text:
\section{Introduction}

Comparative statics in economics concerns how predictions of behavior---be it individual choices,   collective or social choices, or equilibria of games---change as economic conditions indexed by some parameters change.  In many economic problems, predictions are non-unique, so they are represented by a set $S(t)\subset X$ indexed by a  parameter $t\in T$, for some set $X$ of possible predictions.  The key question  is then: \emph{what would it take for set $S(t)$ to ``increase'' as $t\in T$ increases.}  Although there are  typically  well-defined orders  on $X$ and on $T$, there may be no  clear sense of how one set $S'$ ``dominates'' another $S$, given the primitive order $\ge$ defined on $X$.\footnote{While monotone selection---i.e., $S'$ declared to dominate $S$ if $x'\ge x$ for every $x\in S,x'\in S'$---would be most natural and easy to interpret, monotone selection is rather difficult to achieve for individual choices and virtually impossible beyond individual choices such as for equilibria of games.}

  The theory of monotone comparative statics pioneered by \cite{topkis:79, topkis:98} and \cite{milgrom/shannon:94} focuses on the so-called ``strong set order,'' denoted $\ge_{ss}$.  Namely, $S'\ge_{ss} S$ if,  \emph{for any $x\in S$ and $x'\in S'$,  $x \vee x'\in S'$ and $x \wedge x'\in S$, where $x \vee x':=\inf\{x''\in X: x''\ge x, x''\ge x'\}$ and  $x \wedge x':=\sup\{x''\in X: x''\le x, x''\le x'\}$}, and $\ge$ is a partial order on $X$. This notion of induced set order implies an intuitive property, captured by a weaker notion called  ``weak set order'' and denoted by $\ge_{ws}$.  Namely, $S'\ge_{ws} S$ if,  \emph{for each $x\in S$, one can find $x'\in S'$ such that $x'\ge x$, and likewise, for each $x' \in S'$, one can find $x\in S$ such that $x\le x'$.}  Strong set order is stronger than weak set order, although the economic meaning of the difference may not be easy to interpret or motivate.  For ease of discussion, we refer to  \textit{monotone comparative statics in strong set order}  as \textbf{strong monotone comparative statics} (or \textbf{sMCS} in short), whereas we refer to the one in weak set order---the focus of this paper--- as \textbf{weak monotone comparative statics} (or \textbf{wMCS} in short).

As shown by  \cite{topkis:79, topkis:98}, \cite{milgrom/shannon:94}, \cite{quah/strulovici:09} and others, the strong set order proves to be an appropriate notion in the context of individual choices. These authors identify forms of complementarities in  a decision maker's  preference  between her action and the parameters of her decision problem that are  sufficient for her optimal action to satisfy sMCS.  Not only are these conditions intuitive and easy to check, but they are also necessary if one insists that  sMCS  holds for {\it every} subdomain in a suitably-chosen class.\footnote{See \cite{milgrom/shannon:94} for the detail and our discussion in \Cref{sec:ind-choice}.  See \cite{quah/strulovici:09} for a characterization with a weaker condition known as interval dominance for the case in which $X$ is a \emph{chain}, i.e., a totally ordered set.}

Beyond individual choices, however, strong set order proves less useful.  Consider Nash equilibria of a so-called {\it supermodular game}, which exhibits complementarities between a player's strategies and those of her opponents as well as a parameter, say $t$.  \cite{topkis:79, topkis:98}, \cite{vives:90}, \cite{milgrom/roberts:90}, and \cite{milgrom/shannon:94} show that in such a game each player's best-response correspondence  varies monotonically in strong set order with those variables.  Yet, this does not lead to the same sort of monotonic shift for Nash equilibria.   More specifically, appealing to \cite{tarski:55}'s fixed-point theorem, one could, under suitable conditions, guarantee that the largest and smallest Nash equilibria exist and vary monotonically with $t$ (see \cite{milgrom/roberts:94}, for instance).  This result \textit{does} imply monotone comparative statics in weak set order but not in strong set order. \Cref{Fig-Intro} illustrates how a set of Nash equilibria in  a supermodular game may shift when a change in the environment causes a player 1's best response curve to shift out from $B_1$ to $B_1'$. The new set of equilibria dominate the old one in   weak set order but not in strong set order: for instance, $x\vee x'=x$ is not an equilibrium after the change.\footnote{Short of assuming ``uniqueness,'' no obvious way of strengthening the notion of complementarities either across players' strategies or between their strategies and parameters restores monotone comparative statics of equilibria in strong set order.}

\begin{figure}[htb]
	\begin{center}
\begin{tikzpicture} [scale=0.7]
\footnotesize
\draw [->]  (0,0) -- (0,6.75) node [above] {$x_2$};
\draw [->]  (0,0)  -- (8,0) node [right] {$x_1$};
\draw   (0.5,0)
to [out=70,in=200] (3.5,3) to [out=20,in=250] (6.5,6) node [above,xshift=-2pt, yshift=1.6pt] {$B_1$}  ;
\draw  [dashed] (1.2,0)
to [out=70,in=200] (4.6,3) to [out=20,in=250] (7.2,6) node [above, xshift=3pt] {$B_1'$}  ;

\draw [->]     (6.67,6.49) -- (7.05,6.49);
\draw (0,1)
to [out=10,in=240] (3,2.1) to [out=60,in=200] (7.2,5.3) node [right] {$B_2$}  ;

\draw [fill]  (5.9,4.75) circle (0.4ex) node [above left] {$x$};

\draw [fill]  (1.88,1.24) circle (0.4ex) node [below right] {$x'$};

\end{tikzpicture}
\caption[Note]{Failure of sMCS.} \label{Fig-Intro}
\end{center}
\begin{spacing}{1}
\end{spacing}
\end{figure}

Consider next a social choice problem.  One may be interested in the  comparative statics of   Pareto optimal choices by a collection of agents, although, to the best of our knowledge and to our surprise, this   question has never been asked, let alone investigated. One can imagine that when agents' preferences change toward favoring higher actions, their Pareto optimal choices would also likely shift toward higher actions.  But, as we show, the monotonic shift of Pareto optima may hold in weak set order but not in strong set order.

Finally, consider two-sided matching problems where agents on two sides---e.g., men and women, students and schools, and workers and firms---seek to match across the sides  \textit{stably}, i.e. in ways avoiding coalitional deviations, or  ``blocks.''  When the participants have  \textit{substitutable}  preferences, the set of stable matchings can be characterized as a set of fixed points of a certain monotonic operator---which corresponds to Gale and Shapley's deferred acceptance algorithm in a simple setup---while the stable matchings exhibit monotone comparative statics properties when the market conditions change in terms of agents' preferences and/or their entry or exit. Here again, for the reason analogous to the MCS of Nash equilibria, monotone comparative statics holds in weak set order but not in strong set order.

These observations suggest that, for many problems of interest, monotone comparative statics is feasible only in weak set order.  Given this, the current paper asks:  \emph{What would it take to guarantee wMCS?  Namely, what are the minimal properties and structure of the problem that one needs, if the goal is just to establish $S(t')\ge_{ws} S(t)$ whenever $t'\ge t$ and nothing more.}   We show that the conditions required for monotone comparative statics are weaker than existing conditions, sometimes considerably.  Naturally, the notion of complementarities
is  weaker.  More surprisingly,  the lattice structure of domain and the images of relevant operators, taken virtually as given by the existing literature, proves not to be essential and thus can be dispensed with  for results such as existence of equilibria and their monotone comparative statics.

After introducing basic  concepts in \Cref{sec:prel}, we proceed as follows. In  \Cref{sec:ind-choice}, we consider an individual choice problem in which a decision maker chooses an action to  maximize an  objective function over a feasible set.  We  provide sufficient conditions for the optimal actions to exhibit wMCS properties. Specifically, we identify binary relations on a pair of objective functions,  called \textit{weak dominance} and \textit{weak interval dominance}, such that optimal choices are higher in weak set order when the decision maker faces an objective function that dominates another in these senses.
These binary relations are weaker than those required for sMCS by the existing authors (see for example \cite{milgrom/shannon:94} and \cite{quah/strulovici:09}), and they are also necessary if one insists that the individual choices  exhibit wMCS properties for all subdomains of certain richness.

In \Cref{sec:pareto}, we consider Pareto optimal choices for a (finite) set of agents.  Pareto optimal choices are interesting in and of itself, but they can also model  behavior of  an individual whose preference is not complete (\citet{eliaz-ok06} for instance); such an individual may be seen as balancing multiple, possibly conflicting, complete preferences, each represented by a well-defined utility function.  We study conditions on changes of these latter ``component'' utility functions that give rise to wMCS of the associated Pareto optimal choices.  When $X$ is totally ordered, the desired wMCS result is simply ensured by the standard single-crossing property.
When $X$ is a general lattice, by contrast, wMCS requires conditions both on the curvature of individual utility functions and cardinal  complementarity properties.

Next,  \Cref{sec:fixed point} studies fixed points of correspondences. A fixed-point theorem by  \cite{tarski:55} and its extension by \cite{zhou:94} are useful for economic analysis of equilibria. Exploiting the power of a monotonic correspondence, these theorems deliver existence of fixed points without requiring the associated operator to be continuous or its domain to be convex. No less important, these theorems can be readily extended to show that fixed points increase in the weak set order sense when the correspondence shifts up in the strong set order sense.  However, these theorems require strong conditions both on   the correspondence and its domain, which  can severely limit their applicability.
For instance, they require rather rigid lattice properties for the domain and the images of the correspondence, and also require the correspondence to be  \textit{strong set monotonic}.  We develop a new fixed point theorem that relaxes the lattice requirements and requires the correspondence to be monotonic only in the weak set order sense. Under mild topological requirements, fixed points exist. We then establish  weak monotone comparative statics of fixed points   based only on a \textit{weak set}, rather than strong set, monotonic shift of the correspondence.  The fixed points need not form a complete lattice, but minimal and maximal fixed points exist, and they in turn exhibit the  wMCS property.  Further, we show a fixed point can be found via an iterative algorithm, albeit with some subtleties.

  In \Cref{sec:game}, we apply our fixed point theorem to establish existence and wMCS of Nash equilibria in a class of noncooperative games. Naturally, our results apply to a broader class of games with strategic complementarities than have been identified before (see \cite{vives:90}, \cite{milgrom/roberts:90}, and \cite{milgrom/shannon:94}). An advantage of the present approach is that our  class of games exhibits virtually the same set of useful properties as those identified previously while imposing significantly weaker assumptions.  For example, the MCS method can be applied to generalized Bertrand games that include pure Bertrand games previously outside the scope of MCS analysis.  Further,  we establish  that the same powerful results extend to games with strategic complementarities played by agents with incomplete preferences or  coalitions of agents choosing Pareto optimal responses to their opponents.  As an example, we show such preferences may arise in a beauty contest game when players are multidivisional firms.

In \Cref{sec:matching}, we study stable matching.   Tarski's fixed-point theorem has been used to prove existence of stable matchings under substitutable preferences (see \cite{adachi:00}, \cite{fleiner:03}, and \cite{hatfield/milgrom:05}).  The weak assumptions in our fixed-point theorem of \Cref{sec:fixed point} allow us to accommodate agents with very general forms of substitutability,  as well as  indifferences or even incomplete preferences.  Indifferences are natural when agents' preferences arise from coarse priorities; a case in point is  public schools that often place many students in the same priority class.  Incomplete preferences may arise naturally in a multidivisional firm in which multiple divisions may compete for common resources for hiring workers, or in a medical matching with regional caps, where hospitals in the same region may compete for quotas subject to a common cap.  We prove existence of a stable matching and its wMCS properties allowing for such general preferences.  A key step toward this end is a characterization of stable  matching via a fixed point of a t\^{a}tonnement-like operator, and this requires a version of revealed preferences condition.  The standard version, known as the Weak Axiom of Revealed Preference (WARP), however, may not hold for incomplete preferences that sometimes arise in matching situations.  To address this challenge, our characterization only requires a notion of revealed preference that is compatible with incomplete preferences.   This characterization, together with the associated wMCS properties, advances the frontier of matching theory.

All the proofs that are omitted from the main text or  Appendix are provided in  Supplementary Appendix.

 \paragraph{Related Literature.}   The current paper relates to  a large literature of monotone comparative statics. Papers that are related to a specific topic will be discussed in the  relevant section. Here, we make a brief remark on the literature that develops a general methodology on monotone comparative statics.

 \cite{topkis:79, topkis:98}, \cite{vives:90}, \cite{milgrom/roberts:90}, \citet{milgrom/shannon:94}, and \citet{quah/strulovici:09} are among the important contributions that have developed and refined the workhorse methods for comparative statics that are now widely used in economic analysis.  They use the strong set order for monotone comparative statics, which we weaken in the current paper.  Given the general relevance of these papers, they will be discussed in details whenever relevant.

 There are several papers that consider weaker notions of monotone comparative statics.  Similar to us, \citet{acemoglu2015robust} adopt the weak set order in their comparative statics of equilibrium dynamic systems.  In particular, the fixed point theorem they develop using this approach is related to our fixed point theorem, and will be discussed in detail in \Cref{sec:fixed point}. \citet{shannon1995weak} considers a weaker notion of strong set order, which regards a set $S'$ as ``bigger'' than a set $S$ if either $x\vee x'\in S'$ or $x\wedge x'\in S$ (but not necessarily both) for $x\in S$ and $x'\in S'$.\footnote{See also \citet{licalzi1992subextremal} for related results.}  \citet{quah2007comparative}
 introduces a set order that is weaker than strong set order while being stronger than weak set order. Using this order, he provides restrictions on the objective function for monotone comparative statics of the individual's optimal choices when the constraint changes.  While those papers share a broad motivation with ours, none of their results imply ours.

\section{Preliminaries} \label{sec:prel}

This section introduces a set of   notions and terminologies that will be used  for  our comparative statics analysis.

\paragraph{The structural properties of domain.}  Throughout, our domain of choices $X$ is assumed to be  a  \emph{partially ordered set} with regard to some \emph{primitive partial order} $\ge$, namely a binary relation that is \emph{reflexive, transitive} and \emph{anti-symmetric} on $X$.

Some, but not all, results invoke additional order properties.  We say $X$ is a \textit{lattice} if for any $x,x'\in X$, $x\vee x'\in X$ and $x\wedge x' \in X$, or equivalently if $X\ge_{ss} X$.  $X$ is a  \emph{complete lattice} if,  for any $S\subset X$,  $\sup_X S \in X$ and   $\inf_X S\in X$, where $\sup_X S:=\inf\{z\in X : z\ge x, \forall x\in S\}$ and $\inf_X S:=\sup\{z\in X :  z\le x, \forall x\in S\}$.   A subset $S\subset X$ is a \emph{sublattice} of $X$, if, for any $x,x'\in S$, $x\vee_X x'\in S$ and $x\wedge_X x'\in S$,    where  $x\vee_X x':=\inf \{x''\in X:  x'' \ge x \mbox{ and } x''\ge x'\}$ and  $x\wedge_X x':=\sup \{x''\in X: x'' \le x \mbox{ and } x''\le x'\}$.      A subset $S\subset X$ is a \emph{complete sublattice} of $X$ if $\sup_X S'\in S$ and $\inf_X S'\in S$ for all $S'\subseteq S$.\footnote{Some other terminologies are used for the same notion: \cite{topkis:98} uses subcomplete sublattice and \cite{zhou:94} uses closed sublattice.  In particular, the ``closedness'' of \cite{zhou:94} should not be confused with the topological ``closedness'' used in this paper.}   (We will henceforth use $\vee$ and $\wedge$ instead of $\vee_X$ and $\wedge_X$, unless the sup or the inf is being taken over a set other than $X$.)  Finally, a subset $S$ is a \emph{subinterval} of $X$ if  there exist $a\le b, a,b\in X$, such that $S=\{x\in X:   a\le x\le b\}$, denoted equivalently by $[a,b]$.

Some of our results pertaining to existence of maximizers or fixed points invoke topological properties such as compactness of $X$ and upper semicontinuity of an objective function defined on $X$. Whenever such properties are invoked, we invoke a metrizable \emph{natural topology}  under which upper contour sets $U_y:=\{x\in X: x\ge y\}$, $\forall y\in X$, and lower contour sets $L_y:=\{x\in X: x\le y\}$, $\forall y\in X$, are closed, where $\ge$ and $\le$ are our primitive partial order.

\paragraph{Set orders.}  Our partial order induces two set orders,  \emph{weak set order} $\ge_{ws}$ and \emph{strong set order} $\ge_{ss}$.   We say  $S'$ {\it upper weak set dominates} $S$, and write $S'\ge_{uws} S$, if, for each $x\in S$, there exists $x'\in S'$ such that $x'\ge x$; and $S'$ {\it lower weak set dominates} $S$, and write $S'\ge_{lws}S$, if for each $x'\in S'$, there exists $x\in S$ such that $x\le x'$.
And, $S'$ {\it weak set dominates} $S$ if $S'\ge_{ws} S$, i.e.,  if $S'\ge_{uws} S$ and $S'\ge_{lws} S$.
Next, we   say $S'\subset X$ {\it strong set dominates} $S\subset X$ if $S'\ge_{ss} S$.\footnote{One  could  imagine  an even weaker set order than weak set order.  Say  $S'$ \textbf{monotone-selection dominates} $S$ if one can find  $x'\in S'$  and $x\in S$  such that $x' \ge x $.
This order may prove too weak in many contexts, however.  For instance, $S'  =[0,2]$ would dominate $S = [1,3]$ in this sense, although $S'\not\ge_{ws} S$.  In fact, weak set order can be seen as requiring more discipline in  the ability to select from a set:  if $S' \ge_{ws} S$, then for any $\hat x \in S' \cup S$, one should be able to make a monotone selection that involves $\hat x$.}  (Recall that $\ge_{ws}$  and $\ge_{ss}$ were defined in the Introduction.)

As already observed in the Introduction,  strong set order implies weak set order.  The following result further clarifies  their relationship by decomposing  strong set order  into   weak set order and a couple of ``extra properties'' when the choice domain is a lattice (and the compared sets are sublattices):\footnote{One can easily construct examples showing that each property is  indispensable for this characterization.}
\begin{theorem} \label{thm:ss-char}
 Consider  a lattice $X$ and its subsets $S$ and $S'$. Then,  $S' \ge_{ss} S $  if (i)  $S' \ge_{ws}  S$; (ii) $S \cup S'$ is a sublattice; (iii) (sandwich property)  for  any $x \in S$  and $y, z \in S'$  (resp., any $x \in S'$ and $y, z \in S$),  $x \in   [y,z]$ implies $x \in S'$ (resp., $x \in S$).
  Conversely, if $S$ and $S'$ are nonempty sublattices, then   $S' \ge_{ss} S$ implies  the properties   (i) to (iii).
\end{theorem}
\begin{proof}
To prove the first statement, let us consider any $x \in S$ and $x' \in S'$.  By (ii),  $x \vee  x' \in  S \cup S'$. To show $\check x: = x \vee x' \in S'$, suppose not for contradiction. Then, $\check x\in S$ by (ii). By (i), there exists $z \in S'$ such that $z  \ge \check x$. So we have $x' \le \check x \le z$ while $x' ,z \in S'$ and $\check x \in S$. Thus, by (iii), $\check x = x \vee x' \in S'$, a contradiction. To show that $x \wedge x' \in S$ is analogous and hence omitted.

Suppose now that $S' \ge_{ss} S$ where $S$ and $S'$ are nonempty sublattices.
Clearly, (i) holds. To see that (ii)  holds, consider any $x, x' \in S \cup S'$. If either $x, x' \in S$ or $x,x' \in S'$, then  clearly $x \vee x'$ and $x\wedge x'$ belong to $S \cup S'$ due to  the fact that $S$ and $S'$ are sublattices. If $x \in S$ and $x' \in S'$, then $S' \ge_{ss} S$ implies that both $x \vee x' $ and $x \wedge x'$ belong to $S \cup S'$.  To verify (iii), observe that for any $x \in S$ and $y,z \in S'$ with  $  x  \in [y, z]$,  we have  $x =   x \vee y  $ and thus $x \in S'$ since $S' \ge_{ss} S$. Also,  for any $x \in S'$ and $y,z \in S$ with  $ x \in [y,z]$,  we have  $x =   x \wedge z $ and thus $x \in S$.
\end{proof}

 This characterization reveals exactly what one would ``lose'' or ``miss'' by using weak set order instead of strong set order. Those are properties (ii) and (iii).  Observe that these properties apply symmetrically to  compared sets $S$ and $S'$, thus conveying no information about the sense in which  $S'$ dominates $S$. This provides some formal argument that no meaningful loss in substance occurs when one weakens the set order from strong to weak set order.\footnote{Suppose for instance that $X$ is totally ordered.  Then, the difference between two set orders boils down to the sandwich property (iii).  The failure of this property  prevents us from declaring that a set  $\{1,3\}$ dominates a set  $\{0,2\}$ in strong set order, even though we can rank them based on  weak set order.}

\paragraph{Properties of alternative set orders.}
The two set orders $\ge_{ws}$ and $\ge_{ss}$ also exhibit different  order-theoretic or algebraic properties.  First, the strong set order  satisfies the antisymmetry and transitivity (unless  the empty set is involved in the comparison) while it  violates the  reflexivity.\footnote{With the empty set involved, we have $S \ge_{ss} \emptyset \ge_{ss} S $ for any $S \subset X$,  violating the antisymmetry and transitivity.} The weak set order  violates the antisymmetry but satisfies the reflexivity and transitivity, thus forming a \emph{preorder}.   Next, the weak set order is closed under the union operation while  the strong set order is  closed under the  intersection operation:\footnote{However,  the weak set order is not closed under the intersection operation   while  the strong set order is  not closed under the union operation,  as can be easily checked. The proofs of this lemma and \Cref{lem-ws-monotone-operator} are elementary and hence omitted.}
\begin{lemma} \label{lem-ws-set-operator}
For any subsets $S' \ge_{ws} S$ and $T' \ge_{ws} T$, we have $(S' \cup T') \ge_{ws} (S\cup T)$. Also, for any subsets $S' \ge_{ss} S$ and $T' \ge_{ss} T$, we have $(S' \cap T') \ge_{ss} (S\cap T)$.
\end{lemma}

The two set orders can be  used  to  define the monotonicity of  correspondence, $F: X \rightrightarrows Y$, where both $X$ and $Y$ are partially ordered.   We say that $F$ is \emph{upper weak set monotonic} if $F(x') \ge_{uws} F(x)$ for any $x' \ge x$,  \emph{lower weak set monotonic} if $F(x')\ge_{lws} F(x)$  for any $x' \ge x$, and \emph{weak set monotonic} if $F(x')\ge_{ws} F(x)$  for any $x' \ge x$.  Finally, we say that $F$ is \emph{strong set monotonic} if $F(x') \ge_{ss} F (x)$ for any $x' \ge x$. We can see that  the weak set order is preserved under a weak set monotonic  correspondence but  the strong set order is not:\footnote{Strong set order may  not be preserved  even by a strongly monotonic correspondence. Consider $F: X\rightrightarrows Y$, where $X:=\{(1,1),(2,1),(1,2),(2,2)\}, Y:=\{3,4, 5, 6\}$, and $F (x_1,x_2) = \{2x_1+x_2\}$, for  each $(x_1,x_2)\in X$.  Since $F$ is single-valued, i.e., a function, and is monotonic, it is trivially strong set monotonic.  Take  $S =\{(1,1),(2,1)\}$ and
	$T = \{(1,2),(2,2)\}$.  Then, $S\le_{ss} T$. But $F(S) = \{3,5\}\not\le_{ss}\{4,6\}=F(T)$ (due to the failure of the sandwich property).}

\begin{lemma} \label{lem-ws-monotone-operator}
Given a correspondence $F: X \rightrightarrows Y$ and  any  subsets $S' \ge_{ws} S$ of $X$,   $F (S') = \bigcup_{x \in S'} F (x) $ weak set dominates $  F (S) = \bigcup_{x \in S} F (x)$ if $F$ is  weak set monotonic.
\end{lemma}
This property will  later prove useful for drawing payoff implications of comparative statics.

\section{Individual Choices}\label{sec:ind-choice}

In this section, we study wMCS of individual choices.
Consider an individual who chooses  an action $x$ from some set $S\subset X$ by maximizing an objective function $f: X\to \mathbb{R}$.  We are concerned with how her choices
$$M_{S} (f) := \arg\max_{x \in  S} f (x) $$
change when her objective function $f$ shifts from one function $u$ to another $v$.  In particular, we explore  sufficient conditions for her choices to exhibit wMCS---or more precisely, $M_{S} (v) \ge_{ws} M_{S} (u)$---for every subdomain $S$ within a class $\mathcal{X} \subset 2^X$.

The sufficient conditions we look for should ideally be  ``tight'' or ``necessary'' in some sense, and this desideratum  is  achieved by the requirement that the conditions be also necessary for  wMCS for \textit{every} subdomain $S\subset X$ \emph{within a class} $\mathcal{X} \subset 2^X$.
How rich we require that class $\mathcal{X}$ to be involves a tradeoff.   If  $\mathcal{X}$ is very coarse, then the sufficient conditions become weak, but they could become  too dependent on   the ``details'' of the specific subdomain to be of practical value.  If $\mathcal{X}$ is very rich, the conditions become detail-free and robust but at the expense of being strong.  In this regard, we follow two prominent works by  \cite{milgrom/shannon:94} and \cite{quah/strulovici:09}.

  \cite{milgrom/shannon:94} find conditions that guarantee sMCS on the class $\mathcal{X}_{sublat}$ of   all sublattices of $X$, whereas \cite{quah/strulovici:09} find conditions that guarantee sMCS on the class  $\mathcal{X}_{subint}$ of all subintervals of $X$.\footnote{\label{fnote:diff-ms-qs} Note that \cite{quah/strulovici:09} considered the case in which  $X$ is totally ordered. There is also a subtle difference between the two studies:  \cite{quah/strulovici:09} obtain  their characterization by  fixing the constraint set  $S$ in two maximization problems under comparison, while \cite{milgrom/shannon:94} do so by varying  $S$ (in the strong set order sense) together with the objective function. Our study takes the former approach.}  Obviously, the class   of sublattices of $X$ is richer than that of subintervals of $X$ (note a subinterval is a sublattice).\footnote{If $x,x'\in [a,b]$, then $x\wedge  x', x\vee  x'\in [a,b]$.}  So, the condition for monotone comparative statics with respect to the former class will be more robust, albeit stronger, than that with respect to the latter class.

 \subsection{Characterization with Respect to Sublattices of $X$.}
\cite{milgrom/shannon:94} provide canonical  conditions that guarantee sMCS of individual choice on the class  $\mathcal{X}_{sublat}$. Formally, we say $v$ {\bf MS dominates} $u$, and write $v\succeq_{MS} u$, if  (i) $v$  {\it single-crossing dominates} $u$:  for any $x''>x'$,
$u(x'')-u(x')\ge\mathrel{(>)}0 \Rightarrow v(x'')-v(x')\ge\mathrel{(>)}0$; and  (ii) $f=u,v$ is {\it quasi-supermodular}:  for any $x',x''\in X$,
$f(x'')-f(x'\wedge x'') \ge\mathrel{(>)}0 \Rightarrow f(x'\vee x'')-f(x')\ge\mathrel{(>)}0.$
Then, their Theorem 4 proves that the maximizers of $v$ strong set dominate those of $u$ for every sublattice of $X$ if   $v\succeq_{MS} u$.\footnote{As mentioned earlier in  \Cref{fnote:diff-ms-qs}, MS dominance does not quite characterize the sMCS for all sublattices.  To be precise, Theorem 4 of \cite{milgrom/shannon:94} shows that  MS dominance is also necessary (in addition to being sufficient) for $M_{S'} (v)  \ge_{ss} M_{S} (u)$  with $S' \ge_{ss} S$, {\it where $u=v$ is allowed for}.}     Intuitively, (i) means that if it benefits a decision maker to raise the action under utility function $u$, then it does so under $v$ as well, and (ii) means that raising one component of action by a decision maker increases her incentive to raise another component of her action (in the ordinal sense).

These two conditions together imply that:  for any  $x', x'' \in X$,    \begin{align}  \label{sMCS-MS}
 u (x'') \ge\mathrel{(>)}  u (x' \wedge x'') \Rightarrow v (x' \vee x'') \ge\mathrel{(>)} v (x').
\end{align}
It is immediate that sMCS follow from  \eqref{sMCS-MS}: for any sublattice $S$,  if  $x''\in M_{S}(u)$ and  $x'\in M_{S}(v)$, then  $x'\vee x''\in M_{S}(v)$ and $x' \wedge x''\in M_{S}(u)$.

We weaken (\ref{sMCS-MS}) in the following way.  We say $v$ \textbf{weakly dominates} $u$, and write $v \succeq_{w} u$, if, for any $x', x''\in X$,
$x''\not \le x'$,
\begin{align} \label{cond:weak-dominance}
u(x'')\ge\mathrel{(>)}  \max\{u(x'\wedge x''), u(x')\}     \Rightarrow   \max\{v(x''), v(x'\vee x'')\}\ge\mathrel{(>)}  v(x').
\end{align}
This condition is weaker than MS dominance since the hypothesis of (\ref{cond:weak-dominance}) is stronger  and its conclusion is weaker than the corresponding parts of  (\ref{sMCS-MS}).  Therefore, (\ref{sMCS-MS}), and hence $v\succeq_{MS} u$,  implies that $v \succeq_{w} u$.   Note also that weak dominance need not yield sMCS of individual choices.  Suppose  $x''\in M_{S}(u)$ and  $x'\in M_{S}(v)$ for a sublattice $S$, so  the hypothesis of (\ref{cond:weak-dominance}) holds.  Yet, (\ref{cond:weak-dominance}) does not guarantee that $x'\vee x''\in M_{S}(v)$. For wMCS of individual choices on sublattices, however, weak dominance turns out to be just the right condition:
\begin{theorem} \label{thm:w} Suppose that $X$ is a lattice.  Function $v$  weakly dominates $u$ if and  only if, for every  $S \in \mathcal{X}_{sublat}$,
	\begin{equation} \label{0}  M_{S}(u) \le_{ws}M_{S}(v)
	\end{equation}
	whenever both sets are nonempty.
\end{theorem}

\begin{proof}   \noindent \textbf{The ``only if'' direction.}   Fix any sublattice $S\subset X$ and suppose that both $M_{S}(u) $ and $M_{S}(v)$ are nonempty. If $x'' \le x'$  for all $x''\in M_{S}(u) $  and $x'\in  M_{S}(v),$  then trivially $M_{S}(u) \le_{ws}M_{S}(v)$.  Hence, assume $z'' \not \le z'$ for some  $z''\in M_{S}(u) $  and $z'\in  M_{S}(v)$. Clearly, $u(z'')\ge \max\{u(z'\wedge z''), u(z')\}$.  Since $v\succeq_w u$, we then have $\max\{v(z''), v(z'\vee z'')\}\ge v(z')$. Then, the fact that $z' \in M_{S}(v)$ means that either  $z''\in M_{S}(v)$ or $z'\vee z''\in M_{S}(v)$.  Hence, $M_{S}(v)$ upper weak set dominates
	$M_{S}(u)$.  For the lower weak set monotonicity, we invoke the contrapositive of  (\ref{cond:weak-dominance}) involving strict inequalities.  Since  $v(z')\ge \max\{v(z''), v(z'\vee z'')\} $, we must have $\max\{u(z'\wedge z''), u(z')\}\ge u(z'')$, proving that  $M_{S}(v)$ lower weak set dominates
	$M_{S}(u)$.

\vspace{0.2cm}
	 	\noindent \textbf{The ``if'' direction.}  Consider $S=\{x',x'', x'\wedge x'', x'\vee x'' \}$, where   $x''\not\le x'$. Both $M_{S}(u)$ and $M_{S}(v)$ are nonempty because $S$ is a finite set.  Suppose first  $u(x'')\ge \max\{u(x'\wedge x''), u(x')\}$. 	Then,  $\{x'', x'\vee x''\}\cap M_{S}(u)\ne \emptyset$.  We must then have
	$\max\{v(x''), v(x'\vee x'')\}\ge  v(x')$, or else $M_{S}(v)$ does not upper weak set dominate $M_{S}(u)$.   To prove the strict inequality part of  (\ref{cond:weak-dominance}), we consider its contrapositive.  To this end, suppose $\max\{v(x''), v(x'\vee x'')\}\le v(x')$.  Then,    $\{x', x'\wedge x''\}\cap M_{S}(v)\ne \emptyset$.  We must then have $\max\{u(x'\wedge x''), u(x')\}\ge u(x'')$, or else $M_{S}(v)$ does not lower weak set dominate $M_{S}(u)$.  This implies that $u(x'')>  \max\{u(x'\wedge x''), u(x')\}     \Rightarrow   \max\{v(x''), v(x'\vee x'')\}> v(x')$.
\end{proof}

\subsection{Characterization with Respect to Subintervals of $X$.}  The domain of subintervals is coarser than that of  sublattices. Hence, the condition characterizing wMCS in the former domain must be weaker  than weak dominance.  To describe that condition, for any $x', x''\in  X$, we let
$$J(x',x''):=\{x\in X:   x'\wedge x'' \le x  \le  x'\vee x''\}$$
denote the smallest subinterval of $X$ containing them.  Further, we assume that $M_{S}(f)$ is nonempty for every subinterval $S$ of $X$, for $f=u,v$.\footnote{This is guaranteed if $X$ is compact and $f=u,v$ is upper semicontinuous, for instance.}

We say   $v$ {\bf weakly  interval dominates} $u$, and write $v\succeq_{wI} u$, if,  for any $x', x''\in X$ such that  $x''\not \le x'$, $u(x'')\ge u(x)$, and $v(x')\ge v(x), \forall x\in J(x',x'')$,
\begin{align} \label{def-wI}
u(x'')\ge\mathrel{(>)} \, \max_{x\in J(x' \wedge x'', x')} u (x) \Rightarrow  \max_{x\in J(x'', x'\vee x'')} v(x)   \ge\mathrel{(>)}\,  v(x').
\end{align}
Note that   weak interval dominance  is  implied by weak dominance: the hypothesis of (\ref{def-wI}) is stronger  and its conclusion is weaker than the corresponding parts of \eqref{cond:weak-dominance}.  The following result shows that weak interval dominance characterizes wMCS of individual choices on every subinterval.

\begin{theorem} \label{thm:wI}   Suppose that $X$ is a lattice.
	Function  $v$ weakly  interval dominates $u$ if and only if,   for every $S \in \mathcal{X}_{subint}$,
	\begin{equation} \label{*} M_{S}(u)\le_{ws} M_{S}(v).
	\end{equation}
\end{theorem}

\begin{proof}    		\noindent \textbf{The ``only if'' direction.} If $x'' \le x'$ for all $x''\in M_{S}(u) $  and $x'\in  M_{S}(v),$  then trivially $M_{S}(u) \le_{ws}M_{S}(v)$. Hence, assume  $z'' \not\le z'$ for some $z''\in M_{S}(u)$ and  $z'\in  M_{S}(v)$.   Then, since $v\succeq_{wI} u$ and $u(z'')\ge \, \max_{x\in J(z' \wedge z'', z')} u (x) $,  there exists $z'''\in J(z'',  z'\vee z'')$ such that $v(z''')\ge v(z')$.   That $S$ is an interval and $z',z'' \in S$ implies $J (z', z'') \subset S$, which in turn   implies  $z''' \in J(z'',  z'\vee z'')\subset J (z',z'') \subset S$. We must thus have $z'''\in   M_{S}(v)$, since $v(z''')\ge v(z')$ and $z' \in M_{S} (v)$.  Hence,  $M_{S}(v)$ upper weak set dominates $ M_{S}(u)$.

	For the lower weak set dominance, we consider the contrapositive relation involving strict inequalities.  Specifically, choose any $z''\in M_{S}(u)$ and  $z'\in  M_{S}(v)$, and suppose that $z'' \not \le z'$.   Then, since $v\succeq_{wI} u$ and $v (z') \ge \max_{x \in  J (z'', z' \vee z'')} v (x)$, there exists $z''' \in J( z'\wedge z'', z')$ such that $u(z''')\ge u(z'')$. For the same reason as above,  we have $z''' \in J(z' \wedge z'',  z')\subset J (z',z'') \subset S$.  We must then have $z''' \in M_{S}(u)$, since $u (z''') \ge u (z'') $ and $ z'' \in M_{S} (u)$,   proving that $M_{S}(v)$ lower weak set dominates $ M_{S}(u)$.

	\vspace{0.2cm}
 	\noindent \textbf{The ``if'' direction.}  Fix any  $x'',x'$ with $x''\not\le x'$ such that $u(x'')\ge u(x)$ and $v(x')\ge v(x), \forall x\in J(x',x'')$.  Obviously, $u(x'')\ge \, \max_{x\in J(x'\wedge x'', x')}u(x)$.  Suppose to the contrary that $v(x''')< v(x')$,  $\forall x'''\in J (x'',x'\vee x'')$. Then, $M_{J(x',x'')}(v)$ fails to  upper weak set dominate $ M_{J(x',x'')}(u)$, a contradiction.   Next we prove the strict inequality part of the condition, by considering its contrapositive.  Note that $v(x')\ge \max_{x\in J (x'',x'\vee x'')} v(x)$.  Suppose to the contrary that  $u(x''')<u(x'')$, $\forall x''' \in J (x'\wedge x'', x')$. Then,  $M_{J(x',x'')}(v)$ fails to lower weak set dominate $ M_{J(x',x'')}(u)$, a contradiction.  \end{proof}

 \Cref{thm:wI}  parallels  the characterization  result in  \cite{quah/strulovici:09}  for a  totally ordered $X$. For such $X$, their {\it interval dominance order}  characterizes sMCS for all subintervals of $X$.\footnote{Their online appendix considers a general lattice $X$ and provides a set of conditions that are \emph{sufficient}  (but not necessary) for sMCS for all subintervals of $X$.}  In fact, one can extend their interval dominance order to a general lattice $X$.  For such $X$, we say $v$ {\bf interval dominates} $u$, and write $v\succeq_I u$, if, for any $x', x''\in X$, $x''\not \le x'$, such that  $u(x'')\ge u(x)$ and $v(x')\ge v(x), \forall x\in J(x',x'')$,
\begin{align} \label{def-id}
u(x'')\ge\mathrel{(>)}\, u(x'\wedge x'') \, \Rightarrow \,  v(x'\vee x'')\ge\mathrel{(>)}\, v(x').
\end{align}
This condition reduces to Quah and Strulovici's interval dominance order when $X$ is totally ordered.  For the general lattice $X$,  \Cref{thm:I} in   \Cref{sec:appendix-ind-choice} of Supplementary Appendix proves that (\ref{def-id}) characterizes sMCS for every subinterval of $X$ in strong set order.\footnote{This characterization means that the current (generalized) interval dominance order is weaker than the sufficient condition provided in Theorem 1 of  \cite{quah/strulovici:07}: their total-order version of interval dominance and their \textit{I-quasisupermodularity}.  See   \Cref{sec:appendix-ind-choice} of Supplementary Appendix.}  This condition implies weak interval dominance, and hence yields wMCS.

\begin{corollary}  \label{prop:wI-weaker} If  $v\succeq_I u$, then $v\succeq_{wI} u$.
\end{corollary}

\begin{proof}  The statement  follows from the definitions of weak interval dominance and interval dominance.
\end{proof}

\begin{remark}\label{equivalence-remark}  We note that both weak dominance and MS-dominance reduce to single-crossing dominance when $X$ is a totally ordered set, as can be seen by inspection of both conditions. Similarly, weak interval dominance and interval dominance coincide when $X$ is totally ordered. We emphasize that these equivalences do not hold beyond the case of totally ordered $X$.
\end{remark}

\subsection{Properties of Individual Choices.}   One by-product of MS conditions is   that $M_{S}(f)$ forms a sublattice of $X$.  Specifically, if $f$ is quasi-supermodular and $S$ is a sublattice, then $M_{S}(f)$ is a sublattice.  The same is not implied by our wMCS conditions, however.  In fact, $M_{S}(f)$ need not even be a lattice.  The following example illustrates this point.

\begin{example} Let $X=[0,1]^2$, $u(x_1,x_2)= - (x_1+x_2-t)^2$, and
	$v(x_1,x_2)= - (x_1+x_2-t')^2$, for $0<t<t'<1$.
	\begin{figure}[htb]
		\begin{center}
			\begin{tikzpicture} [scale=5.5]

			\draw (0,0) node [below left] {\footnotesize $(0,0)$}  --  (1,0) -- (1,1) node [above right] {\footnotesize $(1,1)$} --   (0,1) --  cycle;
			\draw  [color=blue]  (0.2,0.7)  --  (0.7,0.2) node [pos=0.5, left] {\footnotesize $M_{S}(u)$} ;

			\draw [dotted]  (0,0.9)  -- (0.2,0.7);

			\draw [dotted]  (0.7,0.2)  -- (0.9,0);
			\draw [color=red]  (0.3,0.9)  -- (0.8,0.4) node [pos=0.5,right] {\footnotesize $M_{S} (v)$};
			\draw [dotted]  (0.2,1)  -- (0.3,0.9);
			\draw [dotted]  (0.8,0.4)  -- (1,0.2);

			\draw [fill]  (0.6,0.3) circle (0.05ex) node [right, yshift=1pt,xshift=1pt] {\footnotesize $x'$};

			\draw [fill]  (0.4,0.8) circle (0.05ex) node [left] {\footnotesize $x''$};

			\draw [fill]  (0.6,0.8) circle (0.05ex) node [above right,xshift=-5pt] {\footnotesize $x' \vee x''$};

			\draw [fill]  (0.4,0.3) circle (0.05ex) node [below left, xshift=5pt,yshift=1pt] {\footnotesize $x' \wedge x''$};

			\draw [line width=0.2pt, fill=gray, fill  opacity=0.1, text opacity=1]   (0.2,0.2) --(0.8,0.2)  -- (0.8,0.9) -- (0.2,0.9) --cycle;

			\draw [->, >=stealth]  (0.4,0.1)  node [below] {\footnotesize $S$} to [out=80,in=260] (0.475,0.2) ;
			\end{tikzpicture}
			\caption{Weakly interval dominating shift that is not weakly dominating} \label{fig:wI}
			\end{center}
		\end{figure}
Note that $v$ weakly interval dominates $u$.  Indeed, for any subinterval $S$, $M_{S}(u)$ is the projection of the hyperplane $\{x\in [0,1]^2: x_1+x_2=t\}$ on $S$, and likewise $M_{S}(v)$ is the projection of the hyperplane $\{x\in [0,1]^2: x_1+x_2=t'\}$ on $S$.  See the blue and red lines in \Cref{fig:wI}.  One can easily see that   $M_{S}(v)$ dominates $M_{S}(u)$ in weak set order but not in strong set order.  Note also that neither set forms a lattice, let alone a sublattice of $X$.    Finally, observe that $v$ does not weakly dominate $u$.  Consider a sublattice $Z=\{x', x'', x'\wedge x'', x'\vee x''\}$ consisting of the four dots in  \Cref{fig:wI}. Clearly, $M_{Z}(u)=\{x'\}$ and $M_{Z}(v)=\{x''\}$,  and they are not weak set ordered.
\end{example}

The following proposition establishes some useful properties of individual choices, which will be referred to in our analysis of games.

 \begin{proposition} \label{prop:ind-choice-property}  Assume $X$ is  a partially ordered metric space, and $f$ is upper semicontinuous.  Then, for any compact subset $S$ of $X$, $M_{S}(f)$ is nonempty and compact, and admits maximal and minimal points.\footnote{Minimal points of $S$ are a set $\{x\in S: x' \not< x, \forall x'\in S  \}$ and maximal points of $S$ are a set $\{x\in S: x' \not> x, \forall x'\in S  \}$.}
 \end{proposition}

\section{Pareto-Optimal Choices}\label{sec:pareto}

Consider a set of possible choices $X$ and a finite set $I$ of individuals with utility functions $u=(u_i)_{i\in I}$, where  $u_i: X \to \mathbb{R}$ is a utility function for individual $i$.
We say $y\in X$ {\it Pareto dominates} $x\in X$ given $u$ if $u_i(y)\ge u_i(x)$ for  all $i \in I$ and $u_j(y)> u_j(x)$  for at least one $j \in I$.
The set of {\it Pareto optimal choices} (or POC in short) given $u$ is the set
$\pazocal{P}(u):=\{x\in X: \text{ no }  y\in X \text{ Pareto dominates } x \text{ given } u\}.$
We wish to study conditions enabling wMCS of sets $\pazocal{P}(u)$
with respect to a change in utility functions $u$.

\begin{remark}\rm
	Our main interpretation of the set $I$ is  a collective of individuals. As mentioned in the introduction, however, one could also interpret $I$ as a single decision maker with incomplete preferences. According to that interpretation, the decision maker $I$ is associated with functions $u=(u_i)_{i\in I}$, and the decision maker's choice behavior is described by the Pareto optimal choices $\pazocal{P}(u)$.\footnote{In the context of choice problems under certainty, \cite{ok2002utility} provides sufficient conditions for incomplete preferences to be represented as Pareto optimal choices. Characterization results are  given by \cite{dubra2004expected} and \cite{ok2012incomplete} for problems with lotteries and  uncertainty. See also \citet{eliaz-ok06} who  characterize a class of incomplete preferences.}
\end{remark}

The existence of a Pareto optimal choice follows from standard assumptions.

\begin{proposition}  \label{prop:P-properties}  Assume $X$ is compact and $u_i$ is upper semicontinuous for every $i \in I$. Then, the set $\pazocal{P}(u)$ is nonempty.\footnote{The set of Pareto optimal choices may be empty if $X$ is not compact. For example, let $X=[0,1)$, $I=\{1\}$, and $u_1(x)=x$. Then there exists no Pareto optimal choice because for any $x \in X$, there exists $x' \in X$ with $x'>x$ and hence $u_1(x')>u_1(x)$.}\end{proposition}

\subsection{Monotone Comparative Statics of Pareto Optimal Choices}

In this section, we establish wMCS of Pareto optimal choices.   More specifically, we study the conditions on $\mathbf{u}$ and $\mathbf{v}$ that  yield $\pazocal{P}(v)\ge_{ws} \pazocal{P}(u)$.   A natural conjecture is the condition that causes each individual agent $i$ to prefer a higher action under $v_i$ than under $u_i$.  Will it be enough, for instance, if $v_i$ MS-dominates $u_i$ for each $i\in I$?  It turns out such conditions are not enough for the wMCS of POC even when $X$ is totally ordered:

	\begin{example} \label{example:poc-totally-ordered}
	Let
	$X=(0,1)$ with the Euclidean topology as well as the standard order and
	\begin{align*}
		u_1(x) & =\left\{ 	\begin{array}{l l}
			2-x  & \text{ if } x<1/2\\
			3-x   &  \text{ if  } x\ge 1/2,
		\end{array}
		\right. \mbox{ and }   u_2(x)=1-x \mbox{ for all }x, \\
		v_1(x)& =\left\{ 	\begin{array}
			{ l l }%
			x   & \text{ if } x<1/4\\
			\frac{1}{2}- x   & \text{ if } x\in [1/4, 1/2)\\
			\frac{1}{2}+x & \text{ if } x \ge 1/2 \\
		\end{array}
		\right. \mbox{ and }
		v_2(x)=\left\{ 	\begin{array}
			{ l l }
			x   & \text{ if } x<1/4\\
			\frac{1}{2}- x   & \text{ if } x\in [1/4, 1/2)\\
			\frac{1}{4} (x-\frac{1}{2}) & \text{ if } x \ge 1/2\\
		\end{array}.
		\right. \end{align*}
 See  \Cref{figure003}.	Observe that  the MS conditions are satisfied as  $v_i$ single-crossing dominates $u_i$ for $i=1,2$ and  $X$ is a lattice. However,   $\pazocal{P}(u)=\{\frac{1}{2}\}$ while
	$\pazocal{P}(v)=\{\frac{1}{4}\}$, so $\pazocal{P}(v)$ fails to weak set dominate $\pazocal{P}(u)$.
	\begin{figure}[htp]
		\begin{center}
			\setlength{\unitlength}{0.6cm}\begin{tikzpicture}[yscale=0.85,xscale=0.8] \footnotesize
			\draw[thick,->] (0,0) -- (7.5,0);
			\draw[thick,->] (0,0) -- (0,7);

			\draw[dashed] (0,6) -- (3,5);
			\draw[thick] (3,5) -- (6,4);
			\draw[thick] (0,4) -- (3,3);
			\draw[dashed] (3,3) -- (6,2);
			\draw[thick] (3,5) -- (6,4);
			\draw[dashed] (3,5) -- (3,0);
			\draw[dashed] (6,4) -- (6,0);
			\draw[thick] (0,2) -- (6,0);

			\draw[thick,red] (0,0) -- (1.5,0.5);
			\draw[thick,red] (3,0) -- (1.5,0.5);
			\draw[thick,red] (3,0) -- (6,0.25);

			\node[circle,draw=red, fill=white, inner sep=0pt,minimum size=5pt] at (6,0.25) {};
			\put (7.3,0.6){\textcolor{red}{$v_2$}}
			\draw[thick,red] (3,2) -- (6,3);
			\put (7.3,4.5){\textcolor{red}{$v_1$}}
			\node[circle,draw=red, fill=red, inner sep=0pt,minimum size=5pt] at (3,2) {};
			\node[circle,draw=red, fill=white, inner sep=0pt,minimum size=5pt] at (6,3) {};

			\put (-0.5,8.4){$3$}
			\put (-0.5,5.5){$2$}
			\put (-0.5,2.6){$1$}
			\put (-0.5,-0.6){$0$}
			\put (7.85,-0.6){$1$}
			\put (3.76,-0.6){$\frac{1}{2}$}
			\put (1.83,-0.6){$\frac{1}{4}$}
			\put (9.5,-0.6){$x$}
			\put (5,1.3){$u_2$}
			\put (5,7){$u_1$}

			\node[circle,draw=black, fill=black, inner sep=0pt,minimum size=5pt] at (3,5) {};
			\node[circle,draw=black, fill=white, inner sep=0pt,minimum size=5pt] at (3,3) {};
			\node[circle,draw=black, fill=white, inner sep=0pt,minimum size=5pt] at (6,4) {};
			\node[circle,draw=black, fill=white, inner sep=0pt,minimum size=5pt] at (0,4) {};
			\node[circle,draw=black, fill=white, inner sep=0pt,minimum size=5pt] at (0,2) {};
			\node[circle,draw=black, fill=white, inner sep=0pt,minimum size=5pt] at (6,0) {};
			\node[circle,draw=red, fill=white, inner sep=0pt,minimum size=5pt] at (0,0) {};

			\end{tikzpicture}

			\bigskip
			\caption{\label{figure003} wMCS of POC fails under the MS conditions.}
	\end{center} \end{figure}

What causes the failure of wMCS in this example  is  the non-compactness of $X$. Of course, non-compactness of $X$ may cause nonexistence of POC in light of  \Cref{prop:P-properties}. However, the failure of wMCS in this example is not due to nonexistence.\footnote{Strictly speaking, one may regard nonexistence as an instance of the MCS being vacuous, rather than an instance of its failure.}
Suppose we modified the example to assume  $X = [0,1]$, so as to restore compactness, and assume that the utility functions are continuous at the end points $x =0$ and $1$.  Then, we regain wMCS, as $\pazocal{P}(v)=\{\frac{1}{4},1\} \ge_{ws}  \pazocal{P}(u)= \{0,\frac{1}{2}\}$.\footnote{Since the ``sandwich'' property does not hold, the MCS does not hold in the strong set order sense (\Cref{thm:ss-char}), a point we will come back to in \Cref{ssec:POC-properties}.}  \textsquare
	\end{example}

Indeed, if $X$  is totally ordered, then its compactness is all we need for  MCS conditions for individual choices to yield  wMCS of Pareto optimal choices.    \Cref{thm:MCS-POC-2} below establishes this result.  The theorem uses the following lemma, which holds  for any compact set $X$, not just for totally ordered $X$.\footnote{An astute reader will recognize that the failure of wMCS in \Cref{example:poc-totally-ordered} rests on the failure of this property, namely the possibility of an alternative being Pareto dominated without being Pareto dominated by any Pareto optimal choice.}

\begin{lemma}\label{lemma:dominance-structure} Suppose $X$ is compact  and $u_i$ is upper semicontinuous for every $i \in I$.	If $x \not\in \pazocal{P}(u)$, then $x$ is Pareto dominated by some $x'\in P(u)$. \end{lemma}

With this lemma at hand, we are ready to establish  wMCS  of POC for totally ordered $X$ and its payoff implication. We say    $v  = (v_i)_{ i\in I}$     {\it single-crossing dominates} $u = (u_i)_{i\in I}$ if  $v_i$ single-crossing dominates $u_i$  for each $i\in I$.

\begin{theorem} \label{thm:MCS-POC-2}  Suppose that $X$ is compact and totally ordered.   If $v$ single-crossing dominates $u$, then $\pazocal{P}(v)\ge_{ws} \pazocal{P}(u)$.\footnote{Recall that single-crossing dominance is equivalent to weak dominance when $X$ is totally ordered (\Cref{equivalence-remark}). Therefore, in the present environment,  $\pazocal{P}(v)\ge_{ws} \pazocal{P}(u)$ if $v_i$ weakly dominates $u_i$ for every $i \in I$.}
\end{theorem}

\begin{proof}
To show $\pazocal{P}(v) \ge_{uws} \pazocal{P}(u)$, we let $x\in \pazocal{P}(u)$ and will show there exists $x' \ge x$ such that $x'\in \pazocal{P}(v)$. If $x \in \pazocal{P}(v)$, then the desired conclusion trivially holds.  If $x \not\in \pazocal{P}(v)$, by  \Cref{lemma:dominance-structure} there exists $x'\in \pazocal{P}(v)$ that Pareto dominates $x$ under $v$, so $v_i(x') - v_i(x) \ge 0$ for every $ i \in I \mbox{ and } v_j(x') - v_j(x) > 0$ for some $j \in I$.
If $x' < x$, then because $v$ single-crossing dominates $u$, it follows that $u_i(x') - u_i(x) \ge 0$ for every $ i \in I \mbox{ and } u_j(x') - u_j(x) > 0$  (by the contrapositive of the single crossing property), contradicting $x\in \pazocal{P}(u)$. Since $X$ is totally ordered, this implies $x'\ge x$, as desired. A symmetric argument shows $\pazocal{P}(v) \ge_{lws} \pazocal{P}(u)$, completing the proof.
\end{proof}

The total orderedness of $X$  plays a key role in \Cref{thm:MCS-POC-2},
and the result does not readily extend to a general domain $X$. Indeed,   wMCS of POCs for the general domain $X$ requires a very different  approach with stronger assumptions.  The approach utilizes a novel characterization of POC via sequential utilitarian welfare maximization by \cite{che/kim/kojima/ryan:20}.   For this characterization and the subsequent theorem (\Cref{thm:MCS-POC}), we assume $X$ to be a topological vector space  so that  vector operations on $X$ are well-defined.   The following lemma is an immediate corollary of \cite{che/kim/kojima/ryan:20}.

\begin{lemma}   \label{lem:P-char}
	Suppose  $X$ is  compact and convex, and $u_i$ is upper semicontinuous and concave for each $i \in I$.  Then, $x\in \pazocal{P}(u)$ if and only if there exists a finite sequence $(\phi^t)_{t=1}^T$ of nonzero weights $\phi^t\in \mathbb{R}^{|I|}_+$, for each $t=1,..., T$, with $\phi^T\in \mathbb{R}^{|I|}_{++}$ and $T\le |I|$ such that
	$$x\in X^T, \mbox{ where } X^0:=X \mbox{ and } X^t:= \operatornamewithlimits{arg\,max}_{x'\in X^{t-1}} \sum_{i\in I} \phi^t_i u_i(x')  \mbox{ for all } t=1, ..., T.$$
\end{lemma}

This characterization views Pareto optimal choices as resulting from a sequence of weighted  utilitarian welfare maximizations.\footnote{A simple weighted utilitarian welfare maximization does not characterize Pareto optimality; namely,  not all maximizers of non-negatively weighted sum of utilities are Pareto optimal. See \cite{che/kim/kojima/ryan:20}.} This is useful for our purpose since we can apply the standard machinery of monotone comparative statics to the maximizers of weighted utilitarian welfare functions.  Of course, in light of the above characterization, we must apply the method sequentially and inductively, as will be seen below.

We are now ready to establish a wMCS result for POC. To this end, we introduce several conditions. We say that $v$  {\it increasing-differences dominates} $u$ if,  for each $i\in I$ and    $x'>x$,
$v_i(x') -v_i(x)\ge  u_i(x') -u_i(x)$  and that   $u$ is  {\it supermodular} if $u_i$ is supermodular  for each $i\in I$:  for each
$x,x'\in X$,
$u_i(x\vee x') -u_i(x)\ge   u_i(x') -u_i(x\wedge x')$.\footnote{It is straightforward to see that supermodularity implies quasi-supermodularity while increasing-difference dominance implies single-crossing dominance.}  Just like single-crossing dominance and quasi-supermodularity, increasing-difference dominance and supermodularity guarantee that individual choices exhibit sMCS \citep{topkis:79}. We use them to establish wMCS of POC.

\begin{theorem} \label{thm:MCS-POC}Suppose $X$ is a compact, convex lattice,  and  both $u$ and $v$ are upper semicontinuous, concave and supermodular.  If $v$ increasing-differences dominates $u$, then $\pazocal{P}(v)\ge_{ws} \pazocal{P}(u)$.
\end{theorem}

\begin{proof}  Let $\Upsilon$ be the set of all finite sequences of nonzero weights satisfying the requirements of \Cref{lem:P-char}.  Now fix any sequence of weights $(\phi^t)\in \Upsilon$. Let $\pazocal{P}_{(\phi^t)}(u):=X^T(u),$ where $X^0(u):=X$ and
	$$X^t(u):= \operatornamewithlimits{arg\,max}_{x'\in X^{t-1}(u)} \sum_{i\in I} \phi^t_i u_i(x')  \mbox{ for all } t=1, ..., T,$$
and define $\pazocal{P}_{(\phi^t)}(v):=X^T(v),$ analogously.  We claim $\pazocal{P}_{(\phi^t)}(u)\le_{ss}\pazocal{P}_{(\phi^t)}(v)$.  The proof is inductive.  Note first $X^0(u)=X\le_{ss} X=X^0(v)$, since $X$ is a lattice.  For induction, assume $X^{t-1}(u) \le_{ss}  X^{t-1}(v)$.   Since $u$ and $v$ are supermodular and $v$ increasing-differences dominates $u$,
	$\sum_{i\in I} \phi^t_i u_i(\cdot)$ and   $\sum_{i\in I } \phi^t_i v_i(\cdot)$ are  supermodular,
	and the latter increasing-differences dominates the former.  Then,  since the constraint sets satisfy   $X^{t-1}(u) \le_{ss}  X^{t-1}(v)$ by the inductive hypothesis, we can apply Theorem 4 of \cite{milgrom/shannon:94} to obtain  $X^t(u) \le_{ss}  X^t(v)$.  Completing the induction, we  conclude $\pazocal{P}_{(\phi^t)}(u)\le_{ss}\pazocal{P}_{(\phi^t)}(v)$.

The result follows since
$$ \pazocal{P}(u)=\bigcup_{(\phi^t) \in \Upsilon}\pazocal{P}_{(\phi^t)}(u)\le_{ws}\bigcup_{(\phi^t)\in\Upsilon}\pazocal{P}_{(\phi^t)}(v)=\pazocal{P}(v),$$
where the first and last equalities follow from \Cref{lem:P-char} and the weak set dominance (the inequality) follows from the above observation and the fact that weak set order is closed under the union operation (\Cref{lem-ws-set-operator}).
\end{proof}

Compared with the case where $X$ is totally ordered, both topological and MCS conditions are strengthened here.  First, convexity of $X$ is added to the conditions assumed previously. Its role is to ensure that the utility possibility set (or the projection of $X$ to utility spaces) is convex---a requirement for \Cref{lem:P-char}.  Unlike the single-crossing dominance used for totally ordered $X$, we now require cardinal, and thus stronger, versions of MS conditions for individual payoff functions: {\it supermodularity} of $u$ and $v$, and {\it increasing-differences dominance} of $u$ by $v$.  The reason is that these conditions are preserved, whereas their ordinal versions are not, when we aggregate individual payoff functions to form a (weighted) utilitarian welfare function. Our method crucially uses the fact that a utilitarian welfare function with an arbitrary profile of (non-negative) weights exhibits an MCS property.

It is natural to ask whether these conditions can be relaxed.
Compactness of $X$ or concavity of the utility functions cannot be dispensed with.\footnote{\Cref{example:poc-totally-ordered} shows compactness of $X$ cannot be dispensed with. After our initial draft was distributed, Haoyu Liu obtained an example demonstrating that concavity of the utility functions cannot be dispensed with.}
Whether other properties, namely the convexity of $X$, or the  supermodularity or increasing-differences dominance of the payoff functions, can be weakened remains unresolved. On the one hand, as has been explained above,  our proof strategy utilizes these conditions in an essential manner.
On the other hand, we have not found any counterexample when those conditions are dropped. Whether those conditions are tight or not is an interesting but challenging question, and we submit it as an open question.

The following example illustrates how one may apply the theorem.

\begin{example}[Investment problem for a multidivisional firm]  \label{non-strong-example} \rm  Two agents, 1 and 2,  collectively choose $(x,y)\in [0,1]^2$ facing a ``state'' $\omega=(\omega^A, \omega^B)$. Their payoff functions are:
	\begin{align*}
	u_{1}(x,y; \omega)&=-\left (x-2\omega^A  \right )^2- \left (y - \omega^B  \right )^2,\\
	u_{2}(x,y; \omega)&=-\left (x-\omega^A  \right )^2- \left (y - 2 \omega^B  \right )^2.
	\end{align*}
	One can interpret the collective as a firm, consisting of two divisions, which decides on investments $(x,y)$ in two different technologies, $A$ and $B$.  We may call this pair an \emph{investment plan}.  The firm's objective is to
	adapt the investment plan ``closely'' to the state $(\omega^A,\omega^B)$.\footnote{The firm may be making this investment plan as part of a ``beauty contest'' game facing other firms making similar plans.  Suppose there are benefits for the firms to coordinate their investment plans, say due to network benefits from investing in technologies adopted by other firms.  In this case, the ``state''  $(\omega^A,\omega^B)$ includes the other firms' investment plans.   We shall come back to this example in \Cref{sssec:beauty}, which illustrates how $\omega$ can be ``unpacked'' to generate a full-fledged beauty contest game.  For the current purpose, the current player (consisting of divisions 1 and 2) simply treats $\omega$ as exogenous.}  The two divisions' preferences, while similar, are not fully aligned with each other.  Specifically, division 1 is biased toward $A$ and division 2 is biased toward $B$: each division  enjoys an additional private benefit from its ``pet'' project $Z=A,B$, relative to the state. The firm then chooses a Pareto optimal choice for its divisions.  Let $\mathcal P(\omega):=\pazocal{P}(u (\cdot; \omega))$ denote the set of Pareto optimal choices given parameter $\omega$.

	One can readily confirm that this example satisfies the conditions required by \Cref{thm:MCS-POC}. The set $X$ is a compact convex lattice, and $u_i$ is continuous, concave and supermodular in $(x,y)$, for $i=1,2$.  For $\omega'>\omega$, $u_i(\cdot;\omega')$ increasing-differences dominates $u_i(\cdot;\omega)$ for each $i=1,2$. Then, by  \Cref{thm:MCS-POC}, $\mathcal P(\omega')\ge_{ws} \mathcal P(\omega)$.
\Cref{fig:POC} illustrates this with   $\omega=(1/4,1/4)$  and   $\omega'=(1/3,1/3)$.

	\begin{figure}[htb]
		\begin{center}
			\begin{tikzpicture} [scale=5.5]

			\draw (0,0) node [below left] {\scriptsize$0$}  --  (1,0) node [below] {\scriptsize$1$}  -- (1,1)  --   (0,1) node [left] {\scriptsize$1$} --  cycle;
			\draw  [color=blue]  (1/4,1/2)  --  (1/2,1/4) node [pos=0.4, right] {\footnotesize $\mathcal{P} (\omega)$} ;

		\draw  [color=red]  (1/3,2/3)  --  (2/3,1/3) node [pos=0.4, right] {\footnotesize $\mathcal{P} (\omega')$} ;

			\draw [fill]  (1/4,1/2) circle (0.05ex) ;
\draw    (1/4,-0.01) -- (1/4,0.01)  node [pos=0, below]  {\scriptsize$1/4$};
\draw   (-0.01,1/2) -- (0.01,1/2)  node [pos=0, left]  {\scriptsize$1/2$};

			\draw [fill]  (2/3,1/3) circle (0.05ex) ;

\draw   (2/3,-0.01) -- (2/3,0.01)  node [pos=0, below]  {\scriptsize$2/3$};

\draw    (-0.01,1/3) -- (0.01,1/3)  node [pos=0, left]  {\scriptsize$1/3$};

			\draw [fill]  (1/3,2/3) circle (0.05ex);

\draw   (1/3,-0.01) -- (1/3,0.01)  node [pos=0, below]  {\scriptsize$1/3$};
\draw    (-0.01,2/3) -- (0.01,2/3)  node [pos=0, left]  {\scriptsize$2/3$};

			\draw [fill]  (1/2,1/4) circle (0.05ex);

\draw    (1/2,-0.01) -- (1/2,0.01)  node [pos=0, below]  {\scriptsize$1/2$};
\draw    (-0.01,1/4) -- (0.01,1/4)  node [pos=0, left]  {\scriptsize$1/4$};

			\end{tikzpicture}
			\caption{Change of POCs when preferences shift from $\omega$ to $\omega'$} \label{fig:POC}
			\end{center}
		\end{figure}

\end{example}

\subsection{Properties of Pareto Optimal Choices.}  \label{ssec:POC-properties}

Given  \Cref{thm:MCS-POC-2} and  \Cref{thm:MCS-POC}, a natural question is whether the same assumptions guarantee sMCS, not just wMCS, of POC.  We already saw from \Cref{example:poc-totally-ordered} that this is not true even when $X$ is totally ordered: Recall that  with $X = [0,1]$, we have $\pazocal{P}(u))=\{0,\frac{1}{2}\}$ and    $\pazocal{P}(v)=\{\frac{1}{4},1\}$, so  $\frac{1}{2}\wedge \frac{1}{4}\not\in \pazocal{P}(u)$ and $ \frac{1}{2}\vee \frac{1}{4}\not\in \pazocal{P}(v)$.  Still, the POC forms a lattice in this case.    However, neither the lattice property nor sMCS holds even with the stronger assumptions made for  \Cref{thm:MCS-POC}. \Cref{fig:POC} illustrates this.

Another question of interest is whether in general  $\pazocal{P}(u)$ is closed (or equivalently compact given the compactness of $X$).
	Compactness of POC plays an important role in \Cref{sec:game}. In that section, we consider a game played by  collectives or individuals with incomplete preferences whose best responses comprise POCs.  For existence of   (pure strategy) Nash equilibria and monotone comparative statics for such a game, we need the best response correspondences to be closed-valued.

	The following example shows that   $\pazocal{P}(u)$ is not necessarily closed in our environment.

	\begin{example}[Non-closed POC] \rm
		Let $X=[0,1]$, $I=\{1,2\}$, and
		$$u_1(x)=\left\{ 	\begin{array}{l l}
		x  & \text{ if } x \le 1/2 \\
		1-x   &  \text{ if  } x> 1/2,
		\end{array}
		\right. \mbox{ and }   u_2(x)=\left\{ 	\begin{array}{l l}
		2-x  & \text{ if } x < 1 \\
		3   &  \text{ if  } x=1. \end{array}  \right.
		$$
		This environment satisfies all the assumptions made for  \Cref{prop:P-properties}; In particular, $X$ is compact and both utility functions are upper semicontinuous. See \Cref{figure002}. However, $\pazocal{P}(u)=(0,1/2] \cup \{1\}$, so  the set is not closed.  Note also that $\pazocal{P}(u)$ does not contain a minimal element.\footnote{An example with no maximal element of $\pazocal{P}(u)$ can be obtained from this example by endowing $X$ with the opposite order to the standard one. To see the connection between compactness and existence of maximal/minimal points, refer to the proof of \Cref{prop:ind-choice-property}.}
		\begin{figure} [htp]
			\begin{center}
				\setlength{\unitlength}{0.7cm}\begin{tikzpicture}[>=latex,xscale=0.85,yscale=0.85] \footnotesize
				\draw[thick,->] (0,0) -- (7,0) node [pos=1,right] {$x$};
				\draw[thick,->] (0,0) -- (0,7);

				\draw[thick] (0,4) -- (6,2);

				\draw[dashed] (3,3) -- (3,0);
				\draw[dashed] (6,6) -- (6,0);
				\draw[thick] (0,0) -- (3,1);
				\draw[thick] (3,1) -- (6,0);
				\draw[dashed] (0,1) -- (3,1);
				\draw[dashed] (0,6) -- (6,6);

				\put (-0.5,7.2){$3$}
				\put (-0.5,4.7){$2$}
				\put (-0.5,1.1){$\frac{1}{2}$}
				\put (-0.3,-0.5){$0$}
				\put (7.2,-0.5){$1$}
				\put (3.5,-0.5){$\frac{1}{2}$}
				\put (5,1){$u_1$}
				\put (5,3.5){$u_2$}

				\node[circle,draw=black, fill=white, inner sep=0pt,minimum size=5pt] at (6,2) {};
				\node[circle,draw=black, fill=black, inner sep=0pt,minimum size=5pt] at (6,6) {};
				\end{tikzpicture}
				\caption{Example with no minimal Pareto optimal choice and failure of compactness.} \label{figure002}
			\end{center} \end{figure}
			\textsquare
		\end{example}

This example shows that  conditions assumed in \Cref{prop:P-properties} need not guarantee closedness of  $\pazocal{P}(u)$.  Fortunately, some additional regularity conditions lead to compactness (and hence the existence of maximal and minimal points). To state them, for each $i \in I$ and $x \in X$, let
		$ U_{-i}(x):= \{ y \in  X \, : \,  u_j (y)\ge  u_j(x), \forall  j \in I\setminus\{i\} \}
		$
		denote the set of alternatives that every agent other than $i$ weakly prefers to $x$.
		\begin{proposition}  \label{prop:P-properties-2}  Suppose that $X$ is compact and  that,   for each $i \in I$,   $u_i(\cdot)$ is
			continuous  and the correspondence $U_{-i}(\cdot)$
			is lower hemicontinuous.\footnote{Given the compactness of $X$ (which is  assumed throughout), $U_{-i}(x)$ is lower hemicontinuous if, for each sequence $(x_n)_n$ with $x_n\in X$, $\forall n \in \mathbb N$, that converges to $x$ and for any $z\in U_{-i}(x)$,  there exists $(z_n)_n$ with $z_n\in U_{-i}(x_n)$, $\forall n \in \mathbb N$, that converges to $z$.  \Cref{lem:sufficient-regularity} in \Cref{sec:lower-hemi} of Supplementary Appendix provides sufficient conditions for $U_{-i}(x)$ to be lower hemicontinuous.}
			Then, $\pazocal{P}(u)$ is closed. Further, $\pazocal{P}(u)$ has minimal and maximal points. \end{proposition}

 \section{Fixed Point Theorem} \label{sec:fixed point}
In this section, we present a fixed-point theorem that plays a central role in the remainder of this paper. In addition to establishing the existence of a fixed point, we also offer a new comparative statics theorem for  fixed points and an algorithm for finding them.

Consider a nonempty set  $X$ endowed with a partial order $\ge$ as well as a metric
and a  natural topology induced by them. Throughout,  assume that $X$ is compact with respect to this topology. Let $F: X \rightrightarrows X$ be  a self-correspondence, i.e., a correspondence from $X$ to itself. An element  $x\in X$ is a \emph{fixed point} of $F$ if $x\in F(x)$, and we let $\Fp(F)$ denote the set of all fixed points of $F$.

\subsection{Fixed Point Theorem}

Let $X_+:=\{x\in X: \exists y \ge x \mbox{ s.t. } y\in F(x)\}$ denote the set of points whose image includes a weakly higher point than that point, and similarly let
$X_-:=\{x\in X: \exists y \le x \mbox{ s.t. } y\in F(x)\}$.   We then define a self correspondence $F: X \rightrightarrows X$, defined over a nonempty partially ordered set $X$, to be {\bf upper monotonic}, and write $F\in \mathcal{F}_+$, if
 (i) $F(x)$ is nonempty and closed for each $x\in X$; (ii) $F$ is upper weak set monotonic; and (iii) $X_+$ is nonempty.
Symmetrically, we call $F: X \rightrightarrows X$   {\bf lower monotonic}, and write $F\in \mathcal{F}_-$, if (i)  holds, and (ii) and (iii) are replaced by (ii$'$) $F$ is lower weak set monotonic; and (iii$'$)	$X_-$ is nonempty.

 Now we are ready to present our fixed-point theorem.

\begin{theorem} [Fixed-Point Theorem] \label{main}  The set of fixed points $\Fp(F)$ is nonempty if $F$ is either upper or lower monotonic, i.e., $F\in \mathcal{F}_+\cup \mathcal{F}_-$.\footnote{Examples in \Cref{sec:appendix-fixedpoint-necessity} of Supplementary Appendix show that  none of the conditions required by upper or lower monotonicity can be dispensed with for this result.}   Moreover, $\Fp(F)$ contains a maximal point if $F\in \mathcal{F}_+$ and a minimal point if $F\in \mathcal{F}_-$.
\end{theorem}
\begin{proof}
See \Cref{section-proof-of-fpt}.
\end{proof}

Before proceeding, it is instructive to compare this theorem with \cite{zhou:94}'s fixed-point theorem, which extends \cite{tarski:55}'s fixed-point theorem to accommodate correspondences.   First, we require that $X$ be partially ordered. This condition is considerably weaker than the complete lattice condition required by  \cite{tarski:55} or \cite{zhou:94}.  Second, we do not require  $F(x)$ to be a complete sublattice of $X$, as is assumed by \cite{zhou:94}.  Third, we require $F$ to be  upper or lower weak set monotonic instead of strong set monotonic as in \cite{zhou:94}. Finally, the nonemptiness of $X_+$ (or  $X_-$) is trivially satisfied both in \cite{tarski:55} and  \cite{zhou:94} because they restrict their attentions to the case where $X$ is a complete lattice (which contains   smallest and largest points).   Meanwhile, our theorem requires two topological conditions---compactness of $X$ and closed-valuedness of $F$---absent in \cite{tarski:55} and \cite{zhou:94}.

Compared with the fixed-point theorem  of   \cite{tarski:55}  or \cite{zhou:94}, Theorem \ref{main} thus dispenses with some restrictive order-theoretic assumptions but adds the aforementioned topological assumptions.   Since these latter conditions are satisfied in many economic applications, the current theorem will be useful in many settings in which  \cite{tarski:55}  or \cite{zhou:94} cannot be applied.   In fact, \Cref{conditions} in  \Cref{sec:appendix-fixedpoint} of Supplementary Appendix shows that,
in many problems  of interest,
 the conditions in \Cref{main} are weaker than those of \cite{zhou:94}'s theorem.
For instance, a subset $X$ of a Euclidean space is a compact lattice if and only if it is a complete lattice; since we do not require $X$ to be a lattice, restrictions in the current theorem are strictly weaker.\footnote{More generally, the same conclusion holds for $X$ endowed with the order topology; see  \Cref{conditions} for a formal statement, due to \cite{frink:42}.}

While the conditions required by \Cref{main} are typically weaker than those in extant results, the conclusions obtained are also weaker.  Unlike  Tarski's fixed-point theorem and \cite{zhou:94}'s extension, fixed points need not form a complete lattice in the current case, and the set of fixed points may not even have the largest or the smallest element.  Instead, our theorem shows that the set has a maximal or minimal point.

\begin{remark}\rm
After proving \Cref{main}, we became aware of an earlier contribution by \cite{li:14}, who established the existence of a fixed point under the same set of assumptions as ours. We fully acknowledge his prior contribution here.  Meanwhile, a few remarks are in order. First, our proof is different from, and arguably simpler than, his; see \Cref{section-proof-of-fpt}. Second, we establish existence of maximal and minimal fixed points, a property that \cite{li:14} did not show.  The proof for this is not trivial since the set of fixed points is not necessarily compact; see  \Cref{noncmpact-minimal-example} in \Cref{supp-app-minimal-maximal-fp} of Supplementary Appendix.\footnote{In fact, the same example further shows that the set of maximal/minimal fixed points need not be compact.} Finally, we also establish a comparative statics result on the fixed points, to be presented below as \Cref{thm:comparative statics}, which is novel to our knowledge.
\end{remark}

An important benefit of the fixed-point theorem is the ease with which it can be adapted for monotone comparative statistics.

\begin{theorem}[Comparative Statics] \label{thm:comparative statics} For any pair of self correspondences $F,F'$ defined over a partially ordered set $X$,
\begin{description}
	\item[(i)] $\Fp(F') \ge_{uws} \Fp(F)$ if $\Fp(F) \ne \emptyset$, $F'\in \mathcal{F}_+$, and  $F'(x) \ge_{uws} F(x), \forall x \in X$;
		\item[(ii)] $\Fp(F') \ge_{lws} \Fp(F)$ if $F \in \mathcal{F}_-$, $\Fp(F') \ne \emptyset$,   and  $F'(x) \ge_{lws} F(x), \forall x \in X$.
\end{description}
\end{theorem}

\begin{proof}   Fix any $x^* \in \Fp(F)$. For any $X' \subset X$, let $X'_{\geqslant x^*} := \{ x' \in X' :  x' \geqslant x^* \}$.   Define correspondence $\tilde F': X_{\geqslant x^*} \rightrightarrows  X_{\geqslant x^*} $ by $\tilde F'(x):=F' (x)_{\ge x^*} $ for each $x
	\in X_{\ge x^*}$.  Note that  for any closed $X'$, $X'_{\geqslant x^*}$ is  closed (and thus compact) in the natural topology.    Clearly, $X_+(\tilde F')$ contains $x^*$ and is thus nonempty.  Also, $\tilde F'$ is  closed-valued since, for each $x\in X_{\ge x^*}$, $F'(x)$ is closed and $F'(x)_{\ge x^*}$ is a closed subset of $F'(x)$.  The facts that  $x^* \in F(x^*)$ and  that $F'(x) \ge_{uws} F(x) \ge_{uws} F (x^*)$ for each $x \in X$ imply that for any $x \ge x^*$, there is some $x' \in  \tilde F'  (x)$.   That is, $\tilde F'$  is a nonempty-valued self-correspondence defined on $ X_{\ge x^*}$.   Moreover, $\tilde F'$ is upper weak set monotonic since, for any $x, x' \in X_{\ge x^*}$ with $x' \ge x$ and any $y \in \tilde F' (x) \subset F' (x)$, there exists some $y' \in F'(x')$ such that $y ' \ge y (\ge x^*)$  so that $y' \in F' (x')_{ \ge x^*} = \tilde F' (x')$.  Since $\tilde F'$ satisfies all the conditions for \Cref{main} and $X_{\geqslant x^*}$ is compact, there must exist  a fixed point $ \tilde x \in \tilde F' (\tilde x )$, which means that  $\tilde x \in  F'(\tilde x)$ and $\tilde x \ge x^*$.  This completes the proof for the ``upper'' version of the statement.  The proof of the ``lower'' version is symmetric.  \end{proof}

This result immediately implies the following corollary.
 \begin{corollary}  \label{cor:comparative statics} For any pair of self correspondences $F,F'$ defined over partially ordered set $X$,
$\Fp(F')\ge_{ws} \Fp(F)$ if $F \in\mathcal{F}_-$, $F'\in \mathcal{F}_+$, and  $F'(x) \ge_{ws} F(x)$  for all $x \in X$. \end{corollary}

\citet{acemoglu2015robust}  establish  wMCS of fixed points of a correspondence  and apply it to  dynamic economic models.\footnote{\cite{villas1997comparative} also establishes wMCS of fixed points but only considers single-valued correspondences (i.e., functions).}  A major  difference is that their result  requires the self correspondence $F$ to have the closed-graph property, which is  a stronger than the closed-valuedness we require.\footnote{\citet{acemoglu2015robust}  assume a version of upper hemicontinuity that amounts to a  the closed graph property.
}    Clearly, a correspondence with this property is closed valued, but a closed-valued correspondence need not have this property.  In particular,  if $F$ were a function, this property would force $F$ to be  continuous, whereas closed-valuedness, and thus we, would allow $F$ to be discontinuous.

\subsection{Iterative Algorithm for Finding a Fixed Point.}
An important benefit of a monotonic operator is that it gives rise to a constructive algorithm for finding a fixed point.  It is well known that in the  environment of Tarski and Zhou, given some additional continuity property of $F$, the highest fixed point is obtained by iteratively applying the highest selection from the correspondence starting from the highest point $\overline x:=\sup X$, and likewise  the lowest fixed point is obtained by iteratively applying the lowest selection from the correspondence starting from the lowest point $\underline x:=\inf X$.  This property, known as Kleene's fixed-point theorem (see \citet{baranga1991contraction} for instance) and also established for supermodular games by \cite{milgrom/roberts:90} and \cite{milgrom/shannon:94}, is very convenient in practice.

We show that a similar property holds if $X$ satisfies the hypotheses of  \Cref{main}, albeit with some qualifications.  We say $F$ is \textbf{upper hemi-order-continuous} if, for any $(x,y) \in X^2$ and for any sequence $(x_n, y_n)_{n\in \mathbb{N}}$ converging to $(x,y)$, where $(x_n)_{n\in N}$ is either weakly increasing or weakly decreasing  and  $y_n\in F(x_n)$, $\forall n \in \mathbb{N}$, we have $y \in F(x)$.\footnote{Note that the condition is weaker than upper hemi-continuity since the condition is required only for $(x_n)_n$ that is monotone.  The condition can be seen also as a natural generalization of the order continuity defined for a function to a correspondence.  See \cite{milgrom/roberts:90} for an order-continuous function.}

\begin{theorem}  \label{prop:iteration}
Suppose   $F$   is an upper monotonic and upper hemi-order continuous self correspondence defined over a partially ordered set $X$.
	\begin{enumerate}
		\item  For every $ x\in X_+$ there exists a weakly increasing sequence $(x_n)_{n\in \mathbb{N}}$ with $x_1=x$
		 and $x_{n+1}\in\{y\in X: y\in F(x_n), y\ge x_n  \}$ for each $n\in \mathbb{N}$, such that its limit $x_*=\lim_{n\to \infty} x_n$ exists and is a fixed point of $F$.
		\item  Suppose $F':X \rightrightarrows X$ is upper monotonic and upper hemi-order continuous, and $F'(x) \ge_{uws} F(x)$ for each $x$. Then,  for each  $x_F \in \Fp(F)$, there exists   $x_{F'} \in \Fp(F')$ with $x_{F'}\ge x_F$ that can be found by an upward iterative procedure starting with $ x_1=x_F$  for $F'$.\footnote{More specifically, there exists a weakly increasing sequence $(x_n)_{n\in \mathbb{N}}$ such that
		$ x_1=x_F$ and $x_{n+1}\in\{y\in X: y\in F'(x_n), y\ge x_n  \}$ for each $n\in \mathbb{N}$; and its limit $x_{F'}=\lim_{n\to \infty} x_n$ is well defined and is a fixed point of $F'$.}
	\end{enumerate}
	A symmetric conclusion holds if  $F$ is lower monotonic and upper hemi-order-continuous.
\end{theorem}
\begin{proof}  Given the symmetry, we only prove (i) and (ii).   First, since $ x_1\in X_+$, there exists $ x_2\in\{y\in F( x_1): y\ge x_1 \}$.   By upper weak set monotonicity of $F$,  if $x_{n+1}\in F(x_n)$ and if $ x_{n+1}\ge  x_n$,  for any $n\in \mathbb{N}$, then there must exist  $x_{n+2}\in\{y\in F(x_{n+1}): y\ge x_{n+1}\}$.    We thus obtain a weakly increasing sequence $(x_n)_{n\in \mathbb{N}}$.  Since $X$ is a compact metric space, the weakly increasing sequence has a limit $x_*=\lim_{n\to \infty}{x_n}$.   By the upper hemi-order-continuity of $F$,  $x_*\in F(x_*)$, proving (i).  The proof of (ii) follows the same argument, once we redefine the starting point $x_1=x_F$ of the iterative procedure for operator $F'$.
\end{proof}

Recall that upper hemi-order-continuity  is trivially satisfied if $X$ is finite.  Hence, \Cref{prop:iteration} suggests a convenient and fast  algorithm to identify a fixed point for finite $X$, even without the standard set of assumptions required by the traditional Tarski approach.

One may recall that in the setting of Tarski and Zhou, a monotonic algorithm starting from the largest and smallest elements finds the largest and smallest fixed points, respectively, and may wonder if maximal and minimal points can be found in this way in our context.  The following example provides a negative answer to that  question.\footnote{\Cref{iteration-difficulty} in  \Cref{supp-iterative-algorithm} of Supplementary Appendix illustrates additional difficulty with iterative procedures.}

\begin{example}  Suppose $X=\{1,2,3\} \times \{1,2\}$ and $F: X \rightrightarrows X$ is defined by:  $F((1,1))=\{(1,2),(2,1)\},  F((2,1))=\{(1,2), (3,2)\},$ $F((1,2))=\{(2,1), (3,2)\},$ $F((2,2))=\{(2,2),(3,2)\},$ $F((3,1))=\{(3,2)\}$, and $F((3,2))=\{(3,2)\}$.   Note that $F$ is both upper and lower monotonic.  There are two fixed points $\{(2,2), (3,2)\}$.  Suppose that one iterates $F$  as suggested in \Cref{prop:iteration}, starting with the lowest point $x_1=(1,1)$.  Then,  no matter which point one chooses along the iteration, the only fixed point one can reach is $(3,2)$.  But this is not a minimal fixed point;  $(2,2)$ is the unique minimal fixed point and smaller than $(3,2)$. The minimal fixed point $(2,2)$ cannot be reached from any iterative application of $F$ starting from $(1,1)$. See  \Cref{figure2}.
\begin{figure}[htp] \begin{center}
\setlength{\unitlength}{2cm}\begin{tikzpicture}
			\put (-0.25,-0.25){$(1,1)$};
			\put (0.75,-0.25){$(2,1)$};
			\put (1.75,-0.25){$(3,1)$};
			\put (-0.5,1.1){$(1,2)$};

\put (1,1.1){$(2,2)$};
\put (2.1,0.8){$(3,2)$};

\filldraw [black] (0,0) circle (2pt);
\filldraw [black] (0,2) circle (2pt);
\filldraw [black] (2,0) circle (2pt);
\filldraw [black] (2,2) circle (2pt);
\filldraw [black] (4,0) circle (2pt);
\filldraw [black] (4,2) circle (2pt);
\draw[thick,->] (0,0+0.1) -- (0,2-0.1);
\draw[thick,->] (0+0.1,0) -- (2-0.1,0);
\draw[thick,->] (0+0.1,2) -- (2+0.05,0+0.1);
\draw[thick,->] (2-0.1,0+0.1) -- (0+0.1,2-0.1);
\draw[thick,->] (2+0.1,+0.1) -- (4-0.1,2-0.1);
\draw[thick,->] (4,0+0.1) -- (4,2-0.1);
\draw[thick,->] (2+0.1,2) -- (4-0.1,2);
\draw [thick, ->] (0+0.1,2+0.1) to [out=30,in=150] (4-0.1,2+0.1);
\draw[thick,->] (4,2+0.1) to [looseness=15, out= 90, in=0] (4+0.1,2);
\draw[thick,->] (2,2-0.1) to [looseness=15, out= 270, in=180] (2-0.1,2);
\end{tikzpicture}
\vskip 0.3cm
\caption{Every iteration starting from $(1,1)$ fails to reach a minimal fixed point.}\label{figure2}
\end{center}\end{figure}

\end{example}

\section{Application to Game Theory}   \label{sec:game}

In this section, we apply our monotone comparative statics results and the fixed-point theorem to strategic  environments to establish the existence and  comparative statics of Nash equilibria. Our equilibrium theory parallels  \cite{milgrom/shannon:94}, who apply their sMCS conditions for players' best responses to establish an analogous result to ours---i.e., wMCS of Nash equilibria---in games with strategic complementarities.\footnote{The comparative statics in \cite{milgrom/shannon:94} is restricted to extremal---i.e., largest and smallest---equilibria. \cite{van2007monotone} also establish the comparative statics for  extremal Nash equilibria in Bayesian games of strategic complementarities.   Likewise, \citet{sobel2019iterated} studies the comparative statics  for bounds of the set of strategies that survive  iterated eliminations of dominated strategies in ID-supermodular games (a more general class of games than supermodular games). Technically,  these studies rely   on the iteration of  best response operators whose convergence requires the order continuity of the objective function. By contrast, we do not assume order continuity for our existence and comparative statics of fixed points and instead make use of a novel fixed point theorem.}  Crucially, their results rest on Tarski's fixed point theorem.\footnote{In fact, they use the iterative version of Tarski.  This requires them to assume order upper semi-continuities  of payoff functions, in addition to the lattice conditions required by Tarski.  Our theory dispense with the continuity assumption, along with others not required by our fixed point theorem.}  Since the conditions required by our fixed-point theorem are typically, sometimes considerably, weaker than those required by Tarski (or Zhou),  our existence and monotone comparative statics can be established more broadly.  As observed in our Introduction, the substantive content of monotone compariative statics is not compromised by this  weakening.

 Consequently, our approach applies to a broader class of games, called \emph{games with weak strategic complementarities},  in which strategy sets may not form lattices and the best response correspondences may not form sublattices and are required to be  only weak set  monotonic.
{ Utilizing our results in \Cref{sec:fixed point}, we establish  the  existence and comparative statics of Nash equilibria even in games  where only weak set monotonicity (and not strong set monotonicity) holds.  We  then apply our results to a couple of games that feature weak strategic complementarities. }

\subsection{Games with  Weak Strategic Complementarities}
Consider a normal-form game $\Gamma=(I, X, B)$, where $I$ is a finite set of players, $X:=\bigtimes_{i\in I} S_i$ is a Cartesian product of strategy sets $S_i$, and $B=(B_i)_{i \in I}$ is a Cartesian product of correspondences $B_i: S_{-i} \rightrightarrows S_i$ that we shall interpret as the ``best'' responses  for player $i$.
 We assume that $S_i$ is partially ordered for each $i$ and any Cartesian product, e.g., $X$ or $S_{-i}$, is partially ordered by the product order based on the relevant partial orders.    We further assume that each $S_i$  is  a  compact metric  space inducing a natural topology and let $X$ be endowed with the product topology. Finally, we assume that each $B_i$ is a nonempty- and closed-valued correspondence.  We shall refer to a game $\Gamma$ satisfying these properties {\bf regular}.  A strategy profile $s=(s_i)_{i \in I}$ is a Nash equilibrium if $s_i \in B_i(s_{-i})$ for every $i \in I$.  We let $\Eq(\Gamma)$ denote the set of all Nash equilibria of $\Gamma$.

 \begin{remark}\rm Note that we do not necessarily require the best response correspondence $B_i$ to maximize  a utility function $u_i: X \to \mathbb{R}$, or
 	\begin{align}
 	B_i(s_{-i}):= \arg\max_{s_i\in S_i} u_i(s_i, s_{-i}).\label{br-correspondence}
 	\end{align}
 	Indeed, our analysis also applies when, for instance, the best response correspondence is defined as the set of Pareto optimal choices by  a group of agents or by a  set of ``multi-selves'' in the case of an agent with incomplete preferences (see \Cref{prop:weak-strategic-complement} below). Rest assured that the best response correspondence in the traditional sense  (\ref{br-correspondence}) also works: it is  nonempty- and closed-valued, as long as $u_i$ is upper semicontinuous in $s_i$.\footnote{The closedness can be seen as follows. Let $v_i^*:=\max_{s_i\in S_i} u_i(s_i, s_{-i})$, and let $\{s_i^n\}_{n \in \mathbb N}$ be a sequence with $s_i^n\in B_i(s_{-i})$ for each $n=1,...$, converging to some $s^*_i\in S_i$.  Then, by the upper semicontinuity of $u_i$, $u_i(s^*_i,s_{-i})\ge \limsup_{n\to \infty}u_i(s^n_i,s_{-i})= v_i^*$.}  Either way,  \Cref{prop:weak-strategic-complement} below provides conditions on $u_i$ for best response correspondences to possess desired properties (giving rise to existence and wMCS of Nash equilibria).
 \end{remark}

We call a game  $\Gamma$  \textbf{a game with  upper weak strategic complementarities}, and  write $\Gamma\in \mathcal{G}_{+}$,  if it is regular and satisfies  the following conditions:
 \begin{description}
\item [(a)]  for each $i\in I$,  $B_i$ is upper weak set monotonic;
 	\item [(b)] there exists $\underline{s}=(\underline{s}_i)_{i \in I} \in  X$  such that for each $i$, $  s_i'  \in  B_i(\underline{s}_{-i})$ for some $s_i'\ge \underline{s}_i$.
 \end{description}
 Conditions  (a) and  (b) correspond to those required by the Fixed Point Theorem (\Cref{main}) for general correspondences. Condition (a) is  satisfied if players are economic agents  who possess the  preferences  we imposed for the comparative statics results  in  \Cref{sec:ind-choice} and \Cref{sec:pareto} as will be discussed later.  Condition (b) is  vacuously satisfied if there exists a smallest element in each player's strategy space, e.g., if the strategy space is a complete lattice.    Symmetrically, the class $\mathcal{G}_{-}$ of  {\bf games with lower weak strategic complementarities}  are defined analogously. We call $\mathcal{G}_{WSC}:= \mathcal{G}_{+}\cup \mathcal{G}_{-}$ \textbf{games with weak strategic complementarities.}

 With these preliminary concepts and results at hand, we now provide  general existence and comparative statics results:

\begin{theorem}  \label{thm:NE-existence}
\begin{description}
\item [(i)] Nash equilibria $\Eq(\Gamma)$ of a  game $\Gamma\in \mathcal{G}_{WSC}=\mathcal G_{+}\cup \mathcal G_{-}$ are nonempty.
\item [(ii)] Consider two games,  $\tilde \Gamma=(I, X,   \tilde B)$ and $\Gamma=(I, X,  B ) $. Suppose $\Eq(\Gamma)\ne \emptyset$ (for which $\Gamma\in \mathcal{G}_{WSC}$  is sufficient),  $\tilde \Gamma\in \mathcal{G}_{+}$, and  $\tilde B_i(s_{-i}) \ge_{uws}   B_i(s_{-i})$ for every $i \in I$ and $s_{-i} \in S_{-i}$. Then,   $\Eq(\tilde \Gamma)\ge_{uws} \Eq(\Gamma)$. (A symmetric result based on the lower weak set comparison also holds.)
  \end{description}
\end{theorem}

\begin{proof} Note first that  $B_i (s_{-i})$ is nonempty and compact.
Therefore, by  properties (a) and (b) of games in $\mathcal{G}_{+}$,  the mapping  $F: X\rightrightarrows X$  defined by
$F(s):= \{  s' \in X : s_i' \in B_i (s_{-i}),\forall i \in I \} $ satisfies  the requirement of the upper-monotonic self correspondence. Hence, by  \Cref{main}, we conclude that there exists a fixed point $s^*\in F(s^*)$, which means that the set of Nash equilibria is nonempty. Moreover, observe that $\tilde F(s):= \{  s' \in X : s_i' \in \tilde B_i (s_{-i}),\forall i \in I \} $ upper weak set dominates $F(s)$ for each $s \in X$. Thus, by  \Cref{thm:comparative statics}-(i),  part (ii) follows.
\end{proof}

Unlike here, it is more standard to specify payoff functions rather than best response correspondences as primitives of a game.  The next two propositions illustrate how one may leverage the results from  \Cref{sec:ind-choice} and \Cref{sec:pareto} to apply
\Cref{thm:NE-existence} for this more conventional definition of games.  Let us define an {\bf I-game}
$G=(I, X, u)$, where $u=(u_i)_{i \in I}$ is the profile of the players' payoff functions.\footnote{Here, I-game is a mnemonic  for individual choices, while P-game, to be defined later, is a mnemonic for  Pareto optimal choices.}  We say an I-game $G=(I, X, u)$ {\bf induces} a game $\Gamma=(I, X, B)$ if $B_i(s_{-i})$ maximizes player $i$'s payoff $u_i$ in the sense of (\ref{br-correspondence}), for each $i$ and $s_{-i}\in \bigtimes_{j\ne i}S_j$.  Below we also derive payoff implications of monotone comparative statics of games.  For this purpose, we will need to define the following property: for a subset $J\subset I$ of players,   $u_J=(u_i)_{i\in J}$ is {\bf payoff monotonic}  if  $u_i(s_i, s_{-i})$ is weakly increasing in $s_{-i}$, $\forall s_i\in S_i, \forall i\in J$.  This property   is natural in games with strategic complementarities,  as will be illustrated later.
Finally, we  let $\Eq(G)$ denote the set of equilibria in I-game $G$ (by a slight abuse of notation).

\begin{proposition}\label{prop:weak-strategic-complement}
	\begin{description}
		\item[(i)]  An I-game $G=(I, X, u)$ induces a regular game $\Gamma\in \mathcal{G}_{+}\cap\mathcal{G}_{-}$ and thus admits a Nash equilibrium if, for each player $i \in I$, $S_i$ is a compact  complete lattice, $u_i(\cdot, s_{-i})$ is upper semicontinuous for all $s_{-i}$,  and $u_i (\cdot, s_{-i}')\ge_{wI} u_i (\cdot, s_{-i})$ for any $s_{-i}' \ge s_{-i}$.
		\item [(ii)]  Suppose two I-games $G=(I, X, u)$ and $G'=(I, X, v)$ induce regular games in $ \mathcal{G}_{+}\cap\mathcal{G}_{-}$.  If $v_i (\cdot, s_{-i}) \ge_{wI} u_i (\cdot, s_{-i})$ for every  $i$ and $s_{-i}$, then $\Eq(G')\ge_{ws} \Eq(G)$.
		\item [(iii)] Fix I-games $G$ and $G'$ satisfying the conditions of (ii). If $v_J$  is  payoff monotonic, then  $v_J(\Eq(G'))\ge_{ws} v_J(\Eq(G))$, where $v_J(S')$ is the set  of payoffs  for players in $J$  corresponding to $S' \subset X$.

	\end{description}
\end{proposition}
\begin{proof} Part (i) follows from  \Cref{thm:wI},  \Cref{prop:ind-choice-property}, and  \Cref{thm:NE-existence}-(i); $B_i$ is then nonempty- and closed-valued and weak set monotonic.  Part (ii) follows from \Cref{thm:NE-existence}-(ii).  For Part (iii), define $\tilde v_J:=(\tilde v_i)_{i\in J}: X \to \mathbb{R}^{|J|}$ such that, for each $i\in J$,
	$$\tilde v_i(s):=\max_{s'\in S_i} v_i(s', s_{-i}).$$
Due to the payoff monotonicity, $\tilde v_J(s)=(\tilde v_i(s))_{i\in J}$ is a weakly increasing function of $s$. Further, for any $s\in X$,   $\tilde v_J(s)\ge v_J(s)$, and for any $s'\in \Eq(G')$, $\tilde v_J(s')=v_J(s')$.  We then conclude
$$v_J(\Eq(G'))= \tilde v_J(\Eq(G'))\ge_{ws} \tilde v_J(\Eq(G))\ge_{ws} v_J(\Eq(G)),$$
where the first inequality follows from combining Part (ii) and  \Cref{lem-ws-monotone-operator}.
\end{proof}

While the payoff comparison in Part (iii) is made  by fixing the payoff functions at $ v$, the set of equilibrium payoffs also increases in the weak set order---i.e., $v_J (\Eq (G')) \ge_{ws} u_J (\Eq (G))$---if we additionally assume $G'$  ``payoff dominates'' $G$ in the sense that $v_i (\cdot) \ge u_i (\cdot),\forall i \in I$.  An interesting implication of Part (iii) is that even without the payoff dominance,  the change of environment confers a  positive externality  for players whose payoffs are not directly affected:

\begin{corollary}  For two I-games $G$ and $G'$ satisfying the conditions of \Cref{prop:weak-strategic-complement}-(ii), we have $v_J(\Eq(G'))\ge_{ws} u_J(\Eq(G))$ if $v_i=u_i$ for all $i\in J$ and $u_J$ is  payoff monotonic.
\end{corollary}

Similar  to  an I-game, we define a \textbf{P-game}, $G=(I,X, u)$, played by collectives or by players with incomplete preferences, where $u:=(u_i)_{i\in I}$ with each $u_i=(u_{ij})_{j\in J_i}$ representing the payoffs of sub-players that comprise player $i$ and satisfying the condition in  \Cref{prop:P-properties-2}.\footnote{This condition is imposed  to  guarantee that each player $i$'s best response in a P-game, which is the set of Pareto optimal choices for the  collective $i$,  is closed-valued.}   We say that a P-game $G$ {\bf induces} a game $\Gamma=(I, X, B)$ if $B_i(s_{-i})$ are the Pareto optimal choices for $i$'s sub-players $\{ij\}_{j\in J_i}$.
By abuse of notation, we let $\Eq (G)$ denote the set of Nash equilibria in the game induced by P-game $G$.
\begin{proposition}\label{prop:weak-strategic-complement2}
	\begin{description}
		\item[(i)]  A P-game $G=(I, X, u)$ induces a regular game $\Gamma\in \mathcal{G}_{+}\cap\mathcal{G}_{-}$ and thus admits a Nash equilibrium if, for each  $i \in I$, $S_i$ satisfies the conditions required by  \Cref{thm:MCS-POC-2} (\Cref{thm:MCS-POC}, resp.) and  $u_i(\cdot, s_{-i}')$  dominates $u_i(\cdot, s_{-i})$ whenever $s_{-i}' \ge s_{-i}$, in the sense of  \Cref{thm:MCS-POC-2} (\Cref{thm:MCS-POC}, resp.).
		\item [(ii)]  Suppose two P-games $G=(I, X, u)$ and $G'=(I, X, v)$ induce games that belong to $ \mathcal{G}_{+}\cap\mathcal{G}_{-}$.  If $v_i(\cdot, s_{-i})$  dominates $u_i(\cdot, s_{-i})$ for each $i$ and $s_{-i}$ in the sense of  \
\Cref{thm:MCS-POC-2} or  \Cref{thm:MCS-POC}, then $\Eq(G')\ge_{ws} \Eq(G)$.
	\end{description}
\end{proposition}
\begin{proof} Part (i) follows from    \Cref{thm:MCS-POC-2},  \Cref{thm:MCS-POC},  \Cref{prop:P-properties-2}, and  \Cref{thm:NE-existence}-(i); $B_i$ is then nonempty- and closed-valued and weak set monotonic.  Part (ii) follows from  \Cref{thm:NE-existence}-(ii).
\end{proof}

\subsection{Applications}
We now present a couple of games with weak strategic complementarities and show how our results apply to these games.

\subsubsection{Generalized Bertrand Game}

Consider an oligopoly game played by firms $I$.  Each firm $i\in I$ chooses price $p_i\in P_i$ from a finite set $P_i\subset \mathbb{R}_+$ and sells $D_i(p_i, p_{-i})$ units at a cost given by  an increasing,  convex  function $C_i : \mathbb{R}_+ \to \mathbb{R}_+$.  We assume that, for each $i$, the demand function $D_i: P_i \times \prod_{j\ne i} P_j \to \mathbb{R}_+$ satisfies the following conditions:
\begin{itemize}
\item[\mylabel{itm:D1}{(D1)}]  $D_i$ is weakly decreasing in $p_i$ and weakly increasing in $p_{-i}$;
\item[\mylabel{itm:D2}{(D2)}]  For any $p_i<p_i', p_{-i}<p_{-i}'$, and $D_i(p_i,p_{-i})>0$,
\begin{align} \label{elasticity}
\frac{D_i(p_i', p_{-i})}{D_i(p_i, p_{-i})}\le \frac{D_i(p_i', p_{-i}')}{D_i(p_i, p_{-i}')}.
\end{align}
\end{itemize}
The monotonicity in $p_{-i}$ in \ref{itm:D1}  means that the firms' products are substitutes for each other. This substitute property is  strengthened by the condition \ref{itm:D2} which  implies that the demand function $D_i (\cdot, p_{-i})$ becomes more inelastic as $p_{-i}$ increases. We  call the I-game $G_{(C,D)}$, indexed by $C:=(C_i)_{i\in I}$ and $D:=(D_i)_{i\in I}$, a \textbf{generalized Bertrand game}, and write $G_{(C,D)}\in \mathcal B$, if \ref{itm:D1} and \ref{itm:D2} are satisfied.

The finiteness of $P_i$ is to guarantee the existence of a Nash equilibrium in pure strategies, but otherwise plays no  role. The continuous-price version of this game (which can be obtained say by shrinking the grid sizes aribrarily small) is comparable to, and in fact is more general than, the corresponding game consided  by \citet{milgrom/shannon:94}. This latter game assumes \ref{itm:D1} and a stronger version of \ref{itm:D2} without the qualifier ``$D_i(p_i,p_{-i})>0$,'' which assumes $D_i$   to be strictly positive and continuously differentiable at all profile $(p_i, p_{-i})$.  This difference actually matters since our class of games $\mathcal B$ includes a pure Bertrand game, whereas the class of games considered by  \citet{milgrom/shannon:94} does not.  More formally, we call  $G_{(C,D)}$  a \textbf{pure Bertrand game} if, for each $i$, $C_i(q)=c_i q$ for some $c_i\in [0,\max_{p_i\in P_i}p_i]$, and $$D_i(p)=\begin{cases}
\frac{1}{|\arg\min_{j \in I} p_j|},  & \text{if $p_i=\min_{j \in I} p_j$}\\
0 & \text{otherwise}.
\end{cases}$$

\begin{lemma} \label{lem:pure-bertrand} A pure  Bertrand game is a generalized Bertrand game.
\end{lemma}

Our next result shows that  any generalized Bertrand game induces a game with weak strategic complementarities defined by the lower weak set order.    To state this, define  firm $i$'s best response as
\begin{align} \label{bertrand-objective}
B_i(p_{-i}):= \arg\max_{p_i\in P_i} U_i(p_i, p_{-i}),
\end{align}
where $U_i(p_i, p_{-i}):=p_i  D_i(p_i, p_{-i}) - C_i (D_i (p_i,p_{-i})).$
\begin{lemma} \label{br-lws} If $p_{-i}<p_{-i}'$, then $B_i(p_{-i})\le_{lws} B_i(p_{-i}').$  Hence, any game in $\mathcal{B}$ induces a game in $\mathcal{G}_{-}$.
\end{lemma}
 In words,  a generalized Bertrand game induces a game with lower weak strategic complementarities. It is worth noting that  the pure Bertrand game   does \emph{not} induce a game with strategic complementarities, as defined by  \citet{milgrom/shannon:94}, since   each firm's best  response   is \emph{not} strong set monotonic; in fact, one can show that it is not even upper weak set monotonic.\footnote{To see it, consider, for instance, a pure Bertrand game with  two firms which incur constant marginal costs. If  the firm $j$ charges $p_j < c_i$, then any  $p_i > p_j$ is a best response for firm $i$.  However, if the firm $j$ increases its price to $p_j' > c_i$, then no  $p_i > p_j' $ can be a best response.}

We are now in a position to state the monotone comparative statics properties for the class $\mathcal{B}$ of generalized Bertrand games. For this purpose, consider any $G:=G_{(C,D)}$ and $\tilde G:=G_{(\tilde C, \tilde D)}$ both   in $\mathcal{B}$, where $(C,D)$ and $(\tilde C, \tilde D)$ are ordered as follows:   for each $i\in I$, (a) for any $p_i < p_i' $ and any   $p_{-i}$ such that  $D_i (p_i,p_{-i}) >0$, it holds that  $\tilde D_i (p_i, p_{-i}) >0$ and
\begin{align}
\frac{D_i(p_i', p_{-i})}{D_i(p_i, p_{-i})} \le  \frac{\tilde D_i(p_i', p_{-i})}{\tilde D_i(p_i, p_{-i})}; \label{elasticity1}
\end{align}  and  (b)    $ C_i (q') - C_i (q) \le \tilde  C_i (q') -\tilde  C_i (q),$ for any $q '  >  q$.  In words, the firms in game $\widetilde G$ face  more inelastic demands or higher marginal costs (or both) than the firms in game $G$.   The following result is a simple implication of \Cref{thm:NE-existence}.

\begin{corollary} \label{cor:generalized-bertrand} Both $\Eq(G)$ and $\Eq(\tilde G )$ are  nonempty.  Further,
	$\Eq(G)\le_{lws} \Eq(\tilde G)$.
\end{corollary}

To draw a payoff implication from the above comparative statics, let us consider firms facing higher constant marginal costs and higher demands going from $G$ to $\tilde G$: that is,
\begin{align*}  \label{eq:b'}
C_i (q) = c_i q \le \tilde c_i q =\tilde C_i (q) \;  \mbox{ and } \;  D_i (\cdot)\le \tilde  D_i (\cdot). \tag{$b'$}
\end{align*} The latter property  implies the payoff monotonicity for firms whose costs do not increase, so  we   obtain
\begin{corollary} \label{cor:generalized-bertrand-payoff}
If \eqref{eq:b'}   (in addition to (a) above) holds for every $i\in I$, then the set of  equilibrium profits for each firm $i$ with  $\tilde c_i = c_i$ in $\tilde G$ lower weak set dominates that in $G$.
\end{corollary}

\subsubsection{A Beauty Contest Game} \label{sssec:beauty}

Consider  a game played by multi-divisional organizations, each with (incomplete)   preferences described in  \Cref{non-strong-example}.  Specifically, there are $n$ players, $I=\{1,2,\dots,n\}$. Each player $i \in I$ chooses a two-dimensional action $s_i=(x_i,y_i)$ from $S_i:=[0,1]^2$.   As in \Cref{non-strong-example}, it is useful to interpret each player $i$ as a firm consisting of two divisions $i1$ and $i2$, and the action it chooses is an investment plan in two technologies, $A$ and $B$.
As before, each firm $i$ has an (incomplete) preference given by its divisions' payoff functions:
 \begin{align*}
 u_{i1}(x_i,y_i; \omega_i)&=-\left (x_i-2\omega^{A}_i \right )^2- \left (y_i - \omega^{B}_i  \right )^2,\\
 u_{i2}(x_i,y_i; \omega_i)&=-\left (x_i-\omega^{A}_i  \right )^2- \left (y_i - 2 \omega^{B}_i \right )^2,
 \end{align*}
except now that $\omega^{A}_i= \frac{1}{n-1}\cdot \sum_{j \neq i}x_j+\theta^A$ and $\omega^{B}_i= \frac{1}{n-1}\cdot \sum_{j \neq i}y_j+\theta^B$, where   $\theta:=(\theta^A, \theta^B)\in \mathbb{R}^2$ are parameters.  That is, the ``state''  $\omega_i=(\omega^{A}_i, \omega^{B}_i)$ now depends on other players'  actions as well as some exogenous parameters.

Using the interpretation of \Cref{non-strong-example} but  embedding it into a game context, each (multidivisional) firm $i$ prefers to ``match'' its investment plan $(x_i,y_i)$ to the average investment levels of the other firms as well as the exogenous parameter $(\theta^A, \theta^B)$ representing the desirability of the alternative technologies.  This feature of the game makes it a variant of the beauty contest game (see e.g., \citet{morris2002social}). An important ``twist'' added here is the possible preference incongruence between divisions within each firm over alternative technologies:   division $i1$ is biased toward $A$ while division $i2$ is biased toward $B$.  As mentioned earlier, a common approach for handling such incomplete preference is to assume that each firm $i$ chooses a Pareto optimal choice for its divisions, taking as given the expected investment plans by other firms $j\ne i$. Consequently, the best  response correspondence $B_i(s_{-i})$ for each firm $i$ is given by a Pareto optimal correspondence $\mathcal P_i(s_{-i};\theta):= P(u_{i1}(\cdot; \omega_i), u_{i2}(\cdot; \omega_i))$.  Our (modified) beauty contest game  is then denoted by $G_{\theta}=(I,  (S_i)_{i\in I} ,(\mathcal P_i(\cdot, \theta))_{i\in I} )$, indexed by the parameter $\theta$.

We verified in \Cref{non-strong-example} that $u_{ik}(s_i; \omega_i)$ is supermodular in $s_i$, $\forall\omega_i$, $\forall i\in I, k=1,2$,   and $u_{ik}(\cdot; \omega_i')$ increasing-differences dominates $u_{ik}(\cdot; \omega_i)$, whenever $\omega_i'>\omega_i$. Thus, by \Cref{thm:MCS-POC},  $\mathcal P_i(s_{-i};\theta)$ is weak set monotonic in $s_{-i}$ and in $\theta$.  Furthermore, one can check that $u_i$ satisfies the properties required by   \Cref{prop:P-properties-2}, implying that $\mathcal P_i(s_{-i};\theta)$ is closed for each $s_{-i}$.  We thus conclude that the beauty contest game $G_{\theta}$ is a P-game and induces a game of weak strategic complementarities.  Then, by  \Cref{prop:weak-strategic-complement2}, the following result is immediate.

\begin{corollary} If  $\theta'>\theta$, then  $\Eq(G_{\theta'})\ge_{ws}  \Eq(G_{\theta})$.
\end{corollary}

 \section{Application to Matching Theory}\label{sec:matching}

In this section, we apply our theory to matching problems. As we will demonstrate below, the techniques we developed in the  previous sections prove useful for analyzing stable matching under weaker assumptions than  have been employed by the existing research. We first establish the existence of a stable matching building on our fixed-point theorem (\Cref{main}).  We then  obtain comparative statics of stable matchings based on our general wMCS result for fixed points (\Cref{thm:comparative statics}).  Finally, we provide a couple of applications.

The main departure from the existing literature is the generality of agents' choice correspondences that we allow for. Specifically, we relax the two main assumptions in the literature; WARP and substitutability. These relaxed assumptions allow for indifferences or even incompleteness of preferences. This generality plays an important role in our applications.

\subsection{Model and Results}
We begin by presenting our model. There are a finite set $ F$ of firms and a finite set $W$ of workers, as well as  a finite set $X$ of contracts.   Each contract $x \in X$ is associated with one firm $x_F \in  F$ and one worker $x_W \in W$.   We will often write $x$ to denote a singleton set $X' = \{ x \}$.
  Given a  set  $X' \subset X$ of contracts, let  $X'_f=\{x \in X' : x_F=f\}$ and $X'_w=\{x \in X' :  x_W=w\}$ denote the sets of  contracts firm $f$ and worker $w$ are involved with within $X'$,  respectively.
A set of contracts $X' \subset X$ will be called  an \textbf{allocation} if it contains at most one contract for each worker.

Each agent $a \in F \cup W$  is endowed with a \textbf{choice correspondence}: $C_a: 2^X \rightrightarrows 2^X$ where,  for each  $X' \subseteq X $, $C_a  (X')$ is a nonempty family of subsets of $X_a'$. Any element of $C_a (X')$  represents a set of contracts agent $a$ chooses from $X'$.
The choice correspondence $C_a$ induces  the \textbf{rejection correspondence} $R_a: 2^X  \rightrightarrows 2^X$, defined by $R_a(X')=\{ Z  :  Z =X_a' \setminus Y   \mbox{ for some } Y \in C_a(X')\}$.

For any pair of allocations $X'$ and $X''$, we say
that agent $a$ \textbf{weakly  prefers} $X''$ to  $X'$ if $X''_a \in C_a (X'_a \cup X_a'')$, and write  $X'' \succeq_{a} X'$.\footnote{This is the so-called ``Blair order''  introduced by \cite{blair:88}.}   We say  that $a$   \textbf{strictly prefers} $X''$  to $X'$ if $X'' \succeq_{a} X'$ but not  $X' \succeq_{a} X''$, and write $X'' \succ_a X'$.

We focus on the many-to-one matching setup
by assuming that the choice correspondence of each  worker $w$ satisfies the following properties:  for any $X' \subseteq X$,
(i) $X'' \in C_w(X')$ implies    $|X''| \le 1$;  and (ii) $X'' \in C_w(X')$ if $\emptyset \nsucc_w X''$ and
 $ \{x'\} \nsucc_w  X''$  for  any $X'' , \{x'\} \subset X_w'$.\footnote{ We note that our characterization of stable matchings, \Cref{claim-matching-characterization}, does not hold without condition (ii). However, the ``if'' direction of that theorem holds without it, and hence so does
 the existence of a stable matching, \Cref{thm:stable-match}. Details are available upon request.}  In words, each element of a worker's choice correspondence must be a singleton contract, possibly a null set (i.e., the worker has a unit demand), and a contract that is not dominated by any other  contract (including remaining unemployed) must be included in the choice correspondence.

An economy is summarized as a tuple $\Gamma=(F, W, X, (C_a)_{a \in F \cup W} )$.  An allocation $Z$ is \textbf{stable} if
\begin{enumerate}
\item (Individual Rationality) \label{stabilitycondition1}  $Z_a \in C_a (Z)$ for every $a \in F \cup W$,  and
\item (No Blocking Coalition) \label{stabilitycondition2}  $Z_f \in C_f (Z \cup U(Z) )$ for every  $f \in F$, where   $U(Z) := \{ x\in X   :   x \succ_{x_W}  x' ,\forall x' \in  Z_{x_W}   \}$.\footnote{In \Cref{alternative-stability-concept} of Supplementary Appendix, we consider an alternative  notion  of stability and its relation with the present stability notion under Sen's $\alpha$ or WARP.}
\end{enumerate}

The key method for analyzing stable allocations is to associate them with fixed points of a suitably defined correspondence (see \cite{adachi:00}, \cite{fleiner:03}, \cite{echenique/oviedo:04,echenique/oviedo:06}, and \cite{hatfield/milgrom:05}, for example). WARP has been crucial for this purpose.\footnote{See  \cite{hatfield/milgrom:05},  \cite{che/kim/kojima:19}, and  \cite{aygun/sonmez:12}, among others. We note that authors have invoked WARP under different names; the first two sets of authors call it Revealed Preference, while the last set of authors, who highlight the importance of the condition, call it Irrelevance of Rejected  Contracts.}  Formally,  a preference relation for agent $a\in F\cup W$ satisfies WARP if and only if the associated choice correspondence $C_a$  satisfies the following two conditions (see \citet{kreps1988notes}, for instance):
\begin{enumerate}
	\item  {\bf Sen's}  $\alpha$: \label{alpha}  $Y\in C_a(X'') \mbox{ and } Y\subset X'\subset X''  \Longrightarrow  Y\in C_a(X'),$  and
	\item {\bf Sen's}  $\beta$:  \label{beta}  $Y, Y'\in C_a(X')$ and $Y\in C_a(X'')$ for $X'\subset X''  \Longrightarrow  Y'\in C_a(X'').$
\end{enumerate}
In words, Sen's $\alpha$ states that an optimal choice from a ``bigger'' set must be an optimal choice from a ``smaller'' set that contains it.   Sen's $\beta$ attributes non-uniqueness of choice to indifferences:  if  multiple alternatives are optimal from a smaller set and one of them is still optimal from a bigger set, the other(s) must also be optimal from the bigger set.  While the former remains compelling, the latter can easily fail in the context of multidivisional organizations or of incomplete preferences, as the following example demonstrates.

\begin{example}\label{example-incompleteness}
Consider a firm $f$ with two divisions,  $\delta$ and $\delta'$. The firm is subject to a budget constraint that compels it to hire at most one worker across the divisions, but the firm does not have strict preferences over which division hires a worker when both divisions have applicants.     Each division has its own preferences over the workers.
There are  3 workers, $w$, $w'$, and $w''$, who are all acceptable to both  divisions, and division $\delta'$ prefers  $w''$ to $w'$. Then, if workers
$w$ and $w'$ apply to divisions $\delta$ and $\delta'$, respectively, then the choice of the firm from this set of applications $\{(w,\delta), (w',\delta')\}$ would be either $(w,\delta)$ or $(w',\delta')$, where $(w,\delta)$, for instance,  denotes a contract specifying a matching between $w$ and $\delta$. If $w''$ applies to $\delta'$ in addition, then the firm faces a  set of applications $\{(w,\delta), (w',\delta'), (w'',\delta')\}$ and chooses either $(w,\delta)$ or $(w'',\delta')$. Note that $(w',\delta')$ is no longer optimal since the newly available contract $(w'',\delta')$ dominates it for division $\delta'$. At the same time, the other contract $(w,\delta)$ remains optimal since the new contract is not comparable to it.  This choice behavior is quite natural for a multi-divisional organization.  However,   this constitutes a violation of Sen's $\beta$, so it violates WARP  and thus cannot be rationalized by any  complete (possibly weak)  preference relation.
\end{example}

As seen in this example,   violations of Sen's $\beta$  may naturally arise in organizations with multiple divisions because the organization simply  lacks a criterion to compare placement in different divisions (e.g., between $(w,\delta)$ and $(w'',\delta')$ in the above example). We later illustrate that a similar violation may arise in  matching problems with distributional constraints such as Japanese medical match \citep{kamakoji-basic}.  Motivated by these observations,  in what follows we relax WARP by dispensing with Sen's $\beta$.   We will only assume  Sen's $\alpha$, which is compatible with a wide variety of preferences with indifferences or even incompleteness.\footnote{\citet{eliaz-ok06} introduce an axiom called weak axiom of revealed non-inferiority (WARNI) that is  consistent with incomplete preferences. In  \Cref{warni-subsection} of Supplementary Appendix, we show that Sen's $\alpha$ is implied by  WARNI.}

We now proceed with a fixed-point characterization of stable allocations. Let $C_F(X') :=\{\bigcup_{f \in  F} Y_f  :  Y_f \in C_f(X'),\forall f \in  F \}$ and $R_F(X') :=\{\bigcup_{f \in   F} Y_f  :  Y_f \in R_f(X'),\forall f \in  F\}$.  Define $C_W$ and $R_W$  analogously.  Then, a fixed-point mapping (or correspondence)  $T: 2^X \times 2^X \rightrightarrows 2^X \times 2^X$ is defined as follows:  For each $(X', X'')\in 2^X \times 2^X$,   $ T  (X',X'') = ( T_1 (X''), T_2 (X'))$, where
 	\begin{align*}
 	T_1 (X'') & =      \{  \tilde X  \in 2^X  :  \tilde X  = X \backslash \tilde Y     \mbox{ for some } \tilde  Y \in R_W (X'') \},  \\    T_2 (X')   &=  \{  \tilde X  \in 2^X  :   \tilde X  = X \backslash \tilde Y     \mbox{ for some }\tilde   Y \in R_F (X') \}. \end{align*}
Intuitively, we can think of $T$ as iterating on sets $X'$ and $X''$ of contracts available respectively to firms and workers.   For each pair $(X', X'')$,
$T_1$ returns sets of contracts that are available to the firms after removing contracts workers reject out of $X''$,  while $T_2$ returns  sets of  contracts that are available to the workers after removing contracts rejected by firms out of $X'$. Mapping $T$ is similar to fixed-point mappings used in the existing literature such as \citet{hatfield/milgrom:05}, except that it is generalized to handle  choice correspondences rather than choice functions.

 	\begin{theorem} \label{claim-matching-characterization}
 		Suppose that $C_a$ satisfies   Sen's $\alpha$ for each $a \in F \cup W$.    Then, there exists a stable allocation $Z$  if and only if there exists $(X',X'')$ that  is a fixed point of $T$, where $Z \in C_F ( X' ) \cap C_W (X'')$.
 	\end{theorem}

 As will become clear, our fixed-point characterization is crucial for both existence and comparative statics of stable allocations.
 We first use the characterization together with \Cref{main} to  establish existence of stable allocations. To this end, we consider a partially ordered set $( 2^X, \ge)$, where the order $\ge$ is given by ``set inclusion'' operator; i.e., $X'' \ge X'$  if $X'' \supset X'$. The associated upper  and lower weak set orders over families of sets of contracts are defined based on this primitive (set inclusion) order.  The monotonicity of correspondence $f: 2^X \rightrightarrows 2^X$ is  defined accordingly:  that is,  $f$ is upper weak set  monotonic if  for $X'\subset X'' \subset X$,   $Y'\in f(X')$ implies  there exists $Y'' \supset Y'$ such that $Y'' \in f(X'')$; and similarly for lower weak set monotonicity.
 For the product  set $2^X \times 2^X$, we endow the following  order:  $(X'',Y'') \ge (X', Y')$ if $X'' \supset X'$ and $Y'' \subset Y'$.  The monotonicity of  correspondence $f: 2^X \times 2^X \rightrightarrows 2^X \times 2^X$  is then  defined according to this order.

 The next step is to invoke an appropriate assumption on agents' choice correspondences to ensure that $T = (T_1, T_2)$ is weak set monotonic.   Specifically, we assume that, for each $a \in F \cup W$, the choice correspondence $C_a(\cdot)$ is \textbf{weakly substitutable}, i.e.,   \emph{$R_a$ is weak set monotonic.}  A standard notion of substitutability considers a choice function---rather than a choice correspondence---and requires the associated rejection function to be monotonic (e.g., \citet{hatfield/milgrom:05}).  One way to generalize this notion to choice correspondences would be to require that the rejection correspondences be  complete-sublattice-valued and monotonic in the strong set order---the condition \cite{che/kim/kojima:19} labels substitutability.  However, this condition proves too restrictive to accommodate even the most common form of indifferences:
 \begin{example} \label{ex:substitutability} \rm  A firm is indifferent to hiring one of the three workers, $x$, $y$, and $z$, to fill a single position. (Formally, $x$, $y$, and $z$ refer to contracts.) The resulting  rejection correspondence is not sublattice-valued:   	$R_f(\{x,y\})=\{\{x\},\{y\}\}$, but $\{x\}\vee\{y\}=\{x,y\} \not\in R_f(\{x,y\})$.  It is not strong set monotonic, either: $\{ y,z \}  \in R_f (\{ x,y,z\})$, $\{ x \} \in R_f (\{x,y\})$, so $\{y,z\} \vee \{x\} = \{x,y,z\} \notin R_f (\{x,y,z\})$.  We thus conclude that $C_f$ is not substitutable.  Nevertheless, $R_f$ is weak set monotonic, as can be checked easily, so $C_f$ is weakly substitutable.
 \end{example}
It turns out that weak substitutability is sufficient for existence,  as we show now.

\begin{theorem}  \label{thm:stable-match} Suppose that $C_a$ satisfies  Sen's $\alpha$  and   weakly substitutability for each $a \in F \cup W$.\footnote{For existence of a stable matching, we could require weak substitutability only for the firm side, specifically, upper weak set monotonicity of firms' rejection correspondences.  For the worker side, Sen's $\alpha$ together with the unit-demand assumption   implies the lower weak set monotonicity of a worker's rejection correspondence.  To see this, consider any $X'\supset X''$ and let $x \in C_w(X')$ so that $X'\setminus \{x\}\in R_w(X')$.  If $x\in X''$, by Sen's $\alpha$,  $x \in C_w(X'')$, so $X''\setminus \{x\}\in R_w(X'')$, leading to $R_w(X')\ge_{lws} R_w(X'')$.    If $x\not\in X''$, then $X'\setminus \{x\}\supset X''$, so we trivially have $R_w(X')\ge_{lws} R_w(X'')$.  Combining the upper weak set monotonicity of firms' rejection correspondences and the lower weak set monotonicity of workers' rejection correspondences yields upper weak set monotonicity of $T$ according to our order, which is sufficient for existence of its fixed points.  We assume the current (stronger) conditions since they are used for \Cref{stable-match-comparative-statics}.}   Then, a stable  allocation exists.\end{theorem}
 \begin{proof}  See  \Cref{app:matching-section-1}.
\end{proof}

To the best of our knowledge, the  current existence result is the most general of its kind, requiring very weak preferences conditions that allow for both indifferences and incompleteness.
A number of papers---for instance,  \cite{erdil/ergin:08}  and \cite{abdulkadiroglu/pathak/roth:09}---consider  matching under responsive preferences with ties on the side of schools, but  tie-breaking allows  the problem to be reduced to the case with strict priorities in those cases.\footnote{One may wonder if it is always possible to work with choice functions obtained after breaking ties in some manner, rather than
using our general approach based on choice correspondences.  \citet{erdil2019efficiency} show that the tie-breaking approach does not work in general. Specifically, their Remark 4 shows that a weakly substitutable choice correspondence does not necessarily have any substitutable tie-breaking.}
\cite{che/kim/kojima:19} and \citet{erdil2019efficiency} establish the existence of a stable matching with choice correspondences that satisfy weak  substitutability and WARP.\footnote{\citet{erdil2019efficiency} invoke admission monotonicity and rejection monotonicity, which are equivalent to weak substitutability. One can also check their consistency condition is  equivalent to WARP.}  Our result is a generalization of theirs because our condition weakens WARP to Sen's $\alpha$.

An astute reader may notice that no  claim is made in the above theorem about the existence of worker- and firm-optimal stable allocations, which are often shown to exist under substituble preferences.   Indeed, such ``side-optimal'' stable allocations are not guaranteed to exist in the presence of indifferences, let alone incompleteness.\footnote{Recall \Cref{ex:substitutability}.  Suppose every worker prefers to work for $f$ instead of being unemployed.  Then, there are three stable allocations; $f$ hiring any one of three workers.  None of them is worker optimal.}  Formally, this can be attributed to the fact that our fixed-point theorem (\Cref{main}) does not guarantee the lattice structure for the fixed-point set.

We now turn to our  main  result:  monotone comparative statics of stable allocations.  To this end, we say that choice correspondence $C_a$ is \textbf{weakly more permissive} than $C'_a$ if, for each set of contracts $X'$,
$R_a (X') \le_{ws} R_a' (X')$. In words, an agent  with $C_a$ rejects fewer contracts than an agent with $C'_a$. We let $\succeq_a$  and $\succeq_a'$  denote the (possibly incomplete) preferences associated with $C_a$ and $C_a'$, respectively; and similarly for $T$ and $T'$.\footnote{Recall that $\succeq_a$ and $\succeq_a'$ are  the  preferences defined by Blair (partial) order.}

\begin{theorem}\label{stable-match-comparative-statics}
 Suppose that $C_a$ satisfies  Sen's $\alpha$  and   weakly substitutability for each $a \in F \cup W$. Consider two economies $\Gamma=(F, W, X, (C_a)_{a \in  F \cup W} )$
and $\Gamma'=(F, W, X, (C_a')_{a \in F \cup W} )$ such that $C_w$ is weakly  more permissive than $C_w'$ for each $w\in W$ while $C'_f$ is weakly more permissive than $C_f$ for each $f \in F$. Then,
\begin{enumerate}
\item [(i)] for each stable allocation $Z$ in $\Gamma$, there exists a stable allocation $Z'$ in $\Gamma'$ such that $Z_f  \succeq_f Z_f'$  for each $f \in F$ and $Z_w' \succeq_w' Z_w$ for each $w\in W$, and
\item [(ii)] for each stable allocation $Z'$ in $\Gamma'$, there exists a stable allocation $Z$ in $\Gamma$ such that $Z_f \succeq_f  Z_f'$ for each $f \in F$ and $Z_w' \succeq_w' Z_w$ for each $w \in W$. \end{enumerate}
\end{theorem}
 \begin{proof}  See  \Cref{app:matching-section-2}.
\end{proof}

The basic idea of the proof is to utilize the fixed-point characterization of stable allocations by the mapping $T$. We first establish that the fixed-point mapping ``shifts up'' in the weak set order sense  with the change of choice correspondences.  By \Cref{thm:comparative statics}, this implies that the set of fixed points ``increases'' in the weak set order.
This gives rise to the desired comparative statics properties of stable allocations.

\Cref{stable-match-comparative-statics} generalizes various comparative statics results in the existing literature from the cases of choice functions to choice correspondences.\footnote{There are many  comparative statics results for choice functions in various formulations and generality. See   \citet{gale/sotomayor:85b,gale/sotomayor:85}, \citet{crawford:91}, \citet{konishi/unver:05}, \citet{echenique/yenmez:12}, and \citet{chambers/yenmez:13}, for instance.}
 As such, it implies a number of standard results.  For instance,  a stable allocation becomes more favorable to one side when it becomes more ``scarce'' or when there is more competition from the other side:
 \begin{corollary} \label{cor:change-of-economy}
Suppose that a worker exits the market
 or a new firm enters the market.  Then, for each stable allocation in the original market, there exists a stable allocation in the new market in which all the remaining workers are weakly better off and all the existing firms are weakly worse off.  A symmetric  result, though in the opposite direction, holds if a worker enters the market or a firm exits a market.
 \end{corollary}
 The entry/exit of agents in this Corollary corresponds to their choice correspondences  becoming more/less permissive.  For instance, an agent exiting a market corresponds to that agent having a less permissive correspondence than before (in fact, she rejects every contract). Therefore, all remaining agents from the same side become weakly better off and those from the opposite side become weakly worse off in some new stable allocation by \Cref{stable-match-comparative-statics}.

Aside from these standard comparative statics, the generality of  \Cref{stable-match-comparative-statics} enables us to obtain new kinds of comparative statics results. For instance, if the internal constraint of a multidivisional firm is relaxed (e.g., a hiring budget increases), then  all the workers are made weakly better off while all the other firms are made worse off in at least one new stable matching.   A similar monotone comparative statics holds in matching with constraints.  Suppose, for example, in the Japanese medical matching, the maximum number of doctors that can be hired by  hospitals in a region increases.   Then, the choice correspondence representing that region  becomes more permissive, so the doctors are weakly better off in at least one (weakly) stable matching.   These new comparative statics results are formalized and proven in  \Cref{multidivision-subsection} and \Cref{constraints-subsection} of Supplementary Appendix.

\subsection{Applications} \label{sec:matching-applications} The present framework  subsumes environments beyond those analyzed in existing research. Let us describe two applications of our approach in informal manners here.  The formal treatments are relegated to \Cref{multidivision-subsection} and \Cref{constraints-subsection} of Supplementary Appendix.
\begin{enumerate}
\item \textbf{Multidivisional Organization:} Consider an organization, such as a large firm, that has multiple divisions.\footnote{This class of choice correspondences considered here is similar in spirit to a multidivisional choice function with flexible allotments analyzed by \citet{hatfield2017stable}, but neither  is more general than the other.}   Such an organization may face a total hiring budget and may decide to  allot positions across divisions within that budget.   Given the allotted positions, each division chooses the best applicants according to its own linear preference order. The firm with multiple divisions described in \Cref{example-incompleteness} is a concrete example.

In \Cref{multidivision-subsection} of the  Supplementary Appendix, we  construct a choice correspondence that captures these features.
The organization's choice is not necessarily described as a function, but as a correspondence---the organization may find indifferent or incomparable two allotments of positions across different divisions as long as both of them satisfy the organization's internal constraint. This feature leads to the failure of conditions assumed in existing studies, but we show that the organization's choice correspondence still satisfies both Sen's $\alpha$ and weak substitutability. Hence,  \Cref{thm:stable-match}  and \Cref{stable-match-comparative-statics}  allow us to establish  the existence of a stable matching as well as  a wMCS property.
\item \textbf{Matching with Constraints:} Consider a problem of matching with constraints, such as medical match faced with a government-imposed cap on the number of doctors in each region or in each medical specialty. \citet{kamakoji-concepts} present a model of matching with constraints, introduce a concept  called weak stability, and establish the existence of a weakly stable matching.\footnote{Alternative concepts of stability, including weak stability,   are defined by \citet{kamakoji-basic,kamakoji-concepts,kamakoji-iff}. Weak stability has advantages over others such as existence under mild conditions and an axiomatic characterization \citep{kamakoji-concepts}.}

We prove the existence of a weakly stable matching as a corollary of \Cref{thm:stable-match}. The basic idea of the proof is to associate the model of matching with constraints with an auxiliary model of matching with contracts between doctors and the ``hospital side,'' a consortium that jointly chooses among applicants to different hospitals.\footnote{To our knowledge, \citet{kamakoji-basic} is the first to associate matching with constraints to matching with contracts, and this technique has been used in subsequent studies such as \citet{kamakoji-iff,kamakoji-policy}, \citet{goto:14}, \citet{goto:14b}, and \citet{kojima-tamura-yokoo}. However, our approach is different from theirs in at least two respects.  First, all the other works focus on choice functions rather than  choice correspondences, making it impossible to connect their approach to weak stability. Second,
the class of constraints we consider are more general than those studied in any of the above papers.  Both of these differences are crucial for our analysis, and our analysis capitalizes heavily  on the generality of the present model which allows for choice correspondences under Sen's $\alpha$ and weak substitutability.
} Intuitively, we exploit the fact that the hospital side's choice behavior under constraints works in a manner that is analogous to that of a multidivisional organization. Choice behavior of the hospital side is not necessarily a function but a correspondence because there is some degree of freedom as to how many positions are allotted to different hospitals given the joint constraint.  These features can be readily incorporated into our model.
More formally, we verify that the hospital side's choice correspondence satisfies both  Sen's $\alpha$ and weak substitutability.
Moreover, we  establish that a matching is weakly stable in the given model of matching with constraints if and only if a corresponding allocation in the auxiliary model of matching with contracts is a stable allocation.
These results imply that a weakly stable matching exists.

While the existence of a weakly stable matching has been established before, our technique allows us to obtain a novel  comparative statics result with respect to changes in constraints.  While such results were hitherto unvailable, they are a natural consequence of our approach and can be obtained as a corollary of \Cref{stable-match-comparative-statics}.
\end{enumerate}

 \section{Concluding Remarks}

 We have developed a theory of monotone comparative statics based on  weak set order.  The theory together with a novel fixed point theorem with a general monotonic correspondence allowed us to weaken the conditions, and thus expand the scope, of comparative static predictions in a variety of contexts, including individual choice, Pareto optima, game theory and matching theory.

 One could extend the current work in several ways.  Some conditions such as those in \Cref{thm:MCS-POC-2} are sufficient for monotone comparative statics but the extent to which they are necessary is unknown; one could strive to establish their necessity or  further weaken them.  Some tight conditions, such as those provided in \Cref{sec:ind-choice}, could be operationalized further by  finding easier-to-check, possibly stronger, conditions.  We suspect that such an operationalization would be made possible with  further assumptions on the structural properties of the underlying environment;  \cite{quah2007comparative}  and \cite{dziewulski2021comparative} which make use of the geometric structure of constraint sets  and  objective functions, provide examples of such an approach.

Another avenue for extension is to incorporate uncertainty facing individuals in an individual choice or a game context.\footnote{\cite{athey2002monotone}, \citet{quah/strulovici:09}, and \citet{quah2012aggregating} deal with MCS of individual choice under uncertainty, and \cite{vives:90}, \cite{van2007monotone},  \cite{mekonnen2018bayesian}, and \cite{jensen2018distributional} deal with MCS of Bayesian Nash equilibria of games with incomplete information.} Accommodating uncertainty in an MCS analysis  requires a suitable aggregation property which ordinal MCS conditions often fail. Consequently, our weakening of these conditions would likely face similar difficulties.  Nevertheless, weakening the notion of MCS from strong set order to weak set order could make further progress possible, as has been illustrated in the present paper; we hope fruitful research awaits in this area.

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