Y\in A(\mathbf{X})$ is a stable allocation at
$P$ if

  1. (i)

$Y$ is individually rational at $P$;

  2. (ii)

There are no blocking contracts for $Y$ at $P$.

###### Definition 4

A rule $\phi:\mathcal{P}\mathbf{\rightarrow}A(\mathbf{X})$ is stable if for
all $P\mathcal{\in P}$, the allocation $\phi(P)$ is stable at $P$.

Hatfield and Milgrom (2005) show that if all hospitals have substitutable
preferences, then the set of stable allocations is nonempty. Substitutability
is a basic condition, widely used in the matching literature, and means that
the hospitals do not consider the contracts as complementary among themselves.
In other words, no contract ceases being chosen by a hospital because another
contract becomes unavailable for it.

###### Definition 5

A preference $\succ_{h}$ of an agent $h\in H$ satisfy substitutability if
$x\in C_{h}(W,\succ_{h})$ implies $x\in C_{h}(Y,\succ_{h})$ whenever $x\in
Y_{h}\subseteq W_{h}\subseteq\mathbf{X}.$

From now on $\succ_{h}$ will always satisfy substitutability for each $h\in
H$.

Hatfield and Milgrom (2005) also show that if all hospitals have substitutable
preferences, the set of stable allocations has a lattice structure whose
extreme points are characterized as the doctor-optimal stable allocation and
the hospital-optimal stable allocation. They are the allocation that assigns
every doctor the best contract it could obtain through any stable allocation
and the allocation which assigns every hospital the best set of contracts it
could obtain through any stable allocation, respectively. Formal algorithms to
compute the mentioned optimal stable allocations are included in Section 3. In
the rest of the paper, we will focus particularly on the doctor-optimal rule
and hospital-optimal rule which return the doctor-optimal and the hospital-
optimal stable allocations at each $P\in\mathcal{P}$, respectively.

###  2.3 Manipulations and obvious manipulations

The concept of non-manipulability (or strategy-proofness) has played a central
role in studying the strategic behavior of agents. A doctor manipulates a
matching rule if there exists a situation in which this agent obtains a better
result for him by declaring an alternative preference to his true one.

###### Definition 6

Given a rule $\phi:$ $\mathcal{P}\mathbf{\rightarrow}A(\mathbf{X}),$ an agent
$d\in D$ and a preference $P_{d}\in\mathcal{P}_{d}$, the preference
$P_{d}^{\prime}\in\mathcal{P}_{d}$ is a manipulation of $\phi$ at $P_{d}$ if
there is a (sub)profile $P_{-d}\in\mathcal{P}_{-d}$such that

$\phi_{d}(P_{d}^{\prime},P_{-d})\text{ }P_{d}\text{ }\phi_{d}(P_{d},P_{-d}).$
(1)

A rule is strategy-proof if it has no manipulation.

In the context of substitutable preferences in many-to-one matching models
(with and without contracts), any stable matching is susceptible to
manipulation (see Martínez et al., 2004; Hatfield and Milgrom, 2005). So, a
weaker notion than strategy-proofness could be considered to study the
strategic behavior of agents. Given that manipulations can not be completely
avoided, this paper considers the concept of obvious manipulations introduced
by Troyan and Morrill (2020) and looks for stable rules that at least avoid
obvious manipulations which are, in some sense, more easily identifiable by
agents. To describe such manipulations, it is important to specify how much
information each agent has about other agents’ preferences. Troyan and Morrill
(2020) assume that each agent has complete ignorance in this respect and,
therefore, agents focus on the set of outcomes that can be chosen by the rule
given their own reports. Now, a manipulation is obvious if the best possible
outcome under the manipulation is strictly better than the best possible
outcome under truth-telling or the worst possible outcome under the
manipulation is strictly better than the worst possible outcome under truth-
telling. In the context of college admission, Troyan and Morrill (2020) prove
that any stable rule is not obviously manipulable. We will prove that such a
result could be partially extended to our context of matching with contracts
and substitutable preferences.

Given a rule $\phi:\mathcal{P}\mathbf{\rightarrow}A(\mathbf{X}),$ a doctor
$d\in D$ and a preference $P_{d}\in\mathcal{P}_{d}$, we define the option set
left open by $P_{d}$ at $\phi$ as

$O^{\phi}(P_{d})=\\{\phi_{d}(P_{d},P_{-d}):P_{-d}\in\mathcal{P}_{-d}\\}.$

Given $Y\subseteq\mathbf{X}$, denote by $W_{d}(P_{d},Y)$ to the worst
alternative in $Y_{d}$ according to preference $P_{d}$.

###### Definition 7

Let $\phi:\mathcal{P}\mathbf{\rightarrow}A(\mathbf{X})$ a rule, $i\in D$ let
$P_{d}\in\mathcal{P}_{d},$ and let $P_{d}^{\prime}\in\mathcal{P}_{d}$ be a
manipulation of $\phi$ at $P_{d}$. Manipulation $P_{d}^{\prime}$ is obvious if

$W_{d}(P_{d},O^{\phi}(P_{d}^{\prime}))\ P_{d}\ W_{d}(P_{d},O^{\phi}(P_{d})).$
(2)

or

$C_{d}(P_{d},O^{\phi}(P_{d}^{\prime}))\ P_{d}\ C_{d}(P_{d},O^{\phi}(P_{d})).$
(3)

Rule $f$ is not obviously manipulable (NOM) if it does not admit any obvious
manipulation. In other case, $f$ is obviously manipulable (OM)

##  3 Main results

In this section, we present the results of the paper. In all of them, we
assume that hospitals’ preferences are substitutable. For a clear
presentation, we omit to mention this general hypothesis in each statement.

###  3.1 Doctor-optimal rule

Given a profile $P\in\mathcal{P}$, the doctor-optimal stable allocation can be
defined by the Doctor-proposing Deferred Acceptance algorithm (D-DA). At this
algorithm doctors make offers: a doctor proposes its best contract (if any)
among the set of contracts that have not been rejected during the previous
steps, while a hospital accepts its choice set given the set of received
offers. The algorithm stops when all offers are accepted. The output of the
algorithm is the set of contracts (allocation) accepted by the hospitals in
the final iteration. Formally:

The doctor-proposing deferred acceptance algorithm (D-DA)

Input:
A market $(\mathbf{X},P).$
Begin:
Set $X^{1}=\mathbf{X}$ and $t:=1.$
Repeat:
Step 1: Determine the set of contracts that doctors offer in the iteration
$t,$ this is $O^{t}:=C_{D}(X^{t}).$
Step 2: Determine the set of contracts that are (provisionally) accepted by
hospitals in the iteration $t:$ $C_{H}(O^{t}).$
If $C_{H}(O^{t})=O^{t}$, the algorithm stops with output $C_{H}(O^{t}).$
If $C_{H}(O^{t})\varsubsetneq O^{t}$, define
$X^{t+1}=X^{t}-(O^{t}-C_{H}(O^{t}))$, set $t:=t+1$ and repeat steps 1 and 2
End

Let $\overline{\phi}:\mathcal{P}\mathbf{\rightarrow}A(\mathbf{X})$ denote the
doctor-optimal rule, i.e., the rule that returns the doctor-optimal stable
allocation for each preference profile for doctors, $P\mathcal{\in P}$.
Trivially, $\overline{\phi}$ is a stable rule.

Now, we can state and prove the first result of this section.

###### Theorem 1

The doctor-optimal rule
$\overline{\phi}:\mathcal{P}\mathbf{\rightarrow}A(\mathbf{X})$ is NOM.

The following two lemmas are necessary in order to prove Theorem 1. The first
provides elementary properties of the choice sets when the hospitals’
preferences are substitutable. The second states that, at the end of each
iteration of D-DA, each hospital is assigned to its choice set from the set of
all offers received up to that time.

###### Lemma 1

For all $X,Y\subseteq\mathbf{X}$ and $i\in D\cup H$:

  1. (i)

$C_{i}(Y)\subseteq X\subseteq Y$ implies $C_{i}(X)=C_{i}(Y)$.

  2. (ii)

$C_{i}(C_{i}(Y))=C_{i}(Y).$

  3. (iii)

If $i$’s preferences satisfy substitutability, then $C_{i}(X\cup Y)=$
$C_{i}(C_{i}(X)\cup Y).\vskip 3.0pt plus 1.0pt minus 1.0pt$

_Proof._ (i) and (ii) are direct consequences of the definition of choice set.
(iii) Let $i\in D\cup H$. By substitutability, $X\cap C_{i}(X\cup Y)\subseteq
C_{i}(X).$ Therefore, $C_{i}(X\cup Y)\subseteq C_{i}(X)\cup Y\subseteq X\cup
Y.$ This implies $C_{i}(X\cup Y)=C_{i}(C_{i}(X)\cup Y)$ according to (i).
$\square$

###### Lemma 2

Given $P\in\mathcal{P}$, let $O^{t}$ be as in D-DA definition and let define
$O_{A}^{t}:=\cup_{k=1}^{t}O^{k}.$ Then,

$C_{h}(O^{t})=C_{h}(O_{A}^{t})\text{ for all }t=1,...T$ (4)

and

$\overline{\phi}_{h}(P)=C_{h}(O_{A}^{T})$ (5)

for all $h\in H,$ where $T$ is the number of iterations of the D-DA at $P.$

_Proof._ First, we prove (4). Given $P\in\mathcal{P}$, let $t$ be a stage of
the D-DA at $P$, such that $1\leq t\leq T$ . The proof is by induction on $t.$
If $t=1$ the proof is trivial. Now we assume that (4) holds for all $k<t$ and
prove that it also holds for $t.$ Let $x\in C_{h}(O^{t-1}),$ then by
definition of D-DA, $x\in O^{t-1}=C_{D}(X^{t-1})$ and $x\in X^{t}.$ Therefore,
as $X^{t}\subseteq X^{t-1},$ $x\in C_{D}(X^{t})=O^{t}.$ Hence,
$C_{h}(O^{t-1})\subseteq O^{t}.$ This implies
$C_{h}(O^{t})=C_{h}(C_{h}(O^{t-1})\cup O^{t}).$ Then, by induction hypothesis,
$C_{h}(O^{t})=C_{h}(C_{h}(O_{A}^{t-1})\cup O^{t})$ and, consequently, we have
$C_{h}(O^{t})=C_{h}(O_{A}^{t-1}\cup O^{t})$ according to (iii) in Lemma 1.
Therefore, by definition of $O_{A}^{t},$ $C_{h}(O^{t})=C_{h}(O_{A}^{t})$

The proof of (5) follows from (4) when $t=T$ and the definition of
$\overline{\phi}_{h}$. $\square$

Proof of Theorem 1 First, we prove that $\overline{\phi}$ does not admit any
obvious manipulation in the sense of (2).

Given $d\in D$ and $P_{d}\in\mathcal{P}_{d}$, let
$y=W_{d}(P_{d},O^{\overline{\phi}}(P_{d})).$ As $\overline{\phi}$ is
individually rational, $\\{y\\}R_{d}\varnothing$. As $\mathbf{X}$ is finite,
there exists $\widehat{P}_{-d}\in\mathcal{P}_{-d}$ such that
$\\{y\\}=\overline{\phi}_{d}(P_{d},\widehat{P}_{-d}).$ We denote
$\widehat{X}=\overline{\phi}(P_{d},\widehat{P}_{-d}),$ so
$\widehat{X}_{d}=\\{y\\}.$

Let $O^{t}$ be the set of offers made in the Stage $t$ of the D-DA at
$(P_{d},\widehat{P}_{-d})$ and let $O_{A}^{t}=\cup_{k=1}^{t}O^{k}$ for all
$t=1,\ldots,T$, where $T$ the number of iterations of the D-DA at
$(P_{d},\widehat{P}_{-d})$.

Claim 1: Let
$x=\min\limits_{P_{d}}\\{w\in\mathbf{X}_{d}:\\{w\\}P_{d}\\{y\\}\\}$, then
$\\{x\\}$ $P_{d}$ $W_{d}(P_{d},O^{\overline{\phi}}(P_{d}^{\prime}))$ for all
$P_{d}^{\prime}\in\mathcal{P}_{d}\setminus\\{P_{d}\\}.$888If
$\\{w\in\mathbf{X}_{d}:\\{w\\}P_{d}\\{y\\}\\}=\varnothing,$ then
$\overline{\phi}$ would not admit any obvious manipulation trivially.

As $\\{x\\}P_{d}\\{y\\}=\overline{\phi}_{d}(P_{d},\widehat{P}_{-d}),$ there
exists $k^{\prime}\in\\{1,...,T\\}$ in which $x\in O^{k^{\prime}}$ and
$x\notin C_{x_{H}}(O^{k^{\prime}}).$ As $d$ makes at most one offer in each
stage of the algorithm,

$[C_{H}(O^{k^{\prime}})]_{d}=\varnothing.$ (6)

Let $\widetilde{P}_{-d}\in\mathcal{P}_{-d}$ be such that for each $i\in
D\backslash\\{d\\},$ $\widetilde{P}_{i}=[C_{H}(O^{k})]_{i},\varnothing.$ We
proved that
$\\{x\\}P_{d}\overline{\phi}_{d}(P_{d}^{\prime},\widetilde{P}_{-d})$ for all
$P_{d}^{\prime}\in\mathcal{P}_{d}\setminus\\{P_{d}\\}.$

Assume that there exists
$P_{d}^{\prime}\in\mathcal{P}_{d}\setminus\\{P_{d}\\}$ such that
$\overline{\phi}_{d}(P_{d}^{\prime},\widetilde{P}_{-d})=\\{z\\}$ and
$\\{z\\}R_{d}\\{x\\}.$ Let $T^{\prime}$ be the number of iterations of the
D-DA at $(P_{d}^{\prime},\widetilde{P}_{-d}).$ W.l.o.g. we can assume that
$T=T^{\prime}$, in another case we add some artificial steps where each doctor
offers the contract assigned by the D-DA. Let $O^{\prime t}$ be the set of
offers in the Stage $t$ of the D-DA at $(P_{d}^{\prime},\widetilde{P}_{-d})$
and let $O_{A}^{\prime t}:=\cup_{k=1}^{t}O^{\prime k}.$ As
$\\{z\\}R_{d}\\{x\\}P_{d}\\{y\\}R_{d}\varnothing,$ $z\neq\varnothing$ and
$z_{H}\in H$ is well defined. Therefore, by Lemma 2, $\
\overline{\phi}_{z_{H}}(P_{d}^{\prime},\widetilde{P}_{-d})=C_{z_{H}}(O_{A}^{\prime
T})$ and then,

$z\in C_{z_{H}}(O_{A}^{\prime T})$ (7)

Hence, as $z_{D}=d$, by substitutability, $z\in C_{z_{H}}([O_{A}^{\prime
T}]_{d})$. Furthermore, as $C_{z_{H}}([O_{A}^{\prime T}]_{d})$ is in
$A(([O_{A}^{T}]_{d})_{z_{H}})$, $\left|C_{z_{H}}((O_{A}^{\prime
T})_{d})\right|\leq 1.$ Therefore,

$C_{z_{H}}([O_{A}^{\prime T}]_{d})=\\{z\\}.$ (8)

By definition of $\widetilde{P}_{-d},$ $[O_{A}^{\prime T}]_{i}=[O^{\prime
1}]_{i}=[C_{H}(O^{k^{\prime}})]_{i}$ for each $i\in D\backslash\\{d\\}$. Then,

$C_{z_{H}}(\cup_{i\in D\backslash\\{d\\}}[O_{A}^{\prime
T}]_{i})=C_{z_{H}}(\cup_{i\in D\backslash\\{d\\}}[C_{H}(O^{k})]_{i}).$ (9)

Hence by (6) and (9) ,

$C_{z_{H}}(\cup_{i\in D\backslash\\{d\\}}[O_{A}^{\prime
T}]_{i})=C_{z_{H}}(C_{H}(O^{k^{\prime}}))$

Therefore, by (ii) in Lemma 1,

$C_{z_{H}}(\cup_{i\in D\backslash\\{d\\}}[O_{A}^{\prime
T}]_{i})=C_{z_{H}}(O^{k^{\prime}})$ (10)

Then, as $O_{A}^{\prime T}=[O_{A}^{\prime T}]_{d}\cup(\cup_{i\in
D\backslash\\{d\\}}[O_{A}^{\prime T}]_{i}),$ by (iii) in Lemma 1,

$C_{z_{H}}(O_{A}^{\prime T})=C_{z_{H}}(C_{z_{H}}([O_{A}^{\prime T}]_{d})\cup
C_{z_{H}}(\cup_{i\in D\backslash\\{d\\}}[O_{A}^{\prime T}]_{i})).$

Hence by (8) and (10), $C_{z_{H}}(O_{A}^{\prime T})=C_{z_{H}}(\\{z\\}\cup
C_{z_{H}}(O^{k^{\prime}})).$ Then, by (7),

$z\in C_{z_{H}}(\\{z\\}\cup C_{z_{H}}(O^{k^{\prime}})).$ (11)

Furthermore, by (6) and Lemma 2,

$z\notin C_{z_{H}}(O_{A}^{k^{\prime}}).$ (12)

As $\\{z\\}R_{d}\\{x\\}$ and $x\in O^{k^{\prime}},$ $z\in O_{A}^{k^{\prime}}$.
Then, $O_{A}^{k^{\prime}}=\\{z\\}\cup O_{A}^{k^{\prime}}.$ By (iii) in Lemma
1,

$C_{z_{H}}(O_{A}^{k^{\prime}})=C_{z_{H}}(\\{z\\}\cup
O_{A}^{k^{\prime}})=C_{z_{H}}(\\{z\\}\cup
C_{z_{H}}(O_{A}^{k^{\prime}}))=C_{z_{H}}(\\{z\\}\cup
C_{z_{H}}(O^{k^{\prime}}))$

Therefore, by (12), $z\notin C_{z_{H}}(\\{z\\}\cup C_{z_{H}}(O^{k^{\prime}}))$
which contradicts (11).

Then, $\\{x\\}$ $P_{d}\overline{\text{
}\phi}_{d}(P_{d}^{\prime},\widetilde{P}_{-d})$ for all
$P_{d}^{\prime}\in\mathcal{P}_{d}\setminus\\{P_{d}\\}$ and the proof of Claim
1 is completed.

Now, by Claim 1 and definition of $x$, $\\{y\\}$ $R_{d}$
$W_{d}(P_{d},O^{\overline{\phi}}(P_{d}^{\prime}))$ for all
$P_{d}^{\prime}\in\mathcal{P}_{d}\setminus\\{P_{d}\\}.$ Hence,
$\overline{\phi}$ does not admit obvious manipulations in the sense of (2).

Second, we prove that $\overline{\phi}$ does not admit any obvious
manipulation in the sense of (3).

Given $d\in D$ and $P_{d}\in\mathcal{P}_{d}$ let $\overline{x}\in X_{d}$ such
that
$\overline{x}=\max\limits_{P_{d}}\\{x\in\mathbf{X}_{d}:\\{x\\}\succ_{x_{H}}\varnothing\\}.$
Now let $\widetilde{P}_{-d}\in\mathcal{P}_{-d}$ be such that
$\widetilde{P}_{i}=\varnothing$ for all $i\in D\setminus\\{d\\}.$ Then, from
the definition of D-DA, it follows that
$\\{\overline{x}\\}=\overline{\phi}_{d}(P_{d},\widetilde{P}_{-d}).$ This
implies $C_{d}(P_{d},O^{\overline{\phi}}(P_{d}))$ $R_{d}$
$\\{\overline{x}\\}.$

Furthermore, given $P_{d}^{\prime}\in\mathcal{P}_{d}\setminus\\{P_{d}\\}$, if
$y\in X_{d}$ and $\varnothing\succ_{y_{{}_{H}}}\\{y\\},$ as $\overline{\phi}$
is individually rational and $\succ_{y_{H}}$ is substitutable, we have that
$y\notin O^{\overline{\phi}}(P_{d}^{\prime})$.999By sustitubility, $x\in
C_{y_{H}}(Y)$ implies that $x\in C_{y_{H}}(\\{x\\})$. Hence
$\\{x\\}P_{y_{H}}\varnothing.$ Therefore, for all
$P_{d}^{\prime}\in\mathcal{P}_{d}\setminus\\{P_{d}\\}$ we have
$O^{\overline{\phi}}(P_{d}^{\prime})\subseteq\\{x\in\mathbf{X}_{d}:\\{x\\}P_{x_{{}_{H}}}\varnothing\\}$
and, consequently, $\\{\overline{x}\\}$ $R_{d}$
$C_{d}(P_{d},O^{\overline{\phi}}(P_{d}^{\prime})).$ So, for every
$P_{d}^{\prime}\in\mathcal{P}_{d}\setminus\\{P_{d}\\}$, we obtain

$C_{d}(P_{d},O^{\overline{\phi}}(P_{d}))\text{ }R_{d}\text{
}\\{\overline{x}\\}\text{ }R_{d}\text{
}C_{d}(P_{d},O^{\overline{\phi}}(P_{d}^{\prime}))$

This completes the proof of Theorem 1. $\square$

###  3.2 The hospital-optimal rule

Similarly to the previous subsection, given a profile $P\in\mathcal{P}$, the
hospital-optimal stable allocation can be defined by the Hospital-proposing
Deferred Acceptance Algorithm (H-DA) where hospitals make offers. It is
defined as the D-DA by interchanging the roles of $D$ and $H$.

Let $\underline{\phi}:\mathcal{P}\mathbf{\rightarrow}A(\mathbf{X})$ denote the
hospital-optimal rule, i.e., the rule that returns the hospital-optimal stable
allocation for each preference profile for doctors, $P\mathcal{\in P}$.
Trivially, $\underline{\phi}$ is a stable rule. In the context of college
admission Troyan and Morrill (2020) show that $\underline{\phi}$ is
NOM.101010They show that any stable is NOM. Surprisingly, such a result can
not be extended to the context of matching with contracts. In fact, the
opposite result holds: $\underline{\phi}$ is obviously manipulable even in the
particular case of the one-to-one matching model with contracts. This result
reveals a substantial difference between the models with and without contracts
from the point of view of the strategic incentives of agents.

###### Theorem 2

The hospital-optimal rule
$\underline{\phi}:\mathcal{P}\mathbf{\rightarrow}A(\mathbf{X})$ is OM, even in
the one-to-one matching model with contracts.111111The one-to-one model can be
addressed as a particular case of the many-to-one model where $\succ_{h}$ only
has singleton sets as acceptable, for each $h\in H$.

_Proof._ It will be sufficient to show a particular market where a doctor
$d\in D$ has an obvious manipulation at $\underline{\phi}.$ Let
$D=\\{d_{1},d_{2}\\}$, $H=\\{h_{1},h_{2}\\}$ and $\mathbf{X}=\\{x,z,w\\}$ be
the sets of doctors, hospitals, and contracts, respectively. Suppose that
$x_{D}=z_{D}=d_{1},$ $x_{H}=z_{H}=h_{1}$ and $w_{D}=d_{2},$ $w_{H}=h_{2}.$
Assume that $\succ_{h_{1}}=x,y$ and $\succ_{h_{2}}=w.$ Let $P_{2}$ be an
arbitrary preference in $\mathcal{P}_{d_{2}}.$ Let
$P_{1}\in\mathcal{P}_{d_{1}}$ such that $P_{1}=y,x,\varnothing.$ Following
H-DA, $h_{1}$ offers its best contract $x$ to doctor $d_{1}$ and this contract
is accepted by $d_{1}$. Then, $d_{1}$ does not receive more offers, and
$h_{1}$ does not make more offers. Then, the contract $x$ is the only one
contract of $d_{1}$ chosen by $\underline{\phi}$ at profile $(P_{1},P_{2}).$
Hence, $\underline{\phi}_{d_{1}}(P_{1},P_{2})=x.$ Now, let
$P_{1}^{\prime}\in\mathcal{P}_{d_{1}}$ such that
$P_{1}^{\prime}:=y,\varnothing.$ Following H-DA, $h_{1}$ offers its best
contract $x$ to doctor $d_{1}$ and this contract is rejected by $d_{1}$. Then,
$h_{1}$ offers its second-best contract $y$ to doctor $d_{1}$ and this
contract is accepted by $d_{1}$. Then, $d_{1}$ does not receive more offers,
and $h_{1}$ does not make more offers. Then, the contract $y$ is the only
contract of $d_{1}$ chosen by $\underline{\phi}$ at profile
$(P_{1}^{\prime},P_{2}).$ Hence,
$\underline{\phi}_{d_{1}}(P_{1}^{\prime},P_{2})=y.$ Therefore,
$\underline{\phi}_{d_{1}}(P_{1}^{\prime},P_{2})=yP_{1}x=\underline{\phi}_{d_{1}}(P_{1},P_{2}),$
for any $P_{2}\in\mathcal{P}_{d_{2}}.$ Hence, $P_{1}^{\prime}$ is an obvious
manipulation of $\underline{\phi}$ at $P_{d}.$ $\square$

Finally, in the context without contracts, an equivalent result to Theorem 1
can be obtained for $\underline{\phi}$. This extends the result in Troyan and
Morrill (2020) to substitutable preferences.

###### Theorem 3

Assume that for each pair $(d,h)\in D\times H$ there is at most one contract
$x\in\mathbf{X}$ such that $x_{D}=d$ and $x_{H}=d.$ Then, the hospital-optimal
rule $\underline{\phi}:\mathcal{P}\mathbf{\rightarrow}A(\mathbf{X})$ is NOM.

_Proof._ First, we prove that $\underline{\phi}$ does not admit any obvious
manipulation in the sense of (2). Given $d\in D$ and
$P_{d}\in\mathcal{P}_{d}$, let $y=W(P_{d},O^{\underline{\phi}}(P_{d})).$ As
$\underline{\phi}$ is individually rational, $\\{y\\}R_{d}\varnothing$. As
$\mathbf{X}$ is finite, there exists $\widehat{P}_{-d}\in\mathcal{P}_{-d}$
such that $\\{y\\}=\underline{\phi}_{d}(P_{d},\widehat{P}_{-d}).$ We denote
$\widehat{X}=\underline{\phi}(P_{d},\widehat{P}_{-d}),$ so
$\widehat{X}_{d}=\\{y\\}.$

Let $\widetilde{P}_{-d}\in\mathcal{P}_{-d}$ be such that for each $i\in
D\backslash\\{d\\},$ $\widetilde{P}_{i}=\widehat{X}_{i},\varnothing.$

Claim:
$\widehat{X}_{d}R_{d}\underline{\phi}(P_{d}^{\prime},\widetilde{P}_{-d})$ for
all $P_{d}^{\prime}\in\mathcal{P}_{d}\setminus\\{P_{d}\\}.$

Assume on contradiction that there exists
$P_{d}^{\prime}\in\mathcal{P}_{d}\setminus\\{P_{d}\\}$ such that
$\underline{\phi}_{d}(P_{d}^{\prime},\widetilde{P}_{-d})=\\{z\\}$ and

$\\{z\\}P_{d}\\{y\\}.$ (13)

Let $T^{\prime}$ be the number of iterations of the H-DA in
$(P_{d}^{\prime},\widetilde{P}_{-d})$ and let $X^{T^{\prime}}$ has in H-DA’s
definition. As $\\{z\\}P_{d}\\{y\\}R_{d}\varnothing,$ $z\neq\varnothing$ and
$z_{H}\in H$ is well defined. Furthermore, as $z_{D}=d,$ by hypothesis,
$z_{H}\neq y_{H}$. As $\widehat{X}$ is an allocation, $y\in\widehat{X}$ and
$y_{D}=d$, it follows that $x_{D}\neq d$ for each $x\in\widehat{X}_{z_{H}}.$
Hence, by definition of $\widetilde{P}_{-d},$
$\widetilde{P}_{x_{D}}:=x,\varnothing$ for each $x\in\widehat{X}_{z_{H}}$.
Then, contracts in $\widehat{X}_{z_{H}}$ are never rejected in the H-DA at
$(P_{d}^{\prime},\widetilde{P}_{-d})$. Then, by definition of H-DA,
$\widehat{X}_{z_{H}}\subseteq X^{T^{\prime}}.$ As
$\underline{\phi}_{d}(P_{d}^{\prime},\widetilde{P}_{-d})=\\{z\\},$ $z\in
C_{H}(X^{T^{\prime}}).$ Then, by substitutability and the definition of choice
set,

$z\in C_{z_{H}}(z\cup\widehat{X}_{z_{H}})=C_{z_{H}}(z\cup\widehat{X})$ (14)

But (13) and (14) contradict the stability of
$\underline{\phi}(P_{d},\widehat{P}_{-d})$ and the fact that
$\underline{\phi}_{d}(P_{d}^{\prime},\widehat{P}_{-d}t)\neq z$. This finishes
the proof of the Claim.

By Claim,
$W(P_{d},O^{\underline{\phi}}(P_{d}))P_{d}W(P_{d},O^{\underline{\phi}}(P^{\prime}_{d}))$
for all $P^{\prime}_{d}\in\mathcal{P}_{d}\setminus\\{P_{d}\\}$. Therefore,
$\underline{\phi}$ does not admit any obvious manipulation in the sense of
(2). The proof that $\underline{\phi}$ does not admit any obvious manipulation
in the sense of (3) is similar to the one given to $\overline{\phi}$ in
Theorem 1 and therefore it is omitted. $\square$

###  3.3 Quantile stable matchings

In this section, we introduce the class of quantile stable rules (or
mechanisms) defined in Chen et al. (2016) and studied in Chen et al. (2021)
and Fernandez (2020), among others. It is a new and relevant class of rules
that generate stable matchings which can be seen as a compromise between both
sides of the market. The extremal quantile-stable rules are the doctor-optimal
and the hospital-optimal ones. They assign the best stable outcome for one
side and the worst for the other.

To guarantee the existence of quantile-stable rules it is necessary to assume
that hospitals’ preferences satisfy the law of aggregate demand, in addition
to substatibility.121212see Chen et al. (2016).. The law of aggregate demand
establishes that the number of contracts chosen by the agent either rises or
stays the same if the set of available contracts increases.

###### Definition 8

A preference $\succ_{h}$ of an agent $h\in H$ satisfies the law of aggregate
demand (LAD) if

$\left|C_{h}(Y,\succ_{h})\right|\leq\left|C_{h}(Z,\succ_{h})\right|$

for all $Y\subseteq Z\subseteq\mathbf{X.\vskip 6.0pt plus 2.0pt minus 2.0pt}$

LAD is less restrictive than the property of $q$-responsiveness assumed in the
college admission problem (see Pepa Risma, 2015) and trivially holds in the
one-to-one matching model.

Along this subsection, we assume that $\succ_{h}$ satisfies substitutability
and the law of aggregated demand, for each $h\in H.$

###### Definition 9

For each $q\in[0,1]$, the $q$-quantile stable rule
$\phi^{q}:\mathcal{P}\rightarrow A(\mathbf{X})$ is defined as follows: given
$P\in\mathcal{P}$, let $k$ be the number of stable matchings under $P$ (and
$\succ$), then $\phi^{q}(P)$ is the allocation obtained by joining, for each
doctor $d\in D,$ $d$’s $\lceil kq\rceil$-th best stable allocation in $P$
according to order $P_{d}.$131313Here, $\lceil x\rceil$ denotes the lowest
positive integer equal to or larger than $x$.

Observe that $\phi^{0}=\overline{\phi}$ and $\phi^{1}=\underline{\phi}$ are
the extreme quantile stable rules. Chen et al. (2016) show, that $\phi^{q}$ is
a stable rule when $\succ_{h}$ satisfies substatibility and the law of
aggregate demand for each $h\in H$.

###### Theorem 4

A $q$-quantile stable rule $\phi^{q}:\mathcal{P}\rightarrow A(\mathbf{X})$ is
NOM if and only if $q=0$ i.e.$\phi^{q}=\overline{\phi}$ is the doctor-optimal
rule, even in the one-to-one matching model with contracts.

_Proof._ By Theorem 1, $\phi^{0}=\overline{\phi}$ is NOM. Now assume that
$q\in(0,1],$ we will show that a particular market exists where a doctor $d\in
D$ has an obvious manipulation at $\phi^{q}.$ Let $D=\\{d_{1},d_{2}\\}$,
$H=\\{h_{1},h_{2}\\}$ be the sets of doctors and hospitals, respectively. Let
$k$ be a positive integer such that $\lceil kq\rceil=2$ and let
$\mathbf{X}=\\{x^{1},x^{2},\ldots,x^{k},w\\}$ be the set of contracts, where
$x_{D}^{i}=d_{1},$ $x_{H}^{i}=h_{1}$ for each $i=1,\ldots,k$ and
$w_{D}=d_{2},$ $w_{H}=h_{2}.$ Assume that
$\succ_{h_{1}}=x^{k},x^{k-1},\ldots,x^{1}$ and $\succ_{h_{2}}=w.$ Let $P_{2}$
be an arbitrary preference in $\mathcal{P}_{d_{2}}.$ Let
$P_{1}\in\mathcal{P}_{d_{1}}$ such that $P_{1}=x^{1},x^{2},\ldots,x^{k}.$
Observe that the set of all stable allocations under $P$ is $\cup_{j=1\cdots
k}\\{\\{x^{j},w\\}\\}$ if $wP_{2}\varnothing$ and $\cup_{j=1\cdots
k}\\{\\{x^{j}\\}\\}$ if $\varnothing P_{2}w.$ As $\lceil kq\rceil=2,$
$\phi_{d_{1}}^{q}(P)=x^{2}.$ Now, let $P_{1}^{\prime}\in\mathcal{P}_{d_{1}}$
such that $P_{1}^{\prime}=x^{1}.$ Then, the set of all stable allocations
under $(P_{1}^{\prime},P_{2})$ is $\\{\\{x^{1},w\\}\\}$ if
$wP_{2}\varnothing$, and $\\{\\{x^{1}\\}\\}$ if $\varnothing P_{2}w.$ Hence
$\phi_{d_{1}}^{q}(P_{1}^{\prime},P_{2})=x^{1}.$ Therefore,
$\phi_{d_{1}}^{q}(P_{1}^{\prime},P_{2})=x^{1}P_{1}x^{2}=\phi_{d_{1}}^{q}(P_{1},P_{2})$
for any $P_{2}\in\mathcal{P}_{d_{2}}.$ So, $P_{1}^{\prime}$ is an obvious
manipulation of $\phi^{q}$ at $P_{d}.$ $\square$

##  4 Final Remarks

Table 1 summarizes our main findings. Remember that we assume that hospitals’
preferences are substitutable. When preferences are responsive141414Preference
$\succ_{h}$ is responsive with quota $q$, where $q$ is a non negative integer
number, if:  (i) For all $Z\in A(\mathbf{X}_{h})$ such that $\mid Z\mid>q$, we
have $\varnothing\succ_{h}Z$. (ii) For every pair of contracts
$x,y\in\mathbf{X}_{h}\cup\varnothing,$ and every
$Z\subseteq\mathbf{X}_{h}\setminus\\{x,y\\}$ with $\mid Z\mid<q$, whenever
$Z\cup\\{x\\}$ and $Z\cup\\{y\\}$ are in $A(\mathbf{X}_{h})$, we have
$Z\cup\\{x\\}\succ_{h}Z\cup\\{y\\}\text{ if and only if\
}\\{x\\}\succ_{h}\\{y\\}.$ the results are the same as in the one-to-one
model.

| NOM (for doctors) |
---|---|---
| Without Contracts | With Contracts |
|  one-to-one † | many-to-one | one-to-one | many-to-one |
Doctor-optimal | $yes$ | $yes$ | $yes$ | $yes$ |  Th.1
Hospital-optimal | $yes$ | $yes$ | $no$ | $no$ |  Th.3 and Th.2
|  $\phi^{q}(q$-quantile$)$
---
$q\in(0,1)$
$yes$ | $?$ | $no$ | $no$ |  Th.4
†These results are due to Troyan and Morrill (2020) |
Table 1: _Summary of results._

Two remarks arise from the table. First, there are some open questions left by
the present paper: (i) Are all (quantile) stable rules NOM in the many-to-one
matching model without contracts and substitutable preferences for hospitals?
(ii) Are there other NOM stable rules besides the doctor-optimal rule in
matching models with contracts and, if so, how might these rules be
characterized? Second, as we said before, there is a substantial difference
between the models without and with contracts from the point of view of the
strategic incentives of agents. In fact, the hospital-optimal rule does not
admit obvious manipulations in the context without contracts, but it does in
the context with contracts, even in the simplest one-to-one case.

Finally, an additional open problem arises by interchanging the roles between
doctors and hospitals and considering the manipulability of rules by
hospitals. Even in the college admission model, no stable rule exists that is
not manipulable for colleges (see Roth, 1985). Therefore, in this simple
context, the consideration of non-obvious manipulable matching rules looks
like an interesting problem.

## References

  * Arribillaga and Bonifacio (2022) Arribillaga, R. P. and A. G. Bonifacio (2022): “Obvious manipulations of tops-only voting rules,” _arXiv preprint arXiv:2210.11627_.
  * Aziz and Lam (2021) Aziz, H. and A. Lam (2021): “Obvious manipulability of voting rules,” in _International Conference on Algorithmic Decision Theory_ , Springer, 179–193.
  * Chen et al. (2016) Chen, P., M. Egesdal, M. Pycia, and M. B. Yenmez (2016): “Median stable matchings in two-sided markets,” _Games and Economic Behavior_ , 97, 64–69.
  * Chen et al. (2021) ——— (2021): “Quantile stable mechanisms,” _Games_ , 12, 43.
  * Fernandez (2020) Fernandez, M. A. (2020): “Deferred acceptance and regret-free truth-telling,” Working Paper.
  * Gale and Shapley (1962) Gale, D. and L. Shapley (1962): “College admissions and the stability of marriage,” _The American Mathematical Monthly_ , 69, 9–15.
  * Hatfield and Milgrom (2005) Hatfield, J. and P. Milgrom (2005): “Matching with contracts,” _American Economic Review_ , 95, 913–935.
  * Martínez et al. (2004) Martínez, R., J. Massó, A. Neme, and J. Oviedo (2004): “On group strategy-proof mechanisms for a many-to-one matching model,” _International Journal of Game Theory_ , 33, 115–128.
  * Ortega and Segal-Halevi (2022) Ortega, J. and E. Segal-Halevi (2022): “Obvious manipulations in cake-cutting,” _Social Choice and Welfare_ , 1–20.
  * Pepa Risma (2015) Pepa Risma, E. (2015): “Binary operations and lattice structure for a model of matching with contracts,” _Mathematical Social Sciences_ , 73, 6–12.
  * Psomas and Verma (2022) Psomas, A. and P. Verma (2022): “Fair and efficient allocations without obvious manipulations,” _arXiv preprint arXiv:2206.11143_.
  * Roth (1985) Roth, A. (1985): “The college admissions problem is not equivalent to the marriage problem,” _Journal of Economic Theory_ , 36, 277–288.
  * Roth (1982) Roth, A. E. (1982): “The economics of matching: Stability and incentives,” _Mathematics of operations research_ , 7, 617–628.
  * Troyan et al. (2020) Troyan, P., D. Delacrétaz, and A. Kloosterman (2020): “Essentially stable matchings,” _Games and Economic Behavior_ , 120, 370–390.
  * Troyan and Morrill (2020) Troyan, P. and T. Morrill (2020): “Obvious manipulations,” _Journal of Economic Theory_ , 185, 104970.


# Observations and predictions from past lightcones

Martin Lesourd  Black Hole Initiative, Harvard University
<EMAIL_ADDRESS>
###### Abstract.

In a general Lorentzian manifold $M$, the past lightcone of a point is a
proper subset of $M$ that does not carry enough information to determine the
rest of $M$. That said, if $M$ is a globally hyperbolic Cauchy development of
vacuum initial data on a Cauchy surface $S$ and there is a point whose past
lightcone contains $S$, then the contents of such a lightcone determines all
of $M$ (up to isometry). We show some results that describe what properties of
$M$ guarantee that past lightcones do indeed determine all or at least
significant portions of $M$. Null lines and observer horizons, which are well
known features of the de-Sitter spacetime, play a prominent role.

##  1\. Introduction

In Lorentzian geometry, an observer at a given time in a spacetime $(M,g)$ is
represented by a timelike curve with future endpoint $p\in M$, and the past
lightcone $J^{-}(p)\subset M$ of $p$ represents all signals in $M$ that can
reach the observer at $p$. One can then ask the following.

  1. (A)

Can an observer at $p\in M$ know the global structure of $M$ on the basis of
$J^{-}(p)$?

  2. (B)

Can an observer at $p\in M$ make predictions about $M\backslash J^{-}(p)$ on
the basis of $J^{-}(p)$?

In this short note, we describe some of what is known about (A) and (B), we
prove various further results, and we list some natural further questions.
Part of the appeal in (A) and (B) is that they are subject to somewhat
surprising examples. As described below, two (inextendible and globally
hyperbolic) spacetimes $(M^{\prime},g^{\prime})$ and $(M,g)$ can be non-
isometric, in spite of the fact that each member of the countable collection
of past lightcones $\\{I^{-}(p_{i})\\}$ that covers $M$ can be isometrically
embedded into $(M^{\prime},g^{\prime})$ and likewise with $M$ and $M^{\prime}$
interchanged. Here we will show when this cannot happen.
We now recall some basic definitions of causal theory, cf. [1], [14], [16] for
some classic references and [15] for a more recent authorative survey.
A spacetime $(M,g)$ is a connected $C^{\infty}$ Hausdorff manifold $M$ of
dimension two or greater with a Lorentzian metric $g$ of signature
$(-,+,+,...)$, and we will assume a time and space orientation. Since
regularity is not the issue here, for simplicity of expression we take $g$ to
be smooth, but many of the arguments can be extended to lower regularity.
The lightcone structure inherited on the tangent space $T_{p}M$ at each $p$
leads to the notion of a causal curve, which in turn leads to defining
$J^{-}(p)$ (or $I^{-}(p)$) as the collection of all points $q\in M$ from which
there exists a causal (or timelike) curve with future endpoint $p$ and past
endpoint $q$. A causal curve between $p$ and $q\in J^{+}(p)$ is achronal iff
$q\notin I^{+}(p)$. A null line is an inextendible achronal causal curve.
A set is achronal (acausal) iff no two members of it can be connected by a
timelike (causal) curve. The domain of dependence $D(S)$ of a set $S\subset M$
is given by $D(S)=D^{+}(S)\cup D^{-}(S)$ and $D^{+}(S)$ is defined as the
collection of all points $q$ in $M$ such that any inextendible ($C^{1}$) past
directed causal curve passing through $q$ intersects $S$. $\tilde{D}(S)$ is
defined identically except that curves are timelike, rather than causal. From
the perspective of the Cauchy problem of general relativity, $D^{+}(S)$
represents the maximal portion of $M$ that could be determined by initial data
on $S$. If $S$ is closed as a subset of $M$ then
$\overline{D^{+}(S)}=\tilde{D}^{+}(S)$. The future Cauchy horizon is defined
as $H^{+}(S)\equiv\overline{D^{+}(S)}\backslash I^{-}(D^{+}(S))$ and
$H(S)=H^{+}(S)\cup H^{-}(S)$.
$S$ is a partial Cauchy hypersurface if it an edgeless acausal set, cf.
Definition 14.27 of [1] for the definition of $\text{edge}(S)$ for an achronal
set $S$. A spacetime $(M,g)$ is globally hyperbolic iff there exists a partial
Cauchy hypersurface $S$ such that $M=D(S)$. By [2], a smooth globally
hyperbolic spacetime $(M,g)$ is isometric to $(\mathbb{R}\times
S,-f(t)dt^{2}+h(t))$ where $f(t)$ is smooth and $h(t)$ a Riemannian metric on
$S$. From the perspective of causal structure, global hyperbolicity is
equivalent to the spacetime being causal111No closed causal curves. and
$J^{+}(p)\cap J^{-}(q)$ being compact for all $p,q\in M$.
If $S$ is acausal and $S\cap\text{edge}(S)=\emptyset$ (eg., $S$ is a partial
Cauchy hypersurface), then $D(S)$ is non-empty, open, and $D(S)\cap
H(S)=\emptyset$. Note that the openness of $D(S)$ may be ruined if we take $S$
to be achronal rather than acausal.
A spacetime $(M,g)$ is causally simple iff it is causal and $J^{+(-)}(p)$ is
closed for all $p\in M$. Global hyperbolicity is strictly stronger than causal
simplicity.
An isometric embedding of $(M,g)$ into $(M^{\prime},g^{\prime})$ is an
injective (but not necessarily surjective) map $\phi:M\hookrightarrow
M^{\prime}$ such that $\phi$ is a diffeomorphism onto its image $\phi(M)$ and
$\phi^{*}(g^{\prime})=g$. If $\phi$ maps $M$ surjectively onto $M^{\prime}$
then we write $\phi:M\to M^{\prime}$ and we say that $(M,g)$ and
$(M^{\prime},g^{\prime})$ are isometric. A spacetime $(M,g)$ is inextendible
when there exists no isometric embedding $\phi$ of $M$ into $M^{\prime}$ such
that $M^{\prime}\backslash\phi(M)\neq\emptyset$.
A spacetime is future holed if there is a partial Cauchy hypersurface
$S^{\prime}\subset M^{\prime}$ and an isometric embedding
$\phi:\tilde{D}(S^{\prime})\hookrightarrow M$ such that $\phi(S^{\prime})$ is
acausal in $(M,g)$ where $(M,g)$ is any spacetime, and
$\phi(H^{+}(S^{\prime}))\cap D^{+}(\phi(S^{\prime}))\neq\emptyset$. A
spacetime is hole-free if it lacks future and past holes. Minguzzi [12] shows
that causally simple, inextendible spacetimes are hole free.

Acknowledgements. We thank the Gordon Betty Moore Foundation and John
Templeton Foundation for their support of Harvard’s Black Hole Initiative. We
thank Professors Erik Curiel, JB Manchak, Ettore Minguzzi, Chris Timpson, and
James Weatherall for valuable comments that improved the paper.

##  2\. Previous Work

(A). A natural definition coming from [8] (whose terminology is slightly
different) is the following.

######  Definition 2.1.

A spacetime $(M,g)$ is weakly observationally indistinguishable from
$(M^{\prime},g^{\prime})$ just in case there is an isometric embedding
$\phi:I^{-}(p)\hookrightarrow M^{\prime}$ for every point $p\in M$. If this is
also true with $M$ and $M^{\prime}$ interchanged, then the spacetimes are
strongly observationally indistinguishable.

One could replace $I^{-}(p)$ with $J^{-}(p)$, and although this makes no real
difference to any of the results in [8] and [9] or indeed to what follows,
that would capture a somewhat more honest sense of distinguishability because
observers can certainly receive signals from $J^{-}(p)\backslash I^{-}(p)$.
One can also take observers to be inextendible timelike curves $\sigma$ (as is
also done in [8]), but in that case $J^{-}(\sigma)\subset M$ would represent
“all possible observations that could be made supposing that the observer
lasts forever with respect to $M$”, which, though interesting, is stronger
than what happens in practice.
Malament pg. 65-6 [8] gives examples of spacetimes that are strongly
observationally indistinguishable but non-isometric. His examples are globally
hyperbolic, inextendible, and exploit the presence of observer horizons in the
de-Sitter spacetime.
Based on an explicit cut and paste argument, Manchak [9] shows the following.

######  Proposition 2.2 (Manchak [9]).

Given any non-causally bizarre222A spacetime is causally bizarre if there is a
point $p\in M$ such that $M\subset I^{-}(p)$. spacetime $(M,g)$, there exists
a spacetime $(M^{\prime},g^{\prime})$ that is weakly observationally
indistinguishable from $(M,g)$ but not isometric to $(M,g)$.

Although Manchak’s construction of $(M^{\prime},g^{\prime})$ works for any
(non-causally bizarre) spacetime $(M,g)$, it relies on introducing a countably
infinite collection of holes in $(M^{\prime},g^{\prime})$. It is unknown to us
whether Proposition 2.2 holds for strong observational indistinguishability.

Viewed together, [8] and [9] lead one to the following.

Question. Find conditions $\\{A,B,...\\}$ satisfied by $(M,g)$ and
$(M^{\prime},g^{\prime})$ such that:

‘Weakly (or strongly) observationally indistinguishable $+A+B+...$’
$\Leftrightarrow$
‘$(M,g)$ and $(M^{\prime},g^{\prime})$ are isometric’

Proposition 3.6 and Corollary 3.11 below are in this direction.

(B). Geroch [7] defines prediction in general relativity as follows.

######  Definition 2.3.

$p\in M$ is predictable from $q$, written as $p\in P(q)$, iff $p\in D^{+}(S)$
for some closed, achronal set $S\subset J^{-}(q)$. $p\in M$ is verifiable iff
$p\in P(q)$ and $p\in I^{+}(q)\backslash J^{-}(q)$.

Manchak observes the following.

######  Proposition 2.4 (Manchak [10]).

If $P(q)\cap(I^{+}(q)\backslash J^{-}(q))\neq\emptyset$, then $(M,g)$ admits
an edgless compact achronal set.

A slightly different notion of prediction considered in [10] is as follows.

######  Definition 2.5.

$p\in M$ is genuinely predictable from $q$, written $\mathcal{P}(q)$, iff
$p\in P(q)$ and for all inextendible spacetimes $(M^{\prime},g^{\prime})$, if
there is an isometric embedding $\phi:J^{-}(q)\hookrightarrow M^{\prime}$,
then there is an isometric embedding $\phi^{\prime}:J^{-}(q)\cup
J^{-}(p)\hookrightarrow M^{\prime}$ such that $\phi=\phi^{\prime}_{\mid
J^{-}(q)}$.

The idea here is that genuine predictions guarantee that observers with the
same past make the same predictions. By a short cut-and-paste argument,
Manchak observes the following.

######  Proposition 2.6 (Manchak [10]).

Let $(M,g)$ be any spacetime and $q$ a point in $M$. Then
$\mathcal{P}(q)\subseteq\partial J^{-}(q)$.

Thus the domain of genuine predictions from $q$, if non-empty, is on the verge
of being a retrodiction.

##  3\. Some Observations

We assume that spacetimes satisfy the field equations

(3.1)
$G_{g}\equiv\text{Ric}_{g}-\frac{1}{2}g\>\text{Scal}_{g}=T_{\phi,...}-\Lambda\>g$

where $\Lambda\in\mathbb{R}$ is a constant, and with $T_{\phi,...}$ the
stress-energy tensor associated with possible matter fields $\\{\phi,...\\}$
in $M$.333Here we think of the Cauchy problem from the perspective of ‘initial
data’, as opposed to ‘initial and boundary data’, though the latter is more
natural for $\Lambda<0$. The system (3.1) leads to the formulation of a Cauchy
problem on a spacelike initial data set $S$. In the vacuum setting $T=0$,
$\Lambda=0$, the Cauchy problem was shown [3] to be well posed in the sense
that there exists a unique, up to isometry, maximal globally hyperbolic
development of $S$ obtained by Cauchy evolution of the initial data on $S$
according to (3.1) with $T=0$, $\Lambda=0$. This well posedness has been
extended to more general settings and we take it as a pre-condition for the
spacetimes we consider.

######  Definition 3.1.

Fix the constant $\Lambda$ and fix an expression for $T_{\phi,...}$. Given a
spacetime $(M,g)$ and an acausal edgeless connected set $S\subset M$, we say
that $(\tilde{D}(S),g|_{\tilde{D}(S)})$ is a faithful development if it is
uniquely determined, up to isometry, by Cauchy evolution of the initial data
on $S$ according to (3.1). If $(M,g)$ admits a connected acausal edgeless set,
we say that it is locally Cauchy if, for any connected acausal edgeless set
$S$, $(\tilde{D}(S),g|_{\tilde{D}(S)})$ is a faithful development.

######  Remark 3.2.

Note that this is slightly unorthodox in the sense that it is usually $D(S)$,
rather than $\tilde{D}(S)$, which we think of as being determined by $S$.
Given that $S$ is closed for the definition of locally Cauchy, we have
$\overline{D(S)}=\tilde{D}(S)$, and so being locally Cauchy is only slightly
stronger than asking for $D(S)$ to be determined up to isometry.

######  Remark 3.3.

Since we want to guarantee isometric embeddings, we want to rule out examples
of regions that are globally hyperbolic but not determined by Cauchy
evolution. To see a trivial example444Examples like this suggest the following
problem. Given a Lorentzian manifold $(M,g)$ with an arbitrary geodesically
complete Lorentzian metric $g$, what conditions on $M$ and $g$ make it
possible to solve for a function $\Omega:M\to\mathbb{R}$ such that
$(M,\Omega^{2}g)$ is complete and vacuum? smooth functions transformations
$\Omega$ , start with Minkowski spacetime $(\mathbb{R}^{1,n},\eta)$, identify
some open set $O\subset\mathbb{R}^{1,n}$ lying above $t=0$, and modify it by a
conformal transformation $\eta\to\Omega^{2}\eta$. In that case, in spite of
$M\backslash O$ being vacuum, $O$ will in general have a non-vanishing
Einstein tensor $G_{g}$. But since there are Cauchy hypersurfaces (in
$M\backslash O$) for $(\mathbb{R}^{1,n},\Omega^{2}\eta)$ which are exactly
flat555In the language of the constraints, initial data sets of the form
($\mathbb{R}^{n},g_{E},0$)., $(\mathbb{R}^{1,n},\Omega^{2}\eta)$ is not
locally Cauchy if it is not isometric to $(\mathbb{R}^{1,n},\eta)$.

We also make use of the following.

######  Definition 3.4.

Given a partial Cauchy hypersurface $S$ in a spacetime $M$ we say that an open
subset of $M$ is an $\epsilon$-development of $S$, denoted
$D_{\epsilon}(S)(\supset S)$ if there is exists an
$\epsilon>0\>(\in\mathbb{R})$ such that $D_{\epsilon}(S)$ admits a Cauchy
surface $S_{\epsilon}$ every point of which lies at distance $\geq\epsilon>0$
in the future of $S$, as measured by a normalized timelike vector field
orthogonal to $S$.

######  Remark 3.5.

The non-empty interior of a causal diamond $J^{+}(p)\cap
J^{-}(q)\neq\emptyset$ with $p,q\in(\mathbb{R}^{1,n},\eta)$ is not an
$\epsilon$-development because its Cauchy surfaces are anchored at $\partial
J^{+}(p)\cap\partial J^{+}(q)$.

We now observe the following.

######  Proposition 3.6.

Let $(M,g)$ and $(M^{\prime},g^{\prime})$ be inextendible and locally Cauchy.
Suppose that

  1. (i)

$(M,g)$ has a compact Cauchy surface and no null lines,

  2. (ii)

$(M^{\prime},g^{\prime})$ is causal and hole-free.

Then $(M,g)$ and $(M^{\prime},g^{\prime})$ are isometric iff they are weakly
observationally indistinguishable.

Note that by the examples in [8], Proposition 3.6 is false without the
assumption that $(M,g)$ lacks null lines.
In either case of weakly o.i. or strongly o.i., it would be interesting to
settle whether the compactness in (i) is necessary, cf. Proposition 3.13
below.

###### Proof.

The proof of Proposition 3.6 starts by strengthening Theorem 1 of [6].666In
[6], the authors assume that $(M,g)$ is null geodesically complete, satisfies
the null energy condition, and the null generic condition. These assumptions
implies the absence of null lines. It was then observed by Galloway that one
can prove the theorem by instead assuming that $(M,g)$ lacks null lines (cf.
footnote for Theorem 1 of [6]), but since those details never appeared we
include them for completeness.

######  Lemma 3.7.

Let $(M,g)$ be a spacetime without null lines. Then given any compact region
$K$, there exists another compact $K^{\prime}\supset K$ such that if
$p,q\notin K^{\prime}$ and $q\in J^{+}(p)-I^{+}(p)$ , then any causal curve
$\gamma$ connecting $p$ to $q$ cannot intersect $K$.

###### Proof.

Suppose otherwise for some $K_{0}$ and $K_{1}\supset K_{0}$. Then consider a
sequence of ever bigger compact sets $K_{i+1}\subset K^{\prime}_{i}$. By
assumption, each $K_{i}$ will have horismos777The future horismos of $p$ is
defined as $J^{+}\backslash I^{+}(p)$. related outer points $p_{i},q_{i}\notin
K_{i}$, $q_{i}\in J^{-}(p_{i})-I^{-}(p_{i})$ that are connected by a causal
curve, necessarily achronal, that intersects $K_{0}$. In considering larger
compact sets, we make these causal curves longer in the sense of an auxiliary
Riemannian metric. All of these curves intersect $K_{0}$. Taking the limit, by
compactness of $K_{0}$, there is a limit point for the sequence of points
lying in $K_{0}$ for each causal curve linking $p_{i},q_{i}$, which moreover
is in $K_{0}$. By a standard limit curve arguments, cf. Proposition 3.1 of
[1], there passes an inextendible limit curve through this limit point. The
limit curve is also straightforwardly seen to be achronal.888See [11] for
significantly stronger limit curve statements. Thus we have a null line. ∎

With the same proof as in Corollary 1 of [6], Lemma 3.7 implies the
following.999Lemma 3.8 strengthens the main result of [6].

######  Lemma 3.8.

Let $(M,g)$ be a spacetime with compact Cauchy surface that does not admit any
null lines. Then $(M,g)$ admits a point $p\in M$ such that $S\subset I^{-}(p)$
for some Cauchy surface $S$.

###### Proof.

We include this for completeness, but this argument is exactly as in [6]
except that Lemma 3.7 plays the role of Theorem 1 of [6]. Since $(M,g)$ is
globally hyperbolic, there exists a continuous global time function
$t:M\to\mathbb{R}$, such that each surface of constant $t$ is a Cauchy
surface. Let $K=\Sigma$ and $K^{\prime}$ be as in Lemma 3.7. Let $t_{1}$ and
$t_{2}$ denote, respectively, the minmum and maximum values of $t$ on
$K^{\prime}$. Let $\Sigma_{1}$ be any Cauchy surface with $t<t_{1}$ and let
$\Sigma_{2}$ denote the Cauchy surface $t=t_{2}$. Let $q\in
I^{+}(\Sigma_{2})$, $p\in\Sigma_{1}$ and suppose that $p\in\partial I^{-}(q)$.
Since $(M,g)$ is globally hyperbolic, $J^{-}(q)$ is closed and so $p\in
J^{-}(q)\backslash I^{-}(q)$ and thus there is a causal curve connecting $p$
and $q$. It follows from Lemma 3.7 that this causal curve does not intersect
$\Sigma$. However, this contradicts the fact that $\Sigma$ is a Cauchy
surface. Consequently, there cannot exist a $p\in\Sigma_{1}$ and such that
$p\in\partial I^{-}(q)$, i.e., $\partial I^{-}(q)\cap\Sigma_{1}=\emptyset$.
But $I^{-}(q)$ is open and since $\partial I^{-}(q)\cap\Sigma_{1}=\emptyset$,
the complement of $I^{-}(q)$ in $\Sigma_{1}$ is also open. Since we have
$I^{-}(q)\cap\Sigma_{1}$ and $\Sigma_{1}$ is connected, this implies
$\Sigma_{1}\subset I^{-}(q)$. ∎

We now finish the proof of Proposition 3.6. By Lemma 3.8 there is a point
$p\in M$ with $S\subset I^{-}(p)$ where $S$ is a Cauchy surface of $M$.

$\phi(S)$ is compact. By weak observational indistinguishability, there is an
isometric embedding $\phi:I^{-}(p)\hookrightarrow M^{\prime}$. Because $\phi$
is a diffeomorphism onto its image, the map $\phi|_{S}$, $\phi$ restricted to
$S$, is a diffeomorphism of $S$ onto its image and thus $\phi(S)$ is compact
in $M^{\prime}$.

$\phi(S)$ is achronal. Suppose otherwise that $\gamma^{\prime}$ is a past
directed timelike curve from $x^{\prime}$ to $y^{\prime}$ with
$x,y\in\phi(S)$. Extend $\gamma^{\prime}$ to $\sigma^{\prime}$ so that
$\sigma^{\prime}$ is a past directed timelike curve from $q^{\prime}$ to
$x^{\prime}$ to $y^{\prime}$. Note that the isometric embedding forces
$\phi(S)$ to be locally achronal; that is, there is an open neighborhood $O$
around around $S$ with two connected boundary components
$\partial_{+}O(\subset I^{+}(S))$, $\partial_{-}O$ (allocated using the
orientation in $M$), such that no two distinct points in $O$ can be joined by
a timelike curve in $O$. Since $O$ isometrically embeds into $M^{\prime}$, a
locally achronal neighborhood $O^{\prime}$ exists around $\phi(S)$ in
$M^{\prime}$. As such, the curve $\sigma^{\prime}$ must leave
$\partial_{-}O^{\prime}$ and re-enter $O^{\prime}$ via either
$\partial_{-}O^{\prime}$ or $\partial_{+}O^{\prime}$.
In the former case, we can build a closed piecewise smooth timelike curve from
$q^{\prime}$ and back, which violates causality of $M^{\prime}$.
In the latter case, we will obtain a contradiction with the inextendibility of
$M$. Since $I^{-}(S)\subset D^{-}(S)\subset I^{-}(p)$, we must have that
$\sigma^{\prime}$ leaves $\phi(D^{-}(S))$, say at some point
$r^{\prime}\in\partial\phi(D^{-}(S))$, and re-enter $\phi(I^{-}(p))\cap
I^{+}(\phi(S))$. Since the global hyperbolicity of $M$ implies that every
future directed timelike curve in $\phi(I^{-}(p))$ must eventually leave
$\phi(I^{-}(p))$ when sufficiently extended in the future direction, we can
take the re-entry point to lie on the ‘future’ boundary of $\phi(I^{-}(p))$;
that is, which is the endpoint of a timelike curve whose $\phi^{-1}$ pre-image
has endpoint on $\partial I^{-}(p)$. Now consider an open neighborhood
$Z^{\prime}$ of $r^{\prime}$ in $M^{\prime}\cap\partial\phi(D^{-}(S))$.
Consider the open subset $\phi(I^{-}(p))\cup Z^{\prime}$ of $M^{\prime}$. Now
define a new spacetime $M^{\prime\prime}$ by $\phi(I^{-}(p))\cup
Z^{\prime}\cup J^{+}(\partial I^{-}(p))$. We know this can be done because the
global hyperbolicity of $M$ and the locally Cauchy property of $M$ and
$M^{\prime}$ imply that the regions $\overline{I^{-}(p)}$ and
$\overline{\phi(I^{-}(p))}\backslash\partial\phi(D^{-}(S))$ are isometric. We
now have a spacetime $M^{\prime\prime}$ into which $M$ can be isometrically
embedded as a proper subset (in virtue of the extra $Z^{\prime}$ beyond
$\phi(D^{-}(S))$), contradicting the inextendibility of $M$.

$\phi(S)$ is edgeless. Compactness of $S$ and achronality of $\phi(S)$
straightforwardly implies that $\phi(S)$ is edgeless.

######  Remark 3.9.

At this point if $(M^{\prime},g^{\prime})$ is assumed globally hyperbolic,
$\phi(S)$ being an edgeless compact connected achronal set means that
$\phi(S)$ can be taken to be a Cauchy surface of $M^{\prime}$. In that case,
both $(M,g)$ and $(M^{\prime},g^{\prime})$ are representatives of the unique,
up to isometry, maximal globally hyperbolic development of $S$, and are thus
isometric. We will instead show that $(M^{\prime},g^{\prime})$ is globally
hyperbolic.

In $(M,g)$ we can consider an $\epsilon$-development $D_{\epsilon}(S)\subset
M$. Within $D_{\epsilon}(S)$, we can then find an acausal hypersurface
$S_{\epsilon}$ which is still Cauchy for $M$. We know that
$D_{\epsilon}(S)\subset M$ isometrically embeds in $M^{\prime}$ as a small
neighborhood around $\phi(S)$. The image $\phi(S_{\epsilon})$ of
$S_{\epsilon}$, now denoted $S^{\prime}_{\epsilon}$, is acausal in
$M^{\prime}$ (by a causal version of the argument for the achronality of
$\phi(S)$) and edgeless. We now have partial Cauchy surfaces $S_{\epsilon}$
and $S^{\prime}_{\epsilon}$ in $M$ and $M^{\prime}$ respectively.

We have the inclusion $D^{-}(S^{\prime}_{\epsilon})\supseteq
M^{\prime}\backslash I^{+}(S^{\prime}_{\epsilon})$. Since $(M,g)$ is globally
hyperbolic and inextendible, we have $D^{-}(S_{\epsilon})\supseteq
J^{-}(S_{\epsilon})$. Since $J^{-}(S_{\epsilon})$ isometrically embeds into
$(M^{\prime},g^{\prime})$, if $D^{-}(S^{\prime}_{\epsilon})$ fails to cover
$M^{\prime}\backslash I^{+}(S^{\prime}_{\epsilon})$, then
$S^{\prime}_{\epsilon}$ must have a past Cauchy horizon
$H^{-}(S^{\prime}_{\epsilon})\neq\emptyset$ in $M^{\prime}$. In that case, by
the locally Cauchy property we can isometrically embed $S^{\prime}_{\epsilon}$
and $\tilde{D}^{-}(S^{\prime}_{\epsilon})$ back into $M$ using $\phi^{-1}$,
and by the global hyperbolicity and inextendibility of $M$, we have that
$D^{-}(\phi^{-1}(S^{\prime}_{\epsilon}))\supset\phi^{-1}(H^{-}(S^{\prime}_{\epsilon}))$,
which contradicts the past hole-freeness of $(M^{\prime},g^{\prime})$.

We have the inclusion $D^{+}(S^{\prime}_{\epsilon})\supseteq
M^{\prime}\backslash I^{-}(S^{\prime}_{\epsilon})$. This follows by the same
argument.

Since we now have $D(S_{\epsilon})=M$ and
$D(S^{\prime}_{\epsilon})=M^{\prime}$, the conclusion follows from locally
Cauchy. ∎

In view of the role of null lines, we note the rigidity theorem of Galloway-
Solis [5].

######  Theorem 3.10 (Galloway-Solis [5]).

Assume that the $4$-dimensional spacetime $(M^{4},g)$

  1. (i)

satisfies (3.1) with $T=0$101010Generalizations to Einstein-Maxwell are
possible. and $\Lambda>0$,

  2. (ii)

is asymptotically de-Sitter111111cf. [5] for definitions of asymptotically de-
Sitter and the associated hypersurfaces $\mathcal{J}^{+(-)}$ in that context.,

  3. (iii)

is globally hyperbolic,

  4. (iv)

there is a null line with endpoints on $\mathcal{J}^{+}$ and
$\mathcal{J}^{-}$.

Then $(M^{4},g)$ isometrically embeds as an open subset of the de-Sitter
spacetime containing a Cauchy surface.

Together, Proposition 3.6 and Theorem 3.10 imply the following.

######  Corollary 3.11.

Given two $4$-dimensional spacetimes $(M,g)$ and $(M^{\prime},g^{\prime})$
assume that

  1. (i)

$(M,g)$ and $(M^{\prime},g^{\prime})$ satisfy (3.1) with $T=0$ and
$\Lambda>0$,

  2. (ii)

$(M,g)$ is inextendible and has a compact Cauchy surface,

  3. (iii)

$(M,g)$ is asymptotically de-Sitter but not isometric to de-Sitter,

  4. (iv)

$(M^{\prime},g^{\prime})$ is inextendible, causal and hole-free.

Then $(M,g)$ and $(M^{\prime},g^{\prime})$ are isometric iff they are weakly
observationally indistinguishable.

Note that the assumptions of Corollary 3.11 just falls short of astrophysical
relevance on account of the assumption that $(M,g)$ be past asymptotically de-
Sitter, which is not supported by current data. A more desirable statement
would be welcome.121212[13] contains results precluding the existence of null
lines based on astrophysically interesting assumptions.

(B). We say that a spacetime $(M,g)$ is Cauchy friendly if it is weakly
locally Cauchy131313Weakly locally Cauchy replaces $\tilde{D}(S)$ with $D(S)$.
and there are no points $p\in M$ such that $J^{-}(p)\supseteq M$. We now show
the following, which guarantees that genuine predictions extend a little
beyond what is suggested in Proposition 2.6.

######  Proposition 3.12.

Given two Cauchy friendly spacetimes $(M,g)$ and $(M^{\prime},g^{\prime})$,
assume that

  1. (i)

there is a partial Cauchy surface $S\subset J^{-}(q)\subseteq D(S)$ for some
point $q\in M$,

  2. (ii)

there is an isometry $\phi:J^{-}(q)\to J^{-}(q^{\prime})$.

Then there is an isometric embedding $\psi:A\hookrightarrow M^{\prime}$ for
some $A\supsetneq J^{-}(q)$ such that

  * •

$\psi_{\mid J^{-}(q)}=\phi$,

  * •

$A$ and $\psi(A)$ contain points in the domain of verifiable prediction of $q$
and $q^{\prime}$.

###### Proof.

First we make some basic observations, and throughout we denote
$S^{\prime}\equiv\phi(S)$.

We have $J^{-}(q)\subsetneq D(S)$. Since $J^{-}(q)$ lies in a globally
hyperbolic set $D(S)$, $J^{-}(q)$ is closed. Moreover, since $S$ is acausal
and edgeless, $D(S)$ must be open. From this it follows that the inclusion of
$J^{-}(q)\subseteq D(S)$ is strict. Suppose otherwise that $J^{-}(q)=D(S)$. In
that case $D(S)$ is both open and closed in $M$, and since $M$ is connected,
that implies $D(S)=M$. But then $M=J^{-}(q)$, in contradiction with Cauchy
friendly. Thus $J^{-}(q)$ is a closed proper subset of the open set $D(S)$.

$S^{\prime}$ is a partial Cauchy hypersurface. Since $\phi$ is a
diffeomorphism onto its image, we know that $S^{\prime}$ is compact. Unlike
the arguments given in Proposition 3.6, $\phi(S)$ is acausal and edgeless by
the fact that $\phi$ is an isometry (as opposed to merely an embedding). Since
$S^{\prime}$ belongs to $J^{-}(q^{\prime})$, if $S^{\prime}$ is acausal in
$M^{\prime}$, then $S$ is also acausal in $M$, which contradicts (ii). Thus
$S^{\prime}$ is a partial Cauchy surface in $M^{\prime}$.

We have $J^{-}(x^{\prime})\cap J^{+}(S^{\prime})\subseteq D^{+}(S^{\prime})$
for any $x^{\prime}\in\phi(J^{-}(q))$. We seek to show that any past
inextendible causal curve in $J^{-}(x^{\prime})$ with with future endpoint
$x^{\prime}\in\phi(J^{-}(q))$ intersects $S^{\prime}$. By assumption, we have
an isometric embedding $\phi:J^{-}(q)\cap D^{+}(S)\hookrightarrow M^{\prime}$
where $S$ is a closed achronal set. Consider any point $x\in J^{-}(q)\cap
J^{+}(S)$. By the isometry $\phi$, we have $\phi(J^{-}(x))=J^{-}(x^{\prime})$.
Note first that by definition, there is a neighborhood of $0\in T_{x}M$ such
that the exponential map of the past non-spacelike vectors in that
neighborhood is contained in $D^{+}(S)$. By the isometry $\phi$, the same is
true for $x^{\prime}$ with respect to $\phi(D^{+}(S))$, in particular there is
a neighborhood $U_{x^{\prime}}$ of $x^{\prime}$ such that $U_{x^{\prime}}\cap
J^{-}(x^{\prime})\subset\phi(D^{+}(S))$. Seeking a contradiction, suppose
there is a past inextendible causal curve $\gamma^{\prime}$ with future
endpoint $x^{\prime}$ that does not intersect $S^{\prime}$. By the property
aforementioned, there is at least a segment of $\gamma^{\prime}$ contained in
$\phi(D^{+}(S))$. The curve defined by
$\gamma\equiv\phi^{-1}(\gamma^{\prime})$ is causal and ends at $x$, and is
thus entirely contained in $D^{+}(S)$. But then $\gamma$ does not intersect
$S$, which is a contradiction.

Similarly, we have $J^{-}(S)\subseteq D^{-}(S^{\prime})$. This follows from
(ii), the isometry $\phi$, and the argument just above.

We have $J^{-}(q^{\prime})\subsetneq D(S^{\prime})$. This proceeds as above,
which now leads to a contradiction with the Cauchy friendliness of
$M^{\prime}$.

We have two closed sets $J^{-}(q^{\prime})$ and $J^{-}(q)$ each strictly
contained in the open sets $D(S^{\prime})$ and $D(S)$. We now seek to show the
existence of $\psi$. Although there may be no isometric embedding of $D(S)$
into $M^{\prime}$, we need only show that it is possible to non-trivially
extend the pre-image of $\phi$ beyond $J^{-}(q)$, which will thus enter
$D(S)\backslash J^{-}(q)$.
Consider now the unique (up to isometry) maximal globally hyperbolic
development $X(S)$ of $S$, where $X(S)$ denotes one representative among all
isometric developments. By the isometry $\phi$, we know that $S$ and and
$S^{\prime}$ are isometric as initial data sets, and thus that $X(S)$ is the
unique (up to isometry) maximal globally hyperbolic development of both $S$
and $S^{\prime}$. By locally Cauchy, it follows that both $D(S)$ and
$D(S^{\prime})$ can be isometrically embedded into $X(S)$.141414Note here that
we could use a weaker version of locally Cauchy here that involves $D(S)$
rather than $\tilde{D}(S)$. Denote these isometries by
$\rho:D(S)\hookrightarrow X(S)$ and
$\rho^{\prime}:D(S^{\prime})\hookrightarrow X(S)$.
It is obvious that $\rho(D(S))\cap\rho^{\prime}(D(S^{\prime}))\neq\emptyset$
and since $\rho$ and $\rho^{\prime}$ are local diffeomorphisms, both
$\rho(D(S))$ and $\rho(D(S^{\prime}))$ are open in $X(S)$, and moreover both
$\rho(J^{-}(q))$ and $\rho^{\prime}(J^{-}(q^{\prime})$ are closed in $X(S)$.
We now define the following set in $X(S)$

$[\rho(D(S))-\rho(J^{-}(q))]\cap[\rho^{\prime}(D(S^{\prime}))-\rho(J^{-}(q^{\prime}))]\equiv
I$

The openness of $\rho(D(S)),\rho^{\prime}(D(S^{\prime}))$, and the fact that
we may choose $\rho,\rho^{\prime}$ such that
$\rho(J^{-}(q))=\rho^{\prime}(J^{-}(q^{\prime}))$ means that the strict
inclusions $D(S)\backslash J^{-}(q),D(S^{\prime})\backslash J^{-}(q^{\prime})$
and $\neq\emptyset$ extend to $\rho$ and $\rho^{\prime}$, i.e.,
$\rho(D(S))\backslash\rho(J^{-}(q))\neq\emptyset$ and
$\rho(D(S^{\prime}))\backslash\rho(J^{-}(q^{\prime}))\neq\emptyset$. It
follows that $I\neq\emptyset$.
We can now identify the set $A\equiv\rho^{-1}[\rho(J^{-}(q))\cup I]$ as having
the desired properties, i.e. there exists an isometric embedding of
$\psi:A\hookrightarrow M^{\prime}$ such that $\psi|_{J^{-}(q)}=\phi$. ∎

We can also consider what happens after lifting the compactness assumption on
$S$.

######  Proposition 3.13.

Let $S$ be a partial Cauchy hypersurface in $M$ and $D_{\epsilon}(S)$ an
$\epsilon$-development of $S$. Let $\phi:D_{\epsilon}(S)\hookrightarrow
M^{\prime}$ be an isometric embedding into a hole-free spacetime $M^{\prime}$.
Then either

  * •

$\phi(S)$ is causal in $M^{\prime}$,

  * •

or $\phi(S)$ is a partial Cauchy hypersurface in $M^{\prime}$.

In the latter case, if $M,M^{\prime}$ are locally Cauchy and $M^{\prime}$ is
inextendible, then there is an isometric embedding $\psi:D(S)\hookrightarrow
M^{\prime}$ with $\psi|_{D_{\epsilon}(S)}=\phi$.

Thus, after basic assumptions like hole-freeness, locally Cauchy and
inextendibility, the only obstruction concerns the acausality of $\phi(S)$. It
may be that $\phi(S)$ is always acausal if $M^{\prime}$ satisfies some
causality assumption, eg. causally simple, causally continuous151515cf. pg. 59
of [1], etc.
Note also that $\phi(S)$ need not be a partial Cauchy hypersurface if we
replace $D_{\epsilon}(S)$ by ‘an open globally hyperbolic subset of $D(S)$’
(delete a half-space at $t=0$ from $(\mathbb{R}^{1,n},\eta)$).

###### Proof.

First we show the second statement. If $\phi(S)$ is a partial Cauchy
hypersurface, we know that $D(S)$ and $D(\phi(S))$ are both open subsets of
$M$ and $M^{\prime}$ respectively, and since $M,M^{\prime}$ are locally
Cauchy, we know that $D(S)$ and $D(\phi(S))$ share a common (isometric) subset
extending beyond $D_{\epsilon}(S)$. Let $\mathcal{D}(S)$ denote the maximal
open globally hyperbolic subset of $D(S)$ for which there is an isometric
embedding $\psi^{\prime}:\mathcal{D}(S)\hookrightarrow M^{\prime}$.
Now suppose that $D(S)$ does not isometrically embed into $M^{\prime}$, i.e.
$\mathcal{D}(S)\subsetneq D(S)$. If $H(\psi^{\prime}(S))\neq\emptyset$ then by
locally Cauchy we can use $\psi^{\prime-1}$ to embed
$\tilde{D}(\psi^{\prime}(S))$ into $M$ and contradict the hole-freeness of
$M^{\prime}$. If $H(\psi^{\prime}(S))=\emptyset$, then
$M^{\prime}=D(\psi^{\prime}(S))$. But in that case, by locally Cauchy,
$M^{\prime}$ isometrically embeds into $M$ as a proper subset of $M$,
contradicting the inextendibility of $M^{\prime}$.
Now we show the first part: if $\phi(S)$ is acausal, then it is a partial
Cauchy hypersurface in $M^{\prime}$. Supposing that
$\text{edge}(\phi(S))\neq\emptyset$, we will show that
$\text{edge}({\phi(S)})\cap\phi(S)=\emptyset$, and we then show that this
implies $(M^{\prime},g^{\prime})$ is holed.
Let $q^{\prime}$ be a point in $\text{edge}(\phi(S))\cap\phi(S)$. In that case
denote $q=\phi^{-1}(q^{\prime})\in S$ and take a future directed timelike
curve $\sigma$ from $q$ to $q_{\epsilon}\in I^{+}(S)\cap D_{\epsilon}(S)$, and
set $\sigma$ to be past inextendible in $M$. Then there is a timelike curve
$\sigma^{\prime}=\phi(\sigma)\subset M^{\prime}$ passing through $q^{\prime}$.
Take $U_{i}(q^{\prime})$ to be a system of increasingly small neighborhoods
$U_{i}(q^{\prime})\supsetneq U_{i+1}(q^{\prime})$, each containing points in
$I^{+}(q^{\prime})$ and $I^{-}(q^{\prime})$, such that
$\\{U_{i}(q^{\prime})\\}$ has accumulation point $q^{\prime}$. Define a
collection of curves $\\{\gamma_{i}^{\prime}\\}$ by taking $\sigma^{\prime}$,
removing from $\sigma^{\prime}$ the portion $\sigma^{\prime}\cap
U_{i}(q^{\prime})$, and replacing that portion with timelike segments with
endpoints in $I^{+}(q^{\prime})$ and $I^{-}(q^{\prime})$ which miss $\phi(S)$.
Although this might produce only piecewise smooth timelike curves, the curves
$\\{\gamma_{i}^{\prime}\\}$ can be approximated by $C^{1}$ causal curves
(still missing $\phi(S)$), which we relabel as $\\{\gamma_{i}^{\prime}\\}$.
Now consider $\\{\phi^{-1}(\gamma_{i}^{\prime})\\}$. This defines a collection
of $C^{1}$ causal curves in $D_{\epsilon}(S)$ that approach $\sigma$ but which
do not intersect $S$.
We now recall a well known fact. In any spacetime $L$, there exists a
sufficiently small neighborhood $N(x)\subset L$ around some point $x$ such
that $J^{+}(y)\cap J^{-}(z)$ is compact for all $y$, $z$ $\in N(x)$. Since a
sufficiently small $N(x)$ is causal, every point in a spacetime lives in a
small globally hyperbolic neighborhood.
Consider such a globally hyperbolic neighborhood $N(q)\subset M$ centered at
$q$. Without loss of generality, we can take $N(q)$ to have Cauchy surface
$S\cap N(q)$. For some sufficiently large $n\in\mathbb{N}$, the causal curves
$\\{\phi^{-1}(\gamma_{i\geq n}^{\prime})\\}$ lie in $N(q)$ and are
inextendible therein. But since these do not intersect $S$, we contradict the
global hyperbolicity of $N(q)$.
It now follows that $\text{edge}(\phi(S))$, if not empty, lies outside of
$\phi(S)$. By standard results in causal theory, $H(\phi(S))$ is ruled by null
geodesics intersecting $\text{edge}(\phi(S))$. By the definition of
$D_{\epsilon}(S)$, we can use $\phi$ to pull back
$H(\phi(S))\cap\phi(D_{\epsilon}(S))$ into $D_{\epsilon}(S)\cap M$. By the
definition of $D_{\epsilon}(S)$, it is then clear that
$\phi^{-1}\left[H(\phi(S))\cap\phi(D_{\epsilon}(S)\right]\cap
D(S)\neq\emptyset$, and thus $(M^{\prime},g^{\prime})$ is holed. ∎

## References

  * [1] Beem, J., Ehrlich, P., Easley, K., Global Lorentzian geometry, second ed., Monographs and Textbooks in Pure and Applied Mathematics, vol. 202, Marcel Dekker Inc., New York, 1996.
  * [2] Bernal, A. N., Sánchez, M., On smooth Cauchy hypersurfaces and Geroch’s splitting theorem, 2003, Commun.Math.Phys. 243:461-470.
  * [3] Choquet-Bruhat, Y., Geroch, R., Global aspects of the Cauchy problem in general relativity, 1969, Commun.Math.Phys. 14:329–335.
  * [4] Earman, J., Glymour, C., Stachel, J. (eds.), Foundations of Space-Time Theories. Minnesota Studies in the Philosophy of Science, vol. 8., University of Minnesota Press 1977.
  * [5] Galloway, G. J. G., Solis, D., Uniqueness of de Sitter space, 2007, Class.Quant.Grav. 24:3125-3138.
  * [6] Gao, S., Wald, R. M., Theorems on gravitational time delay and related issues, 2000, Class. Quant.Grav. 17:4999-5008.
  * [7] Geroch, R., Prediction in General Relativity, in Earman, J., Glymour, C., Stachel, J. (eds.) Foundations of Space-Time Theories, Minnesota Studies in the Philosophy of Science, vol. 8., University of Minnesota Press 1977.
  * [8] Malament, D., Observationally Indistinguishable Space-Times, in Foundations of Space-Time Theories, Minnesota Studies in the Philosophy of Science, vol. 8., University of Minnesota Press 1977.
  * [9] Manchak, JB., Can We Know the Global Structure of Spacetime ?, 2009, Studies in History and Philosophy of Modern Physics 40:53–56.
  * [10] Manchak, JB., Is prediction possible in general relativity ?, 2008, Foundations of Physics 38(4):317-321.
  * [11] Minguzzi, E., Limit curve theorems in Lorentzian geometry, 2008, J.Math.Phys. 49:092501.
  * [12] Minguzzi, E., Causally simple inextendible spacetimes are hole free, 2012, J.Math.Phys. 53:062501.
  * [13] Minguzzi, E., Chronological spacetimes without lightlike lines are stably causal , 2009, Commun. Math. Phys. 288:801-819.
  * [14] Minguzzi, E., Sanchez, M., The Causal Hierarchy of Spacetimes, 2008, arXiv: 0609119.
  * [15] Minguzzi, E., Lorentzian causality theory, 2019, Living Rev Relativ 22, 3.
  * [16] O’Neill, B., Semi-Riemannian Geometry, San Diego: Academic Press 1983.