text:
Q: get website information by URL in Node With reference to the stack overflow question:

Get website information with jQuery

I want to accomplish a similar task in Node.js to get the information of the website like: Image and Name when I provide the url.
I am aware about the HEAD Request in Node but it does not pull that information, rather it returns HTML Headers.
How can I do so?
A sample example can be when we send a url in Slack as a message it pull the Image and meta and displays automatically.
With a small research I ended up making a request like this:
var request = require('request');
request("http://stackoverflow.com", {method: 'HEAD'}, function (err, r es, body){
console.log(res.headers);
});

and got basic header info:
{"cache-control":"private","content-length":"245925","content-type":"text/html; charset=utf-8","x-frame-options":"SAMEORIGIN","x-request-guid":"02d28d15-13e3-4d2e-8495-fa0ff5922224","accept-ranges":"bytes","date":"Mon, 06 Mar 2017 14:44:52 GMT","via":"1.1 varnish","connection":"close","x-served-by":"cache-lcy1130-LCY","x-cache":"MISS","x-cache-hits":"0","x-timer":"S1488811492.680621,VS0,VE89","x-dns-prefetch-control":"off","set-cookie":["prov=fbdc85f1-c72a-bcc4-8538-2572d0580c74; domain=.stackoverflow.com; expires=Fri, 01-Jan-2055 00:00:00 GMT; path=/; HttpOnly"]}

But it does not have image and Name information which I am precisely requiring.


1.Go to myblueprint.ca/peel – Login (if you forget your password, submit your email and request a new password).
2. You will be prompted to enter your OEN. This is a 9-digit number found on your student card or an old report card.
3.Go to "High School Planner" and complete the planner for next year – ensure the course codes are correct and you have the prerequisites as these are the courses submitted for course selection.** Plan as far ahead as possible.
4.Submit the courses you chose in your High School Planner. To submit your courses, click the green submission button on the top right of your High School Planner. **The program will guide you step by step through the course selection and submission process.
5.Print your course selection sheet and take it home to be signed. Return the signed sheets to your teacher as soon as possible.
6.Complete your IPP as much as possible by clicking on the IPP tab at the top right of the screen. The IPP tab should say 100% complete.
Click HERE to View Your Student Profile.
Course selection is a continual process and should be based on an individual student's interests, abilities and goals. To assist with this process, students are strongly encouraged to speak to their teachers about proposed classes, read course descriptions online or in the Common Course Calendar, talk to their counsellors and parents, and speak to students who have previously taken these classes. Students choose their classes online—a password is required which can be obtained through the school.
Once a student chooses a course, they are expected to remain enrolled in that course, unless there are special circumstances. Students are encouraged to speak to their teachers and counsellors, should any difficulties and/or challenges arise.


Can Any Of the April 2019 Movie Releases Beat Avengers: Endgame At the Box Office?
Some say there are no stupid questions, but in this case, that is a stupid question. Avengers: Endgame will rule the April 2019 box office, but before it releases in theaters, there are other movies to spend your money on.
It's April 2019 movie lovers! Do you know what that means? Avengers: Endgame releases in theaters at the end of the month. It is by far the most anticipated April 2019 movie release, for those who want to know how the non-dusted Avengers are going to bring back the fallen heroes and right what Thanos did to the world — or fail trying. But Avengers: Endgame is not the only movie releasing in April 2019; in fact, there a ton of movies for fans of superheroes and those who prefer other types of films (or both!).
As for the box office winner, the April 2019 movie releases before Avengers: Endgame on the 26th don't stand a chance to make as much money as the sequel to Avengers: Infinity War, but they may still fare well. By release date, here are the top movie contenders for each weekend in April 2019.
The first weekend of April will be quite a battle for box office winner. The two likely top spots, given the live-action Dumbo did not perform all that well opening weekend and will likely drop quite easily, are Pet Sematary and Shazam!. Two questions linger: Will audiences want to see a remake of Pet Sematary? and Can a kid-focused Marvel movie like Shazam! bring in as much money as other MCU movies? I'm betting on Shazam! taking the weekend because even if it seems more geared towards kids, they love going to the movies, and will pack a theater. The second and third spots will likely belong to Dumbo and Pet Sematary, with all the rest, including The Best of Enemies, taking what is left.
As much as I would love to see High Life take the box office, it's not going to happen. Shazam! will likely keep the top spot but Missing Link and Hellboy may give it some good competition. It's a tad surprising that Hellboy has this release date because it will be incredibly difficult for it to make a big splash at the box office considering the competition and looming threat of Avengers: Endgame. It makes you wonder: Is Hellboy terrible? And then there is Little, which could prove to be a surprising money maker if it is funny and word of mouth spreads fast (remember, comedies don't need $100 million dollar opening weekends to be a success).
It had to occur to many of us that any film studio who would release a movie on the same weekend as Avengers: Endgame or right before it must be insane. Well, maybe the studio heads are not because what comes out in theaters after the April 12 weekend is sparse, to say the least. Holdovers will rule the box office until next weekend when the entire world goes to see Avengers: Endgame.
Did developers build new movie theaters around the world to accommodate the opening weekend of Avengers: Endgame? They should have. Get your tickets now – this movie is going to break box office records we didn't even know were breakable.
That does it for the very much predictable April 2019 movie box office. May should be more interesting. And don't forget what is streaming in April on Netflix and Hulu (there's plenty of new original movies and series).


text:
UPDATE: Head-on M-28 crash under investigation in Alger County
Published: Nov. 23, 2021 at 8:06 PM EST|Updated: Nov. 24, 2021 at 4:30 PM EST
MUNISING TOWNSHIP, Mich. (WLUC) - UPDATE at 4:30 p.m. Wednesday:
Investigators have not released any new information Wednesday about a head-on crash on M-28, but a family member of one of the drivers says the injuries were serious.
Cassie Ploof says her uncle is hospitalized at UP Health System-Marquette. He's in intensive care on a ventilator and is sedated. Ploof says her aunt and cousin are taking turns visiting. Ploof asks anyone who is able to help her family at this time to contact her at 906-360-9306.
This story will be updated when more information is available.
A head-on collision east of Wetmore Tuesday morning sent multiple people to the hospital.
According to the Alger County Sheriff's Office, the crash happened on M-28 about three miles east of Wetmore around 10:30 a.m. Investigators say an eastbound vehicle lost control on icy roads and went into the path of a westbound vehicle. They collided head-on.
The Alger County Sheriff's Office says "all parties involved" were taken to UP Health System-Marquette by Alger EMS. Their conditions are stable, investigators say. A press release from investigators did not say how many people were in the vehicles.
The Alger County Sheriff's Office and Michigan State Police investigated the crash. Other responding agencies included the Munising Twp. Fire Department.

redpajama_set_name: RedPajamaCommonCrawl


text:
\section{Introduction}

A common economic problem is deciding where a public facility should be located to service a population of agents with heterogenous preferences. For example, a government needs to decide the location of a public hospital, or library. More abstractly, the `location'  may represent a type or quality of a service.  For example, a government may have a fixed hospital location but must decide on the type of service the hospital will specialize, and, in particular, whether the service will be targeted to those suffering from acute, moderate, or mild severity of a certain illness. In such problems, participants may benefit by misreporting their preferences, and this can be  problematic for a decision maker trying to find a socially optimal solution. This leads to the mechanism design problem of providing optimal, or approximately optimal, solutions while also being \emph{strategyproof}, i.e., no agent can profit from misreporting their preferences regardless of what others report.\footnote{We focus on the `mechanism design without money' problem where the use of money is assumed to not be permitted. This is a natural assumption for environments where the use of money is considered  unlawful (e.g., organ donations) or  unethical (e.g., political decision making, or locating a public good).} We call this the \emph{facility location problem}.

A large literature has studied the facility location problem under the assumption that the facility does not face capacity constraints. When the facility is not capacity constrained, all agents can benefit from the facility and hence it is modeled as a \emph{public good}.\footnote{A public good is non-rivalrous and non-excludable.}  Under this assumption, the mechanism design problem is explored in several classic papers~\cite{Blac48,Gibb73,Gibb77a,Satt75,Moul80,BoJo83}, and more recently in  algorithmic mechanism design~\cite{PrTe13,NiRo01,FFG16}.

To the best of our knowledge, an unexplored setting for the mechanism design problem is where the public facility is capacity constrained.\footnote{There is a distinct setting sometimes referred to as the `constrained facility location' problems~\cite{SuBo15} where the feasible locations for the facility are constrained. The algorithmic problem, of locating multiple capacity constrained facilities when agents are not strategic, has also been studied~\cite{BrCh89,KPTW01,Vygen05:Approximation}.} Capacity constraints limit the number of agents who can benefit from the facility's services. Such constraints are ubiquitous in practice: a hospital is capacity constrained by the number of beds and doctors, and a library may have limited seating.  When present, capacity constraints introduce a particular form of \emph{rivalry} to the facility, since once the facility reaches its capacity limit additional agents are prevented from using, and hence benefiting, from the facility.

A number of new strategic challenges arise for the mechanism designer when the public facility is capacity constrained but is still non-excludable. For example, when the mechanism designer chooses a location for the facility, we cannot stipulate which agents  will be served, instead these decisions are made by
participants, through strategic interactions once the facility has been located. That is, the ex-post Nash equilibrium of a subgame induced by the facility location determines the agents who ultimately benefit from the facility and those who do not. This introduces a technical challenge, because it leads  agents to have interdependent utilities, since the utility for a particular location depends on who else will use the location (and in turn on their preferences). Furthermore, the reporting game is made in anticipation of the extensive-form game and ex-post Nash equilibrium, and  the designer must consider mechanisms that are strategyproof in this broader game-theoretic context.

In this paper, we initiate the study of the capacity constrained facility location problem from the viewpoint of mechanism design. In our model, $n$ agents are located in the $[0,1]$ interval, and there is a single facility to be located, this facility is able to service at most $k$ agents, where $k$ is some positive integer. When $k\ge n$ the capacity constraint is of no effect, and the capacity constrained facility location problem is equivalent to the classic problem. Agent locations are privately known, and, given a facility location, the ex-post Nash equilibrium of an induced subgame determines which agents are served.  The mechanism designer's problem is to design mechanisms that are strategyproof and maximize social welfare. In our model, we take strategyproof to  mean ex-post dominant-strategy incentive compatible (DIC\xspace) at the reporting stage. That is, conditional on the ex-post Nash equilibrium being attained in the induced subgame, an agent never benefits ex-post from misreporting their location to the mechanism regardless of what other agents report, and regardless of other agents' true locations. For ease of exposition, a mechanism that is DIC\xspace at the reporting stage will simply be said to be DIC\xspace. Unlike the classic facility location problem where the facility is not capacity constrained, the social welfare optimal mechanism is not DIC\xspace  except when the capacity constraint is trivial, i.e., $k=1$ or $n$. As a result, we follow the approach of Procaccia and Tennenholtz~\cite{PrTe13} and consider the approximate mechanism design problem. We adopt the worst-case approximation measure for social welfare, and ask what is the best approximation achievable  with a DIC\xspace mechanisms and how does this vary as a function of the capacity constraint?

The literature studying the facility location problem without capacity constraints, or simply $k=n$, provides a number of important results. Gibbard-Satterthwaite~\cite{Gibb73,Satt75} showed a powerful impossibility result: when agents can have unrestricted preferences there need not exist any strategyproof mechanism. As a result, more recent works typically restrict agent preferences' over the location of the facility to be single-peaked and sometimes in addition symmetric.\footnote{A single-peaked preference is symmetric if equidistant locations on either side of the ideal, or `peak', location are always equally preferred.} We focus on the case where, conditional on the agent being served, the agent has preferences that are both single-peaked and symmetric. When the objective of the mechanism designer is to maximize social welfare, i.e., utilitarian welfare, the standard median mechanism is both strategyproof and social welfare optimal~\cite{Blac48}. More generally, a goal of the social choice literature has been to characterize the complete family of strategyproof mechanisms. Closest to our setting, Border and Jordan~\cite{BoJo83} provide a partial characterization of strategyproof mechanisms  via the family of Generalized Median Mechanisms (GMMs). Border and Jordan show that a mechanism is strategyproof and unanimity respecting\footnote{Unanimity respecting simply means that if there is a unanimously most preferred facility location then the mechanism must locate the facility at this location.} if and only if it is a GMM, and that the family of GMMs is strictly smaller than the complete family of strategyproof mechanisms.\footnote{We note that in a slightly different setting, where the single-peaked preferences are possibly asymmetric, GMMs provide a complete characterization of strategyproof and `peak only' mechanisms (Proposition 3 of Moulin~\cite{Moul80}). Example~\ref{Example: DIC hard} in the present paper provides an example of a mechanism that is strategyproof in the Border and Jordan~\cite{BoJo83} setting but not the Moulin~\cite{Moul80} setting.} This has left a gap in the literature to characterize the complete family of strategyproof mechanisms and understand the difference in strategyproof mechanisms that are GMMs and those that are not. Figure~\ref{Figure: Summary of border jordan 1983} schematically illustrates this gap.\\

\textbf{Our Contributions:} We introduce a new mechanism problem, the capacity constrained facility location problem. This problem is a natural variant of the classic facility problem where the facility is assumed to face capacity constraints. A conceptual contribution  is to formalize the effect of capacity constraints when the facility is non-excludable but cannot service all agents. We do this by defining an extensive-form game involving the mechanism designer and agents. First, agents report their preferences to the designer, and then the facility is located by the mechanism. Once the facility is located a subgame is induced where agents strategically choose whether or not to attempt to be served by the facility. The ex-post Nash equilibrium determines which  agents are served by the facility and which are not.
We seek mechanisms that are strategyproof in this broader game-theoretic context, i.e.,  ex-post dominant-strategy incentive compatible; that is, conditional on the ex-post Nash equilibrium being achieved in the subgame, no agent can benefit from misreporting their location regardless of what other agents report and regardless of other agents' true locations.

 Our main theoretical contribution is a complete characterization of DIC\xspace mechanisms for the capacity constrained facility location problem. We show that a mechanism is DIC\xspace if and only if it belongs to the established family of mechanisms called the Generalized Median Mechanisms (GMMs), which appear in Moulin~\cite{Moul80} and Border and Jordan~\cite{BoJo83}. Thus, the framework we introduce  surprisingly provides a new characterization of GMMs. This result contributes to a novel perspective to a ``major open question" (Barbar{\`a}, Mass{\'o} and Serizawa~\citep{BMS98}) posed in Border and Jordan~\cite{BoJo83} (further discussion is provided in Section~\ref{Section: related lit}).

 We also provide algorithmic results and study the performance of DIC\xspace mechanisms in optimizing social welfare. We adopt the  worst-case approximation measure, and provide a lower bound on the approximation ratio of any DIC\xspace mechanism. We show that at best the approximation ratio of a DIC\xspace mechanism is $2\frac{k}{k+1}$ when $k\le \ceil{(n-1)/2}$, and $\max\{\frac{n-1}{k+1}, 1\}$ otherwise. Interestingly, this lower bound is achieved by the standard median mechanism (which is also DIC\xspace) when $k\le  \ceil{(n-1)/2}$ or $k=n$, and hence the median mechanism is optimal among all DIC\xspace mechanisms in those ranges. Figure~\ref{figure: illustration DIC lower boundxxy} illustrates these approximation results.

 Finally, we consider an extension of our framework where the
 mechanism designer can also restrict access to the facility, and
 hence dictate which agents are served. This extension is relevant to
 settings where the designer can issue permits, and prevent certain
 agents from accessing the facility. Under an anonymity assumption, we show that no mechanism that both locates the facility
 and stipulates which agents can be served is DIC\xspace.

\begin{figure}[H]
\centering
\begin{tikzpicture}[scale=1.05,
  declare function={
    func(\x)= (\x <= 50) * (2*(\x)/(\x+1))   +
                   (\x>50) * (99/(\x+1))
   ;
    funcy(\x)= (\x <= 50) * (2*(\x)/(\x+1))   +
                          (\x>50) *(2*(\x)/(\x+1))

   ;
       funcz(\x)= (\x <= 81) * (2*(\x)/(\x+1))   +
                       (\x>81) *( 1+ 2*(100-\x+1)/(3*\x-2*(100)-2))
                          ;
  }
]
\begin{axis}[
  axis x line=middle, axis y line=middle,
  ymin=0, ymax=3, ytick={0,1,2,3}, ylabel={$\alpha$-approximation},
  xmin=0, xmax=100, xlabel=$k$,
  xtick={0,1,25,50,75,100},
xticklabels={$0$, 1,$n/4$,$n/2$, $3n/4$,$n$},
  domain=0:100,samples=201,
]

\addplot [blue,ultra thick, domain=1:100] {func(x)};

\addplot [red,ultra thick,dotted, domain=1:100] {funcz(x)};

\legend{DIC\xspace lower bound, median mech. upper bound}
\end{axis}
\end{tikzpicture}
\captionsetup{justification=centering,margin=2cm}
\caption{Worst-case approximation ratio as a function of the capacity constraint, $k$.}
\label{figure: illustration DIC lower boundxxy}
\end{figure}

\textbf{Outline:} Section~\ref{Section: related lit} provides a brief literature review. Section~\ref{sec:model} presents our model and formalizes the objective of the mechanism designer, Section~\ref{Section: characterization} then presents our key characterization result of DIC\xspace mechanisms. Section~\ref{Section: approxim of dic} explores the performance, i.e., approximation results, of DIC\xspace mechanisms. Section~\ref{Section: Excludable} considers an extension of our framework where the mechanism designer is able to dictate which agents are served by the facility. Lastly, we conclude with a discussion in Section~\ref{section: discussion and conlc}.

\section{Related literature}\label{Section: related lit}

A number of papers have considered related mechanism problems where the use of money is not permitted~\citep{AsRo11,AbSo03,PrTe13,Moul80,BoJo83,Gibb73,Satt75,SuBo15,MLY+16}. Most closely related to our paper is~\cite{PrTe13}, where agents with single-peaked preferences are located along the real line and the problem of locating a (non-capacity constrained) public facility is studied with the goal of minimizing two distinct objective functions; the total social cost and the maximum social cost. This problem is often referred to as a single facility location problem, or single facility location game.\footnote{We do not review a large segment of computer science and operations research literature on facility location problems that assumes complete information and hence does not require a mechanism design approach to overcome strategic tensions (for a survey see~\cite{BrCh89}). Furthermore, this literature, when incorporating capacity constraints, typically focuses on the problem of locating multiple capacity constrained facilities that have sufficient capacity to service all agents~\cite{Cygan12:LP,Charikar02:Constant,KPTW01,Vygen05:Approximation}. Instead we review the subset of literature  that assumes strategic agents and takes a mechanism design approach.}  In this paper, we focus on minimizing the first objective function in the new environment where the facility is capacity constrained. In contrast to the setting studied by~\cite{PrTe13}, agents have interdependent utilities, in our model, due to the capacity constraints of the facility and the induced subgame. Accordingly, the mechanism design problem requires consideration of a broader game-theoretic environment where agents face an extensive-form game when reporting preferences.

 Another large body of literature has been concerned with characterizing DIC\xspace mechanisms for the unconstrained facility location problem. The key pioneering works in this area are by Moulin~\cite{Moul80}, and Border and Jordan~\cite{BoJo83}. In one-dimensional space and for symmetric and single-peaked preferences, Border and Jordan~\cite{BoJo83} characterize a general class of DIC\xspace mechanisms which have become to be known as \emph{generalized median mechanisms} (GMM), and in addition show that when the property of unanimity is enforced every DIC\xspace mechanism is a GMM.\footnote{Border and Jordan~\cite{BoJo83} also consider the problem in higher dimensions.} These results differ slightly from the characterization results of Moulin~\cite{Moul80} since the setting studied in~\cite{Moul80} does not restrict the single-peaked preferences to be symmetric.  Characterizing DIC\xspace but non-unanimity respecting mechanisms was posed as an open problem; as stated by Border and Jordan in~\cite{BoJo83} ``\emph{[the characterization] leaves several open problems. The most obvious question is: what happens if the unanimity assumption is dropped?}" Characterizations however, have remained elusive and it has become known as a ``\emph{major open question}"~\citep{BMS98} with only partial progress towards a resolution~\citep{Chin97,BMS98,PPS+97,Weym11}. In this paper we focus on the one-dimensional case where open questions still remain; in particular, the results of~\cite{BoJo83} in one-dimensional space leaves two gaps:
 \begin{enumerate}
 \item there exist non-unanimity respecting DIC\xspace mechanisms that are not GMM, and
 \item there exist DIC\xspace mechanisms that are GMMs but do not respect unanimity.
 \end{enumerate}
Our characterization of DIC\xspace mechanisms via the family of GMM, although considered in a different setting where the facility is capacity constrained, applies more generally to mechanisms that are not unanimity respecting. Hence, we contribute a novel perspective to these gaps in characterization, showing that a mechanism is DIC\xspace for all possible capacity constraint $k\le n$ if and only if it is a GMM. This means that any mechanism in gap (1) is not DIC\xspace when the facility is capacity constrained with $k<n$. Furthermore, the unanimity property is sufficient to ensure that a mechanism that is DIC\xspace in the non-capacity constrained setting remains DIC\xspace when capacity constraints are present.

\section{Model, Basic Properties, and Definitions} \label{sec:model}

\textbf{Model:} Let $N=\{1,\ldots, n\}$ be a finite set of $n$ agents and let $X=[0,1]$ be the domain of agent locations. Each agent $i\in N$ has a location $x_i\in X$, which is privately known,  the profile of agent locations is denoted by $\boldsymbol{x}=(x_1,  \ x_2, \  \ldots \ , \ x_n)$. The profile of all agent except some agent $i\in N$ is denoted by $\boldsymbol{x}_{-i}=(x_1, \ x_2,\  \ldots \ , x_{i-1}, \ x_{i+1}, \ \ldots \ , \ x_n)$. There is a single facility to be located in $X$ by some mechanism. A \emph{mechanism} is a function $M:  \prod_{i\in N} X\rightarrow X$ mapping a profile of locations to a single location.\footnote{We restrict our attention to deterministic mechanisms.} We denote the mechanism's output, or facility location, by $s\in X$.

The facility faces a capacity constraint $k \ : \ k \le n$, which provides a limit on the number of agents that can be served.  A \emph{served} agent attains utility $u_i=1-d(s, x_i)\ge 0$, where $d(\cdot, \ \cdot)$ denotes the Euclidean metric;  an \emph{unserved} agent attains zero utility, $u_i=0$.\footnote{Our characterization results (Section~\ref{Section: characterization}) do not rely on this specific utility function -- we only require that agents weakly prefer to be served than not, and conditional on being served the agent's utility is symmetric and (strictly) single-peaked. However, our approximation results do rely on the choice of utility function.} The set of agents served by the facility's limited capacity is not directly controlled by the mechanism, since the facility is assumed to be non-excludable.\footnote{In Section~\ref{Section: Excludable} we weaken this assumption and consider the problem when the facility can be made excludable.} Instead, this is determined by the equilibrium outcome of a subgame induced by the mechanism's choice of facility location.

Given an instance $\langle \boldsymbol{x}, s, k\rangle$, we assume that the set of agents served by the facility is determined via the ex-post Nash equilibrium\footnote{That is, no agent has an incentive to unilaterally deviate, whatever the preferences of each agent.} of a subgame, $\Gamma_{\boldsymbol{x}}(s,k)$. The subgame $\Gamma_{\boldsymbol{x}}(s,k)$ is as follows. Each agent $i\in N$ chooses an action $a_i\in A= \{\emptyset, s\}$ of whether, or not, to travel from their location $x_i$ to the facility location $s$. Action $a_i=s$ denotes agent $i$'s choice to travel to the facility, and action $a_i=\emptyset$ denotes the agent's choice to not travel to the facility. We denote the profile of agent actions by $\boldsymbol{a}=(a_1, \ a_2, \ \ldots \ , a_n)$. An agent $i$ is \emph{served} by the facility if they travel to the facility, $a_i=s$, and strictly less than $k$ other agents travel to the facility, i.e., $|N(\boldsymbol{a},s)|\le k$ where $N(\boldsymbol{a},s):=\{i\in N\  : \ a_i=s\}$. If instead they travel to the facility and at least $k$ other agents also travel to the facility, i.e., $|N(\boldsymbol{a},s)|>k$, then a tie-breaking rule is used to determine which subset of $k$ agents in $N(\boldsymbol{a},s)$ are served. We assume a distance-based tie-breaking rule ($\triangleright$) whereby agent $i$ has higher priority than agent $j$, denoted $i\triangleright j$, if agent $i$ is closer to the facility than agent $j$, i.e., $d(s, x_i)<d(s, x_j)$; if agent $i$ and $j$ are equidistant, i.e., $d(s, x_i)=d(s, x_j)$, then we apply some deterministic tie-breaking rule.\footnote{This ensure that the binary relation $\triangleright$ is complete.} This distance-based tie-breaking rule can be motivated by a `first-come-first-serve' protocol when the location $s$ is geographical and agents physically travel to the facility to be served. If the location $s$ corresponds to a type, or quality, of service the `first-come-first-serve' protocol is analogous to a `best-fit' tie-breaking protocol that prioritizes agents according to how close the type of service being offered is to their true needs, i.e., $d(s,x_i)$. An agent $i$ with location $x_i$ attains utility $1-d(s, x_i)$ if $a_i=s$ and they are served, if $a_i=s$ and they are not served they attain utility $-d(s,x_i)$, and otherwise $a_i=\emptyset$ and agent $i$ attains zero utility.

Abusing terminology slightly, given a profile of locations $\boldsymbol{x}$ and facility location $s$, we will refer to $k$ highest priority agents with respect to the distance-based tie-breaking rule ($\triangleright$) as the \emph{$k$-closest} agents. We denote this set of agents by $N_k^*(\boldsymbol{x}, s)$. \\

\textbf{Basic properties of the model:} For any instance $\langle \boldsymbol{x}, s, k\rangle$, the subgame $\Gamma_{\boldsymbol{x}}(s, k)$ has an (essentially) unique equilibrium. There always exists an equilibrium where the $k$-closest agents, $N_k^*(\boldsymbol{x}, s)$, choose to travel to the facility and are served by the facility. In instances where one or more of the $k$-closest agents are indifferent between being served and not traveling to the facility, i.e., whenever $d(s, x_i)=1$ for some $i\in N_k^*(\boldsymbol{x}, s)$, multiple equilibria arise. For the purposes of this paper these equilibria are all `equivalent' since every agent attains the same utility in each of the equilibria. Proposition~\ref{Proposition: Basic prop 1} states this basic property. The proof is straightforward and left to the appendix for the interested reader.

\begin{proposition}\label{Proposition: Basic prop 1}
For any instance $\langle \boldsymbol{x}, s, k\rangle$, there exists an equilibrium of the subgame $\Gamma_{\boldsymbol{x}}(s, k)$ and, furthermore, in every equilibrium agent $i\in N$ attains utility $1-d(s, x_i)$ if $i\in N_k^*(\boldsymbol{x}, s)$, and otherwise, attains zero utility.
\end{proposition}

Given Proposition~\ref{Proposition: Basic prop 1}, we can denote agent $i$'s ex-post equilibrium utility from the facility location $s$ by simply $u_i^*(s, \boldsymbol{x}, k)$. A useful observation is that the agent's ex-post utilities are (weakly) single-peaked, this result is stated in Proposition~\ref{Proposition: Basic prop 2}. The proof is straightforward and left to the appendix for the interested reader. Intuitively, the result holds because under the distance-based priority ($\triangleright$) an agent's priority only (weakly) improves when the facility moves from a location $s<x_i$ to a new location $s' \ : \ s<s'\le x_i$ (similarly for $s>x_i$).

\begin{proposition}\label{Proposition: Basic prop 2}
For any agent $i\in N$ and any pair of instances $\langle \boldsymbol{x}, s, k\rangle$\xspace and $\langle \boldsymbol{x}, s', k\rangle$, if $s<s'\le x_i$  or $x_i\le s'<s$ then $u_i^*(s, \boldsymbol{x}, k)\le u_i^*(s', \boldsymbol{x}, k)$.
\end{proposition}

In this paper we are interested in `strategyproof' mechanisms where agents do not have an incentive to misreport their location.  In particular, we use the ex-post \emph{Dominant-strategy Incentive Compatible} (DIC) concept of strategyproofness. That is, a mechanism $M$ is  DIC\xspace if for every agent $i\in N$
$$u_i^*\Big(M(x_i, \hat{\boldsymbol{x}}_{-i}), \boldsymbol{x}, k \Big)\ge u_i^*\Big(M(x_i', \hat{\boldsymbol{x}}_{-i}), \boldsymbol{x}, k \Big)$$
for every $x_i'$, for every $\hat{\boldsymbol{x}}_{-i}$, and for every $\boldsymbol{x}_{-i}$. Note that DIC\xspace implies that, conditional on the ex-post Nash equilibrium being achieved in the subgame $\Gamma_{\boldsymbol{x}}(s, k)$, the mechanism is dominant-strategy incentive compatible at the reporting stage. Formally speaking, the DIC\xspace definition depends on the capacity constraint $k$ however, abusing notation slightly, we omit the $k$ dependence as this will be clear from the context.\\

\textbf{Objective of the mechanism designer:} In this paper we are interested in DIC mechanisms that perform well with respect to \emph{social welfare}, i.e., the sum of agents' equilibrium utilities. As is now standard in the algorithmic mechanism design literature we measure the performance of a DIC mechanism by the worst-case approximation ratio.

Given an instance $\langle \boldsymbol{x}, s, k\rangle$\xspace, denote the optimal social welfare by $\Pi^*(\boldsymbol{x}, k):=\max_{s\in X} \sum_{i=1}^n u_i^*(s, \boldsymbol{x}, k),$ and given a mechanism $M$ let $\Pi_M(\boldsymbol{x}, k)$ denote the social welfare attained by the mechanism, i.e.,
\begin{align*}
\Pi_M(\boldsymbol{x}, k)&:=\sum_{i=1}^n u_i^*(s, \boldsymbol{x}, k) &&\text{where $s=M(\boldsymbol{x})$.}
\end{align*}
The mechanism $M$ is an $\alpha$-approximation if
\begin{align}\label{Equation: approx 1}
\max_{\boldsymbol{x}\in \prod_{i=1}^n X}\Bigg\{\frac{\Pi^*(\boldsymbol{x},k)}{\Pi_M(\boldsymbol{x}, k)}\Bigg\}\le \alpha,
\end{align}
the LHS of (\ref{Equation: approx 1}) is referred to as the \emph{approximation ratio}. A mechanism (or family of mechanisms) is said to have a \emph{lower bound}, $\bar{\alpha}$, on the approximation ratio if
\begin{align}\label{Equation: approx 2}
\bar{\alpha}\le \max_{\boldsymbol{x}\in \prod_{i=1}^n X}\Bigg\{\frac{\Pi^*(\boldsymbol{x},k)}{\Pi_M(\boldsymbol{x}, k)}\Bigg\}.\end{align}
We refer to a mechanism $M$ that attains the optimal social welfare for all instances $\langle \boldsymbol{x}, s, k\rangle$\xspace, and hence is an $\alpha=1$-approximation, as an \emph{optimal mechanism}.  Again, the optimal mechanism definition depends on the capacity constraint $k$ however, abusing notation, we will omit the $k$ dependence as this will be clear from the context. Note that the optimal mechanism need not, and in general will not, be DIC\xspace for a given $k$.\\

\begin{remark}\label{Remark: special case}
When $k=n$ our model reduces to the well-known facility location problem studied in~\cite{PrTe13,Moul80,Blac48}. Accordingly, this case ($k=n$) is fully resolved: the `median' mechanism which always locates the facility at the median reported location is both optimal and DIC\xspace.  However, the case for $k<n$ has not been studied before -- this is the focus of the present paper.
\end{remark}

To illustrate how the case where $k<n$ differs from the standard $k=n$ setting we provide an example. The example considers a mechanism that is DIC\xspace when $k=n$ but for any capacity constraint $k<n$ is not DIC\xspace.

\begin{example}\label{Example: DIC hard}
Let $M$ be the mechanism such that $M(\boldsymbol{x})=\text{arg}\min_{s\in \{ 1/4, \ 3/4\}} d(s, x_i)$ for some $i\in N$, tie-breaking in favor of $s=1/4$ if necessary. That is, the mechanism locates the facility at either location $1/4$ or $3/4$ depending on which is closest to agent $i$'s report.

First, notice that the mechanism $M$ is DIC\xspace when $k=n$. If $k=n$ then  every agent $i$ is always served by the facility and hence attains utility $1-d(s, x_i)$ for any facility location $s$. It is immediate that agent $i$ can never strictly benefit from misreporting their location.

However, when $k<n$ the mechanism is not DIC\xspace. To see this, consider an instance where agent $i$ is located at $3/8$ and all other agents are located at $1/4$. When agent $i$ truthfully reports, the facility is located at $1/4$ and is not served -- leading to zero utility. On the other hand, misreporting to $x_i'\in (1/2,1]$ leads to the facility location $3/4$ and agent $i$ is the closest agent to the facility. In this case agent $i$ attains strictly higher utility equal to $1-d(3/4, 3/8)>0$. Thus, the mechanism is not DIC\xspace for any $k<n$.\hfill $\diamond$
\end{example}

\subsection{A complete characterization of DIC\xspace mechanisms}\label{Section: characterization}

We begin by defining a family of mechanisms called \emph{Generalized Median Mechanisms (GMM)}. This family was introduced by Border and Jordan~\cite{BoJo83} for the $k=n$ setting, and provides a partial characterization of DIC\xspace mechanisms. The main result of the present paper shows that GMMs provide a complete characterization of mechanisms that are (1) DIC\xspace for all $k\le n$, and (2) DIC\xspace for some $k<n$.

\begin{definition}{[Generalized Median Mechanism (GMM)]}
A mechanism $M$ is said to be a \emph{Generalized Median Mechanism} (GMM) if for each $S\subseteq N$ there are constants $a_S$, such that for all location profiles $\boldsymbol{x}$
\begin{align}\label{equation: GMM}
M(\boldsymbol{x})=\min_{S\subseteq N} \max\Big\{\max_{i\in S}\{x_i\}, a_S\Big\}.
\end{align}
\end{definition}

To build some intuition, we highlight some well-known GMM mechanisms:
\begin{enumerate}
\item The \emph{median mechanism}\footnote{A mechanism that always outputs the median of the reported location profile i.e., the $\floor{(n+1)/2}$-th smallest report.} is attained from (\ref{equation: GMM}) by setting $a_S=1$ for all subsets $S\subseteq N$ with $|S|<\floor{(n+1)/2}$ and $a_S=0$ otherwise.
\item The \emph{$s$-constant mechanism}\footnote{A mechanism that always outputs the location $s$.} for some location $s\in X$, i.e., the mechanism that always outputs location $s$, is attained from (\ref{equation: GMM}) by setting $a_\emptyset=s$ and $a_S=1$ for all other (non-empty) subsets $S\subseteq N$.
\item The \emph{agent $i$ dictatorship mechanism}\footnote{A mechanism that always outputs the location of agent $i$'s report.} is attained from (\ref{equation: GMM}) by setting $a_S=0$ for $S=\{i\}$ and $a_S=1$ for other subsets $S\subseteq N$.
\end{enumerate}
An example of a mechanism that is not a GMM is the dictatorial-style mechanism considered in Example~\ref{Example: DIC hard}.

The main result of the present paper is the following characterization: A mechanism $M$ is DIC\xspace for some $k< n$ if and only if $M$ is DIC\xspace for every $k\le n$ if and only if $M$ is a GMM. This result is stated in Theorem~\ref{Corollary: equivalence results}.

\begin{theorem}\label{Corollary: equivalence results}
Let $M$ be a mechanism. The following are equivalent:
\begin{enumerate}
\item $M$ is a GMM,
\item $M$ is DIC\xspace for some $k<n$,
\item $M$ is DIC\xspace  for every $k\le n$.
\end{enumerate}
\end{theorem}

We present the proof via a series of propositions, and utilize a characterization of Border and Jordan~\citet{BoJo83}. Before presenting these propositions we illustrate the contribution of Theorem~\ref{Corollary: equivalence results}, benchmarked against the results of~\citet{BoJo83}: where GMM are shown to be a strict subset of DIC\xspace mechanisms when $k=n$. Below, in Figure~\ref{Figure: Summary of border jordan 1983}, we present the result of~\citet{BoJo83}. Figure~\ref{Figure: illustration of our results} illustrates our contribution. When considering the capacity constrained problem, with $k<n$, the family of DIC\xspace mechanisms coincides precisely with the GMM family.

\begin{figure}[H]
\centering
\begin{minipage}{.5\textwidth}
\centering
   \begin{tikzpicture}[scale=0.8,
dot/.style = {circle, inner sep=0pt, minimum size=1mm, fill,
              node contents={}}
                        ]
\def(0,0) coordinate (a) circle (2.75cm){(0,0) coordinate (a) circle (2.75cm)}
\def(0,-0.24) coordinate (b)  circle (2.3cm){(0,-0.24) coordinate (b)  circle (2.3cm)}
\def(0,-0.9) coordinate (b)  circle (1.5cm){(0,-0.9) coordinate (b)  circle (1.5cm)}
    \begin{scope}
\fill[gray] (0,-0.24) coordinate (b)  circle (2.3cm);

    \end{scope}
\draw (0,0) coordinate (a) circle (2.75cm);
\draw (0,-0.24) coordinate (b)  circle (2.3cm);
 \node[circle] at (0, 2.35)  {DIC};
   \node[circle] at (0, 0.8)   {GMM};
    \end{tikzpicture}
    \caption{Setting where $k=n$~\cite{BoJo83}.}
    \label{Figure: Summary of border jordan 1983}
    \end{minipage}%
\begin{minipage}{.5\textwidth}
\centering
   \begin{tikzpicture}[scale=0.8,
dot/.style = {circle, inner sep=0pt, minimum size=1mm, fill,
              node contents={}}
                        ]
\def(0,0) coordinate (a) circle (2.75cm){(0,0) coordinate (a) circle (2.75cm)}
\def(0,-0.24) coordinate (b)  circle (2.3cm){(0,-0.24) coordinate (b)  circle (2.3cm)}
\def(0,-0.9) coordinate (b)  circle (1.5cm){(0,-0.9) coordinate (b)  circle (1.5cm)}
    \begin{scope}
\fill[white] (0,0) coordinate (a) circle (2.75cm);
\fill[gray] (0,-0.24) coordinate (b)  circle (2.3cm);

    \end{scope}
\draw (0,-0.24) coordinate (b)  circle (2.3cm);
   \node[circle] at (0, 0.8)   {DIC $\equiv$ GMM};

    \end{tikzpicture}
     \caption{Setting where $k<n$.}
    \label{Figure: illustration of our results}
    \end{minipage}
    \end{figure}

First, we present a result of~\citet{BoJo83} characterizing the family of GMMs via a property of the mechanism that they call `uncompromising'. Informally speaking, an uncompromising mechanisms means that an agent cannot influence the mechanism output in their favor by reporting extreme locations. The most obvious mechanism satisfying this property is the median mechanism. Formally, a mechanism $M$ is said to be \emph{uncompromising} if for every profile of locations $\boldsymbol{x}$, and each agent $i\in N$,  if $M(\boldsymbol{x})=s$ then
\begin{align}\label{eqaution: uncompromising 1}
x_i>s &\implies M(x_i',\boldsymbol{x}_{-i})=s\qquad \text{ for all } x_i'\ge s &&\text{ and,}\\
x_i<s &\implies M(x_i' , \boldsymbol{x}_{-i})=s \qquad \text{ for all } x_i'\le s.\label{eqaution: uncompromising 2}
\end{align}

\begin{lemma}[Border and Jordan~\cite{BoJo83}]\label{theorem: generalized median}
A mechanism $M$ is uncompromising if and only if it is a GMM.
\end{lemma}

Note that Lemma~\ref{theorem: generalized median}, although proved in the setting where $k=n$,  does not rely on any strategic properties of the mechanism and so applies more generally to our setting of interest where $k\le n$.

We now prove our first proposition towards the characterization result. Proposition~\ref{proposition: uncompromise connections} says that, every GMM is DIC\xspace for any $k\le n$.

\begin{proposition}\label{proposition: uncompromise connections}
Every GMM is DIC\xspace for any $k\le n$.
\end{proposition}

\begin{proof}
Fix $k\le n$ and let $M$ be a GMM. For the sake of a contradiction suppose that $M$ not DIC\xspace. That is, for some agent $i$ with location $x_i$, there exist a profile of other agent locations $\boldsymbol{x}_{-i}$, and reports $\hat{\boldsymbol{x}}_{-i}$ such that for some $x_i'\neq x_i$
\begin{align}\label{equation: uncompromising truthful}
u_i^*(M(x_i', \hat{\boldsymbol{x}}_{-i}), \boldsymbol{x}, k)> u_i^*(M(x_i, \hat{\boldsymbol{x}}_{-i}), \boldsymbol{x}, k).
\end{align}
Define $s'=M(x_i', \hat{\boldsymbol{x}}_{-i})$ and $s=M(x_i, \hat{\boldsymbol{x}}_{-i})$. It is immediate from (\ref{equation: uncompromising truthful}) that $s\neq x_i$ and $s\neq s'$. Without loss of generality we assume that $x_i>s$. By assumption, $M$ is a GMM and hence by Lemma~\ref{theorem: generalized median} satisfies the uncompromising property. It follows that $x_i'<s$, since otherwise $x_i' \ge s $ and (\ref{eqaution: uncompromising 1}) would imply $s'=s$ contradicting (\ref{equation: uncompromising truthful}).

Case 1: Suppose $s<s'$. Then $x_i'<s'$ and the uncompromising property (\ref{eqaution: uncompromising 2}) implies that
$$M(x_i'', \boldsymbol{x}_{-i})=s' \qquad \text{for all } x_i'' \le  s'.$$
If $x_i''\in [s, s']$ the uncompromising property implies that $M(x_i'', \boldsymbol{x}_{-i})=M(x_i, \boldsymbol{x}_{-i})$, i.e., $s'=s$, which contradicts (\ref{equation: uncompromising truthful}). Thus, we conclude that $x_i''<s$.

Now consider a new instance where agent $i$ has true location $y_i=\varepsilon\in (0, s)$, all other agents have true location $y_j=0$ but collectively report $\hat{\boldsymbol{x}}_{-i}$. If agent $i$ reports $y_i=\varepsilon$ then the facility location is $s'$ and $i$ attains utility $1-d(s', \varepsilon)$. If instead agent $i$ reports $y_i'=x_i$ then the facility location is $s<s'$ and $i$ attains strictly higher utility $1-d(s, \varepsilon)$. Thus, the mechanism is not $k$-DIC -- a contradiction.

Case 2: Suppose $s>s'$. Since $x_i>s>s'$, it follows from the single-peaked property (Proposition~\ref{Proposition: Basic prop 2}) that $u_i^*(s, \boldsymbol{x}, k)\ge  u_i^*(s', \boldsymbol{x}, k)$. This contradicts (\ref{equation: uncompromising truthful}).
\end{proof}

We now prove our second proposition towards the characterization result. Proposition~\ref{lemma: SP translation} says that, if a mechanism is DIC\xspace for some $k<n$ then it is DIC\xspace for $k=n$. Thus, the DIC\xspace requirement  is more restrictive for $k<n$ than for $k=n$ -- meaning that the capacity constraints induce new strategic concerns for the mechanism designer.

\begin{proposition}\label{lemma: SP translation}
If a mechanism $M$ is DIC\xspace, for some $k<n$, then it is DIC\xspace for $k=n$. The converse is not true.
\end{proposition}

\begin{proof}
We prove the contrapositive. Suppose that $M$ is not DIC\xspace for $k=n$. That is, for some agent $i$ with location $x_i$ there exists a report $x_i'$, a profile of other agent reports $\hat{\boldsymbol{x}}_{-i}$, and a profile of other agent locations $\boldsymbol{x}_{-i}$ such that
\begin{align}\label{equation: sp translation}
u_i^*(M(x_i', \hat{\boldsymbol{x}}_{-i}), \boldsymbol{x}, n)>u_i^*( M(x_i, \hat{\boldsymbol{x}}_{-i}), \boldsymbol{x}, n).
\end{align}
Let $s'=M(x_i', \hat{\boldsymbol{x}}_{-i})$ and $s=M(x_i, \hat{\boldsymbol{x}}_{-i})$. When $k=n$ all agents are served and so (\ref{equation: sp translation}) simplifies to
\begin{align}\label{Equation: k<n implies n}
1-d(s', x_i)>1-d(s, x_i).
\end{align}

Now we consider the same profile of reports but for an arbitrary $k<n$. Furthermore, suppose all agents have location equal to $x_i$ and agent $i$ has highest priority ($\triangleright$), i.e., after tie-breaking. The mechanism output is independent of agent true locations and so we still attain $M(x_i', \hat{\boldsymbol{x}}_{-i})=s'$ and $M(x_i', \hat{\boldsymbol{x}}_{-i})=s$. Furthermore, since $i$ has highest priority (recall that the priority is distance-based but in this instance  all agents are equidistant for every facility location) they are always served for every facility location.  In particular, the utility from reporting truthfully is $1-d(s, x_i)$ and misreporting is $1-d(s', x_i)$ -- the latter provides strictly higher utility, as per (\ref{Equation: k<n implies n}). We conclude that the mechanism is not DIC\xspace, and since $k<n$ was chosen arbitrarily it holds for all $k<n$.

The final statement in the proposition was shown in Example~\ref{Example: DIC hard}.
\end{proof}

We now prove our third and final proposition, which completes the characterization result. Proposition~\ref{theorem: capacitated SP equivalence} says that, if a mechanism is DIC\xspace for some $k<n$ then it is a GMM.

\begin{proposition}\label{theorem: capacitated SP equivalence}
If a mechanism $M$ is DIC\xspace, for some $k<n$, then it is a GMM.
\end{proposition}

\begin{proof}
Let $M$ be a mechanism that is DIC\xspace for some $k<n$.

First, consider an instance where an arbitrary agent $i$ has location $x_i$, and the other agents report $\hat{\boldsymbol{x}}_{-i}$. If $i$ reports truthfully the mechanism outputs some location that we denote as $s$, i.e.,
\begin{align}\label{Equation: truthful report DIC capacatited leads to GMM s}
s:=M(x_i, \hat{\boldsymbol{x}}_{-i}).
\end{align}
If $s=x_i$ then consider an alternate location and profile of other agents' reports so that the equality does not hold. If no such location and report profile exists then the mechanism always coincides with agent $i$'s report; that is, the mechanism is the agent $i$ dictatorship mechanism, which is a GMM.

Now suppose $s\neq x_i$, and without loss of generality assume $s<x_i$. By assumption $M$ is DIC\xspace, for some $k<n$, and so it must be that for all $x_i'$
\begin{align}\label{equation: sp translationx1}
u_i^*( s, \boldsymbol{x}, k)\ge u_i^*(M(x_i', \hat{\boldsymbol{x}}_{-i}), \boldsymbol{x}, k),
\end{align}
where $\boldsymbol{x}$ denotes the location profile of all agents.

We now show that deviations by agent $i$ satisfy the uncompromising property, i.e., for any $x_i'\ge s$ $M(x_i', \hat{\boldsymbol{x}}_{-i})=s$. To do so, we analyze different cases and sequential refine the possible values of $M(x_i', \hat{\boldsymbol{x}}_{-i})$, we the derive a contradiction to eventually conclude that $M(x_i', \hat{\boldsymbol{x}}_{-i})=s$.\\

\underline{Case 1:} Suppose all other agents have location $s$. When agent $i$ truthfully reports $x_i$ the facility location is $s$ and they attain zero utility. Now consider some report $x_i' \ge s$, leading to facility location
$$s_{x_i'}:=M(x_i', \hat{\boldsymbol{x}}_{-i}).$$
If $s_{x_i'}\in (\frac{s+x_i}{2}, 1]$ for any $x_i'\ge s$ we attain a contradiction, since this agent $i$ would be served from this report and attain strictly more utility than being truthful. We conclude that
$$s_{x_i'}\in [0,s)\cup \{s\}\cup (s, \frac{s+x_i}{2}) \qquad \text{ for all } x_i'\ge s.$$

\underline{Case 2:} Suppose all other agents have location $1$, noting that $s<x_i\le 1$. In the event that $x_i=1$ (in which case all agents are equidistant from every facility location), assume agent $i$ has the highest priority in the tie-breaking rule ($\triangleright$). When agent $i$ truthfully reports their location they are served and attain utility $1-d(s, x_i)$. To avoid a contradiction of (\ref{equation: sp translationx1}), it must be that $s_{x_i'}\le s$. Thus, we conclude
\begin{align*}
s_{x_i'}&\in [0,s)\cup \{s\}&& \text{ for all } x_i'\ge s.
\end{align*}

For the sake of a contradiction suppose there exists some $x_i''\ge s$ such that
\begin{align}\label{equation: sp translationx3}
s_{x_i''}&\in [0,s).
\end{align}
Consider a new instance where agent $i$'s location is $y_i=x_i''$ (note that $x_i''\ge s$), all other agents have location $1$, and the other agents report $\hat{\boldsymbol{x}}_{-i}$ (the same profile of reports as per (\ref{equation: sp translationx1})). In the event that $y_i=x_i''=1$ (in which case all agents are equidistant from every facility location), assume agent $i$ has the highest priority in the tie-breaking rule ($\triangleright$). If agent $i$ reports their location $y_i$ the facility location is $s_{y_i}=s_{x_i''}<s$, as per (\ref{equation: sp translationx3}), and they attain utility $1-d(s_{y_i}, y_i)$. But now misreporting to $y_i'=x_i$ then as per (\ref{Equation: truthful report DIC capacatited leads to GMM s}) the facility location is $s$ where
$$s_{y_i}< s \le y_i,$$
leading to utility $1-d(s, y_i)$. This is a contradiction of the mechanism being DIC\xspace, since $d(s, y_i)<d(s_{y_i}, y_i)$; that is, agent $i$ by reporting $y_i'$ instead of their true location $y_i$  attains strictly higher utility. We conclude that $s_{x_i'}=s$ for all $x_i'\ge s$. Thus, the mechanism is uncompromising and hence a GMM.
\end{proof}

\section{Approximation of DIC\xspace mechanisms}\label{Section: approxim of dic}

Given the characterization result (Theorem~\ref{Corollary: equivalence results}) of the previous section, there is no distinction between the family of mechanisms that are DIC\xspace for some $k<n$, and the family of mechanisms that are DIC\xspace for all $k\le n$: both families are equal to the GMM family. Accordingly, we will now simply refer to a mechanism as being DIC\xspace.

\subsection{Optimal mechanism is not DIC\xspace}

We first show that, in general for $k<n$, the optimal mechanism is not DIC\xspace. Note that this result contrasts with the $k=n$ setting where the median mechanism is both optimal and DIC\xspace (Remark~\ref{Remark: special case}).

\begin{theorem}\label{Theorem: optimal not dic}
The optimal mechanism is DIC\xspace if and only if $k\in \{1,n\}$.
\end{theorem}

\begin{proof}
The backward direction of the theorem statement is straightforward: If $k=1$ then for any $i\in N$ the agent $i$ dictator mechanism, where the mechanism output always coincides with agent $i$'s report, is both optimal and DIC\xspace. This is trivial and we do not provide further details.  If $k=n$ then the median mechanism is both optimal and DIC\xspace. This result has long been known and can be found in~\citet{Blac48,Moul80,PrTe13}.

We now prove the forward direction using the contrapositive. Let $k\notin \{1,n\}$ and partition the agent into $\floor{n/k}$ groups of size $k$, denoted by $N_{t}$ for $t=1, 2, \ldots, \floor{n/k}$, and one group of size $n-\floor{n/k}$, denoted by $N_{\floor{n/k}+1}$.  We now identify $\floor{n/k}+1$ locations in $[0,1]$, let
$$y_t=  \frac{t}{\floor{n/k}+1}\qquad \text{ for } t=1, 2, \ldots, \floor{n/k}+1.$$

Consider a scenario such that for each $t=1, 2, \ldots, \floor{n/k}+1$, all but one agent in $N_t$ is located at $y_t$ and a single agent is located at $y_t- t\, \varepsilon$ for some sufficiently small $\varepsilon>0$. In each instance denote the single agent located at $y_t- t\, \varepsilon$ by $i_t\in N_t$.

In this scenario it is immediate the optimal welfare is attained by locating the facility at location $y_1$, leading to a social welfare of $k-\varepsilon$ and agent $i_1$ attain utility $1-\varepsilon$.

Now in a new scenario where agent $i_1$ is located at $y_1-3\varepsilon$ the optimal mechanism must locate the facility at $y_2$. In this case agent $i_1$ attains utility zero. However, if agent $i_1$ misreport their location to $y_1-\varepsilon$ then (as shown above) the facility location will be $y_1$ and they will attain strictly higher utility $1-\varepsilon$. That is, the optimal mechanism is not DIC\xspace for $k\notin \{1,n\}$.
\end{proof}

Despite Theorem~\ref{Theorem: optimal not dic} stating a stark impossibility result, we note that absent strategic manipulations by the agents the optimal mechanism can  be efficiently computed. Remark~\ref{Remark: Computational result} says that,  for any $k\le n$ the optimal mechanism's output and corresponding welfare can be computed in polynomial time.

\begin{remark}\label{Remark: Computational result}
The optimal facility location and welfare can be computed in polynomial time for any $k\le n$.
\end{remark}

We sketch an informal argument for Remark~\ref{Remark: Computational result}. Order the agents $i\in  N$ such that $x_i\le x_{j}$ if and only if $i\le j$. It is straightforward to show that an optimal solution has two features (1) the facility serves a  \emph{contiguous} set of $k$ agents, i.e., if agent $i$ and $i+2$ are served then agent $i+1$ is served, and (2) the facility is located at the median of these $k$ served agents. Given these features, it is immediate that a polynomial-time procedure exists by simply comparing the welfare produced by, the at most $n$, sets of $k$ contiguous agents.

\subsection{Lower bound on DIC\xspace approximation}

Utilizing the characterization result of DIC\xspace mechanisms via the family of GMMs, we provide a lower bound on the approximation ratio for all DIC\xspace mechanisms.

Theorem~\ref{theorem: uncompromising at best 2} shows that at best a DIC\xspace mechanisms provides a $2\frac{k}{k+1}$-approximation when $k\le \ceil{(n-1)/2}$, and otherwise provides at best an $\max\{\frac{n-1}{k+1},1\}$-approximation. This lower bound on the approximation ratio is illustrated in Figure~\ref{figure: illustration DIC lower boundxxy}.

\begin{theorem}\label{theorem: uncompromising at best 2}
Let $n\ge 2$. A DIC\xspace mechanism is at best an $\alpha$-approximation with $\alpha=2\frac{k}{k+1}$ when $1\le k\le \ceil{(n-1)/2}$, and $\alpha=\max\{\frac{n-1}{k+1},1\}$ otherwise.
\end{theorem}

\begin{proof}
Let $M$ be a DIC\xspace mechanism, and consider a scenario where all $n$ agents have distinct locations contained in the interval $I=(1/2-1/2\varepsilon, \ 1/2 +1/2\varepsilon)$ for some sufficiently small $\varepsilon>0$. Denote the profile of agent locations by $\boldsymbol{x}$, and the mechanism's corresponding output by $s=M(\boldsymbol{x})$.

We consider two cases.

\underline{Case 1:} Suppose $s\notin I$ and without loss of generality assume $s<1/2-1/2\varepsilon$. Now suppose that sequentially agents $i=1, 2, \ldots, n$ have their locations changed and kept at $x_i=1$, and consider the sequence of facility locations produced by the mechanism $s_1, s_2, \ldots, s_n$. By the uncompromising property (satisfied by $M$ since it is a GMM) the location of the facility never changes from $s$. That is, $s_n=s$ despite every agent having location at $1$. The optimal social welfare in this scenario is clearly $k$, however, the mechanism provides welfare of
\begin{align*}
k(1-d(s, 1))&=k\, s< k (1/2-1/2\varepsilon)\rightarrow k/2 &&\text{as $\varepsilon\rightarrow 0$.}
\end{align*}
Thus, the approximation ratio is at best $k/(k/2)=2$.

\underline{Case 2:} Suppose $s\in I$ and without loss of generality assume $s\le 1/2$. Let $\lambda_1, \lambda_2$ be the number of agents with locations strictly less than $s$, and strictly above $s$, respectively. Note that $\lambda_1+\lambda_2\in \{n-1, n\}$. Similar to Case 1, suppose the $\lambda_1$ agents instead had location at $0$ and the $\lambda_2$ agents had location at $1$ -- by the uncompromising property the facility location is unchanged.

To attain the bound on the approximation ratio we consider two subcases where $k\le \ceil{(n-1)/2}$ and $k>\ceil{(n-1)/2}$.

 In the first subcase ($k\le \ceil{(n-1)/2}$): the optimal welfare is $k$, since either $\lambda_1$ or $\lambda_2$ exceeds $k$ meaning that $k$ agents can be served at either 0 or 1. The mechanism's welfare is at most
\begin{align*}
1+(k-1) (1-d(s,0))&< 1+(k-1) (1/2-1/2\varepsilon)\rightarrow 1/2 + k/2&&\text{as $\varepsilon\rightarrow 0$.}
\end{align*}
Thus, the approximation ratio is at best $k/(1/2+k/2)=2 \, k/(k+1)$.

In the second subcase ($k> \ceil{(n-1)/2}$): the optimal welfare is at worst $\ceil{(n-1)/2}$, i.e., when the facility serves either $\lambda_1$ or $\lambda_2$ agents  (whichever is larger) from location 0 or 1. The mechanism's welfare is at most
\begin{align*}
1+ (k-1)(1-d(0, s))&<k-(k-1)(1/2-1/2\varepsilon)\rightarrow k/2+1/2&&\text{as $\varepsilon\rightarrow 0$.}
\end{align*}
Thus, the approximation ratio is at best $\ceil{(n-1)/2}/(k/2+1/2)$, but
\begin{align*}
\ceil{(n-1)/2}/(k/2+1/2)&\ge \frac{(n-1)/2}{(k+1)/2}= \frac{n-1}{k+1}.
\end{align*}
Furthermore since $k> (n-1)/2$ it follows that $ \frac{n-1}{k+1}<2$. Of course, this bound is only meaningful when $n-1/k+1 >1$.

We conclude that when $k\le \ceil{(n-1)/2}$ the approximation ratio is at best $2\frac{k}{k+1}$ and otherwise is at best $\max\{\frac{n-1}{k+1},1\}$.
\end{proof}

\subsection{Optimized approximation ratio for DIC\xspace Mechanism}

We now  analyze the performance of the median mechanism for general $k\le n$. In instances where $k\in\{1,n\}$, the median mechanism is both optimal mechanism and DIC\xspace (Theorem~\ref{Theorem: optimal not dic}). Furthermore, this mechanism is DIC\xspace for all $k\le n$ since the median mechanism is a GMM (Theorem~\ref{Corollary: equivalence results}).

Theorem~\ref{Theorem: median performance} says that the median mechanism is an $\alpha$-approximation where $\alpha=2\frac{k}{k+1}$ when $k\le \floor{(n+1)/2}$, and $\alpha=\min\{2\frac{k}{k+1}, 1+2\frac{n-k+1}{3k-2n-2}\}$ otherwise. In particular, this means that the median mechanism is optimal among DIC\xspace mechanism for $k\le \floor{(n-1)/2}$ since the  approximation-ratio matches the lower bound found in Theorem~\ref{theorem: uncompromising at best 2}. These approximation results are illustrated in Figure~\ref{figure: illustration DIC lower boundxxy}.

\begin{theorem}\label{Theorem: median performance}
Let $n\ge 5$. The median mechanism is an $\alpha$-approximation with $\alpha=2\frac{k}{k+1}$ for $k\le \floor{(n+1)/2}$, and $\alpha=\min\{2\frac{k}{k+1}, 1+2\frac{n-k+1}{3k-2n-2}\}$ otherwise.%
\end{theorem}

\begin{proof}
Let $n\ge 5$. Throughout the proof let $i_m$ denote the agent with median location (choose the agent arbitrarily if multiple such agents exist), and let $s_m$ denote the median location. The median mechanism provides welfare
\begin{align*}
\Pi_M(\boldsymbol{x}, k)=\max_{N' \in N_{k}} \sum_{i\in N_{k}} (1-d(s_m, x_i))=1+\max_{N' \in N_{k-1, i_m}} \sum_{i\in N_{k-1, i_m}} (1-d(s_m, x_i)),
\end{align*}
where $N_k$ is the set of all $k$-sized subsets of $N$ and $N_{k-1, i_m}$ is the set of all $(k-1)$-sized subsets of $N\backslash \{i_m\}$. This follows since the subset of agents served are always the $k$-closest to the facility location. Hence, given a facility location, the served subset is welfare maximizing. Furthermore, the median location coincides with at least one agent's location, i.e., agent $i_m$.\\

First, we provide an upper bound on the approximation-ratio for all $k$. The median mechanism locates the facility at the $\floor{(n+1)/2}$-th location and hence there are $\floor{(n+1)/2}-1$ agents with locations (weakly) below and $\ceil{(n+1)/2}-1$ with locations (strictly) above. A lower bound on the median mechanism's welfare is attained when the agents below and above the median location at located at 0 and 1, respectively. Thus,
$$\Pi_M(\boldsymbol{x}, k)\ge 1+ (k-1)\max\{1-d(s_m, 0),\ 1-d(s_m,1)\},$$
and since either $d(s_m,0)\le 1/2$ or $d(s_m, 1)\le 1/2$ it follows that $\Pi_M(\boldsymbol{x}, k)\ge (k+1)/2$. This leads to an upper bound on the approximation-ratio of $k/((k+1)/2)=2\frac{k}{k+1}$ for all $k$, since the optimal welfare is always bounded above by $k$.\\

Now we attain a tighter upper bound for certain values of $k$. To do so, we bound the median welfare using the optimal welfare. Let $s^*$ be the location of the facility under the optimal mechanism. Let $N_m^*$ denote the set of $k$ agents served under the median mechanism,  and let $N^*$ denote the set of $k$ agents served under the optimal mechanism. We have
\begin{align*}
\Pi_M(\boldsymbol{x}, k)&\ge  \sum_{i\in N^*} \big(1-d(s_m, x_i)\big)\\
& =\sum_{i\in N^*} \Big(1-d(s_m, x_i) -d(s^*, x_i) +d(s^*, x_i) \Big)\\
&=\Pi^*(\boldsymbol{x}, k)-\sum_{i\in N^*} \Big(d(s_m, x_i)-d(s^*, x_i)\Big).
\end{align*}
Clearly, the lower bound is smallest when $s_m\neq s^*$, without loss of generality assume that $s_m<s^*$. Let $N_1^*, N_2^*$ be a partition of $N^*$ such that $|N_1^*|, |N_2^*|\le \floor{(n+1)/2}$ and all agents in $N_1^*$ have location in $[0,s_m]$ and agent in $N_2^*$ have location in $[s_m, 1]$. Such a partition of $N^*$ exists since the location $s_m$ coincides with the $ \floor{(n+1)/2}$ highest location.
Using this partition we further bound the median mechanism's welfare:
\begin{align*}
\Pi_M(\boldsymbol{x}, k)&\ge  \Pi^*(\boldsymbol{x}, k)-\sum_{i\in N_1^*} (d(s_m, x_i)-d(s^*, x_i))-\sum_{i\in N_2^*} (d(s_m, x_i)-d(s^*, x_i))\\
&\ge  \Pi^*(\boldsymbol{x}, k)- |N_1^*| \max_{x\in [0, s_m]} \Big(s_m- s^*-2x\Big)-|N_2^*| \max_{x\in [s_m,1]}\Big(x_i-s_m-|s^*-x_i|\Big)\\
&\ge  \Pi^*(\boldsymbol{x}, k)-|N_1^*| (s_m-s^*)-|N_2^*| (s^*-s_m)\\
&\ge  \Pi^*(\boldsymbol{x}, k)-(|N_2^*|-|N_1^*|) (s^*-s_m)\\
&\ge  \Pi^*(\boldsymbol{x}, k)-(|N_2^*|-|N_1^*|).
\end{align*}
We now attain our lower bound by considering the maximum value of $|N_2^*|-|N_1^*|$. For $k\le \floor{(n+1)/2}$, the value can only be guaranteed to be no larger than $k$ -- leading to a trivial zero lower on $\Pi_M(\boldsymbol{x}, k)$. However, for $k>\floor{(n+1)/2}$ we attain a more useful bound by noting that
$$(|N_2^*|-|N_1^*|) \le \floor{(n+1)/2}-(k-\floor{(n+1)/2})=2\floor{(n+1)/2}-k\le n+1-k.$$
This lead to an approximation-ratio upper bound of
$$\max_{\boldsymbol{x}\in \prod_{i=1}^n X}\Bigg\{\frac{  \Pi^*(\boldsymbol{x}, k)}{  \Pi^*(\boldsymbol{x}, k)-n-1+k}\Bigg\}=\max_{\boldsymbol{x}\in \prod_{i=1}^n X}\Bigg\{1+\frac{ n+1-k}{  \Pi^*(\boldsymbol{x}, k)-n-1+k}\Bigg\}.$$
Furthermore, for any instance $\Pi^*(\boldsymbol{x}, k)\ge k/2$ since at least as much welfare is attained by locating the facility at $s=1/2$. Thus, an upper bound on the approximation ratio is
\begin{align*}1+\frac{ n+1-k}{  k/2-n-1+k}=1+2\frac{ n+1-k}{  3k-2n-2}.
\end{align*} \end{proof}

\section{Extension: Location-Allocation Mechanisms }\label{Section: Excludable}

In this section we consider an extension of our framework where the mechanism designer is able to dictate which agents are served by the facility. Note that this extension introduces an underlying assumption that the facility is excludable. In practice, a designer may be able to dictate which agents are served by issuing permits or, when costs are not  prohibitive, checking the identities of agents attempting to benefit from the facility.

Previously, a mechanism $M \ : \prod_{i\in N} X\rightarrow X$ was defined as a function mapping a profile of locations to a single facility location. In our extension, a mechanism not only locates the facility but also chooses a subset of at most $k$ agents to be served by the facility, if they so choose. We denote these extended mechanisms by
$$M_A \ : \prod_{i\in N} X\rightarrow X \times N_k,$$
where $N_k=\{A\subseteq N \ : 0<|A|\le k\}$. We call these mechanisms \emph{location-allocation} mechanisms, to distinguish them from the (location-only) mechanisms considered in earlier sections of the present paper. The output of the mechanism is a pair $(s,A)\in X\times N_k$ where $s\in X$ denotes the facility location and $A\in N_k$ denotes the subset of agents allocated to the facility. Abusing notation slightly we will denote the mechanism output from a location profile $\boldsymbol{x}$ by $s_{\boldsymbol{x}}$ and $A_{\boldsymbol{x}}$ where $M_A (\boldsymbol{x})=\big(s_{\boldsymbol{x}},\ A_{\boldsymbol{x}}\big)$. An agent $i\in A$ is guaranteed to be served by the facility if they so choose, whilst an agent $i\notin A$ is never served.

We omit the details, but it is immediate that the modified subgame $\Gamma_{\boldsymbol{x}}(s, k, A)$ has an essentially unique ex-post Nash equilibrium where all agents $i\in A$ are served by the facility and the remaining agents are not. Thus, we assume that agent $i$ reports their location to the mechanism designer with the understanding that they will be served by the facility if and only if $i\in A$, as per the ex-post Nash equilibrium. Note that the strategyproof concept, DIC\xspace, in this section still coincides with the concept used in the earlier sections, albeit with the modified subgame explained above.

We first remark that the revelation principle~\cite{Gibb73} does not apply. A location-only mechanism, based on the profile of agent reports, $\hat{\boldsymbol{x}}$, outputs a facility location $s$ -- that depends on $\hat{\boldsymbol{x}}$ -- and a subset of $k$ agents are then allocated to the facility, via the ex-post Nash equilibrium, $A\subseteq N$ -- this subset depends on the agent true locations $\boldsymbol{x}$ and not the reports $\hat{\boldsymbol{x}}$. In contrast, a location-allocation mechanism outputs both a facility location and an allocation of $\le k$ agents to the facility depending on agent reports $\hat{\boldsymbol{x}}$, and not true locations $\boldsymbol{x}$. Thus, an agent misreporting their location -- in a way that does not affect the facility location -- will never affect whether or not they are served by the facility under a location-only mechanisms. However, under a location-allocation mechanism the agent may potentially benefit from the misreport if they are now allocated to the facility by the mechanism.

We now show that no `reasonable' location-allocation-mechanism is DIC\xspace. In particular, we only enforce one criteria, which is a weak form of \emph{anonymity}. Informally speaking, we require that the location-allocation mechanism allocates agents to the facility independently of their label if their report is distinct from all other agents. The usual definition of anonymity is not directly applicable since with a deterministic mechanism, if all agents report identical locations the mechanism must discriminate against at least $n-k$ agents who will not be included in the allocation set $A$.

To formally define our anonymity condition we first introduce the notion of an $i$-identifiable location profile. This is simply a profile where agent $i$ is uniquely identified by their report.

\begin{definition}[$i$-identifiable location profile]Let $i\in N$. A location profile $\boldsymbol{x}$ is \emph{$i$-identifiable} if
$$x_i\neq x_j \qquad \text{  for  all } j\in N\backslash \{i\}.$$
\end{definition}

We now define our anonymity condition, which we call {\em allocation-anonymous} since the condition only applies to the allocation set rather than the facility location. Informally speaking, the allocation-anonymous condition requires that for every $i$-identifiable location profile, whether or not agent $i$ is allocated to the facility does not depend on $i$'s label. Given that allocation-anonymity only applies to $i$-identifiable location profiles the condition is relatively weak.

\begin{definition}[Allocation-anonymous]\label{allocation anon definition} The mechanism $M_A$ is said to be \emph{allocation-anonymous} if for  every distinct $i,j\in N$ and every $i$-identifiable location profile $\boldsymbol{x}$, the modified profile $\boldsymbol{x}'$ such that $x_\ell=x_\ell' $ for all $\ell\neq i,j$ and
$$x_i'=x_j \qquad \text{ and }\qquad x_j'=x_i,$$
we have
$$i\in A_{\boldsymbol{x}} \iff j\in A_{\boldsymbol{x}'},$$
where $A_{\boldsymbol{x}}$ is such that $M_A (\boldsymbol{x})=\big(s_{\boldsymbol{x}},\ A_{\boldsymbol{x}}\big)$.
\end{definition}

We now show that if we restrict our attention to allocation-anonymous mechanisms there is no DIC\xspace location-allocation mechanism.

\begin{theorem}\label{theorem: impossibility of location-allocation}
Let $k<n$, any location-allocation mechanism $M_A$ that is allocation-anonymous is not DIC\xspace.
\end{theorem}

\begin{proof}
For the sake of a contradiction suppose that $M_A$ is a location-allocation mechanism that is both allocation-anonymous and DIC\xspace.

First consider a location profile $\boldsymbol{x}$ where $x_i=3/4$ for all $i\in N$, and denote the output of the mechanism by $M_A(\boldsymbol{x})=(s,A)$. Let $i^*$ be some agent such that $i^*\in A$ and $j^*$ some agent such that $j^*\notin A$. In this outcome agent $j^*$ attains utility zero, since $j^*\notin A$.

Now consider another location profile $\boldsymbol{x}'$ such that $x_i'=3/4$ for all $i\in N\backslash \{j^*\}$ and  $x_{j^*}'=1/2$. Note that the profile $\boldsymbol{x}'$ can be achieved via a unilateral deviation from the profile $\boldsymbol{x}$ by agent $j^*$. Denote the mechanism's output from this location profile by $M_A(\boldsymbol{x}')=(s', A')$.  We consider two cases and derive a contradiction in each case.\\

Case 1: Suppose $j^*\in A'$ and suppose that $\boldsymbol{x}$ is the true location of all agents. In this case agent $j^*$ by misreporting their location to $x_{j^*}'=1/2$ strictly profits, since under the profile $\boldsymbol{x'}$ we have $j^*\in A'$ and their utility is now $1-d(s',\frac{3}{4})>0$ rather than zero. Thus, we have a contradiction. \\

Case 2: Suppose $j^*\notin A'$ and suppose that the agents have true locations $y_\ell=3/4$ for all $\ell \in N\backslash \{i^*\}$ and $y_{i^*}=1/2$. Denote the mechanism's output from location profile $\boldsymbol{y}$ by $M_A(\boldsymbol{x}')=(s'', A'')$. Notice that $\boldsymbol{x}'$ is a $j^*$-identifiable location profile and the profile $\boldsymbol{y}$ satisfies the condition in Definition~\ref{allocation anon definition}, and so by the allocation-anonymous property we require that
$$i^*\in A'' \iff j^*\in A'.$$
Thus, we infer that $i^*\notin A''$ and attain zero utility under the location profile $\boldsymbol{y}$. Now suppose agent $i^*$ unilaterally deviates and reports the location $y_{i^*}'=3/4$. In this case, the location profile coincides with the profile $\boldsymbol{x}$ where $x_\ell=3/4$ for all $\ell\in N$. But recall that $M_A(\boldsymbol{x})=(s,A)$ and $i^*$ was taken to be some agent such that $i^*\in A$. Thus, under this unilaterally misreport agent $i^*$ is now served and attain strictly positive utility of $1-d(s, y_{i^*})>0$. This is a profitable deviation and contradicts our assumption that the mechanism $M_A$ was DIC\xspace. We conclude that there is no location-allocation mechanism that is both allocation-anonymous and DIC\xspace.
\end{proof}

The above impossibility result means that the extensive-form approach taken in the main body of this paper is crucial for DIC\xspace mechanisms that are non-dictatorial. The use of an extensive-form game and corresponding ex-post Nash equilibria to decide the allocation of agents to the facility reduces the incentive compatibility constraints faced by the mechanism designer. Furthermore, the result suggests that the excludability of the facility presents a greater challenge for incentive compatibility than rivalry.

\section{Discussion and Conclusion}\label{section: discussion and conlc}

We now conclude the paper with a brief discussion of future research directions.

\underline{Extensions to multiple facilities:} In the present paper we focused on the case of a single capacity constrained facility location problem. Extending the capacity constrained to multiple facilities presents a number of challenges. Firstly, the subgame induced from a profile of facility location will lead to multiple equilibria that are not welfare (nor utility) equivalent. Furthermore, even when ignoring the multiplicity of equilibria issues, the mechanism design problem is drastically more complicated -- as is the algorithmic problem of finding the optimal facility locations (see Brimberg et al.~\cite{BKE-C+01}). A recent contribution by Golowich, Narasimhan and Parkes~\cite{GNP18} explores the mechanism design problem for multiple facilities without capacity constraints.

\underline{Weakening DIC\xspace:} A natural direction to consider is weakening the strategproofness concept (DIC\xspace) that we use in the present paper. The DIC\xspace requirement is very strong: agents must attain maximal ex-post utility from reporting their location no matter what other agents report, and other agents' true locations.  The weaker notion of \emph{ex-post Incentive Compatible (IC)} may be interesting to be explore for both characterization and performance results. This notion requires that agents attain maximal ex-post utility from reporting their location no matter the other agents' true locations, but conditional on the other agents reporting truthfully. It is straightforward to construct IC mechanisms that out-perform the median mechanism for certain parameter ranges.

\textbf{Conclusion:} In this paper we initiated the study of the capacity constrained facility location problem from a mechanism design perspective. We formalized a model that allows the subset of served agents to be endogenously derived from equilibrium outcomes. Our main contribution is a complete characterization of all DIC\xspace mechanisms via the family of GMM mechanisms. This characterization also provides a novel perspective to an open problem in regard to GMM mechanisms, posed in~\cite{BoJo83}. Our second contribution is an analysis of the performance of DIC\xspace mechanisms with respect to social welfare -- where we also show that the well-known median mechanism is optimal among DIC\xspace mechanism for certain parameter ranges. Finally, we show that extending the space of mechanisms to allow the mechanism to allocate agents to the facility leads to a  stark impossibility result. Namely, there is no allocation-anonymous DIC\xspace mechanism which both locates the facility and stipulates the subset of agents to be served.

\bibliographystyle{ACM-Reference-Format}

redpajama_set_name: RedPajamaArXiv


Giselle, a peasant girl, has fallen in love with Count Albrecht, who has told her he is a villager named Loys. Her discovery of his true identity has devastating consequences. With its irresistible combination of a heart-breaking story and exquisite choreography, Giselle is the perfect way to discover classical ballet.


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Exposed Beauty: Tasmanian Make-over Competition!
Exposed Beauty: clever title for a make-over and photo shoot competition! And networking between like-minded business owners… we think that's pretty clever too! That's why we loved the Exposed Beauty competition conducted by a group of four entrepreneurial women from the Devonport region in Tasmania. To encourage others to embrace that kind of community spirit, we'd like to share the stories of Prue, Cacia, Katie and Belle.
Working together, this fab four launched a competition on the SPY Photography Facebook page. Women were asked to nominate themselves (or a friend or relative) for the chance to win a make-over. Claiming natural beauty shines through in a professional photo-shoot, the campaign proved very successful.
The top 10 finalists were presented to the public at Mystique Boutique. From those finalists the lucky winner was announced and rewarded with a new outfit, hair styling and make-up in readiness for her photography session. With a glamorous new look, the winner (Kristy) fronted the camera and came away with a selection of Exposed Beauty portraits to treasure.
Mystique Boutique is located in The Hub Arcade in Devonport. The store is going into its eleventh year under the current management. The manager Prue Clarke has used all her creativity and enthusiasm to develop Devonport fashion boutique Mystique from a small retail store into one of the most popular fashion destinations in the city.
As the retail industry is very challenging, especially with the current economy, Prue understands the need to pull together with other business owners and support each other with great little promotions like Exposed Beauty. It is now a necessity to develop your business online and Mystique Boutique is hoping to make this their next success story.
Hair Gallery Railton is a small, modern salon in the town of Railton, on the North West Coast. Opened by Cacia Rand in February 2010, the business has seen consistent growth and in March 2011 employed another senior hairdresser, Tanya Waldon.
Hair Gallery Railton is a Keune branded salon, one of Europe's leading hair cosmetic laboratories. Keune products are of the highest standard, free from PPD, and provide high quality, long-lasting results. For a professional look at home, and to maintain healthy hair, a large selection of professional retail products are also available at the salon.
Specialising in cutting, colouring and perming and offering both modern and classic styles to suit your individual look, Hair Gallery Railton can create a look to suit your requirements. Also available to clients is the Brasil Cacau Smoothing Therapy, a keratin treatment for shiny, smooth and frizz-free hair. This is a formaldehyde free keratin product with results lasting up to four months. The salon also offers facial waxing and lash/brow tinting at great prices.
Hair Gallery Railton is open Monday-Friday from 9am, with after-hours and Saturdays available by appointment. Located in the main street of Railton, just 10 minutes from Latrobe, offering free parking and affordable salon prices, Hair Gallery Railton invites you to make an appointment today.
Katie Hooper is located in the beach side town of Turners Beach and specialises in make-up artistry and the hair removal technique of threading.
Katie is a professionally trained freelance make-up artist having trained at the Academy of Make-up in Melbourne (where she lived for 5 years before moving back to Tasmania in April 2011). Katie was trained by some of the best names in the industry and specialises in several areas of make-up including weddings, special occasion and photography and also offers make-up parties.
Katie is also the official make-up artist for Spy Photography and has also worked with the ABC on a documentary called "The Forgotten Australians".
Katie is one of the very few people in Tasmania to be trained in the ancient hair removal technique of threading. Threading is an Indian and Middle Eastern technique that removes hair by using a twisted piece of cotton. It is an acquired skill, which one must learn from a skilled practitioner. It is a skill which needs time and practice in order to perfect. The technique can also be called a form of art, as it requires creating clean lines with precision to enhance the appearance of the face itself.
Belle Young is the founder, owner and photographer for SPY Photography, and 30-year-old mother of two young boys.
Belle earned her full accreditation with the Australian Institute of Professional Photographers in 2011, after a full assessment of her portfolio, and continues to improve and expand her skills as an ongoing passion and determination to be a leader in her field on the North West Coast of Tasmania.
Since February 2008, Belle has been capturing the unique personality of each person, big and small, both naturally and spontaneously, in addition to offering real estate photography.
For more information about this network of Tasmanian regional business owners contact any one of the Exposed Beauty team.
Here at Albury TAFE, we have a hair & beauty department. To provide the students with clients, emails are circulated to the staff requesting them to act as models for the various treatments. So far I have resisted volunteering myself for a male leg wax or (even worse) a Brazilian!
Good luck girls in what you are doing.
PLEASE NOTE: Belle Young will be changing her business name from Spy Photography to Belle Young Photography in 2012.


In 2015, the Canadian Supreme Court ruled — unanimously — that patients with severe medical problems could legally ask their doctors to help end their lives.
None of that, however, is okay with Catholics who believe euthanasia is a sin. Last year, we posted about an 87-year-old man who was in enormous pain and wanted to end life on his own terms. However the staff at St. Paul's Hospital in Vancouver refused to assist him.
He spent the last several hours of his life trying to transfer to a new hospital — the ambulance was three hours late and the ride was bumpy enough to amplify his pain — where someone could take care of him because the taxpayer-funded Catholic institution had abdicated its responsibility. He eventually died at Vancouver General Hospital with the help of a physician, but not before experiencing hours of unimaginable pain.
The thing about some of these Catholic hospitals, however, is that the staffers don't always agree with the Catholic Church. Doctors work there for a variety of reasons, and most of them have no reason to worry that their expertise would ever go against Catholic doctrine.
The majority of people, 52 per cent, said that assisted dying must be available to all Canadians in any institution that receives public funding, regardless of that institution's historical or religious affiliations.
The hospital's president, Bruce Roe, said the survey showed staff want to be able to continue supporting patients.
So there was overwhelming support within the hospital for various aspects of assisted dying to be made available to patients — and a majority of the staff wanted the hospital to follow the law and make the procedure available for patients who qualified for and requested the help.
No wonder the board passed the policy.
Unfortunately, St. Boniface Hospital doesn't get the final say in the matter. Despite being publicly funded, the hospital is managed by the Catholic Health Corp. of Manitoba, and the policy had to go through them.
So the CHCM did something downright nefarious.
They appointed 10 new members to the St. Boniface Hospital board of directors, which already had 16 people on it, and asked for a revote.
Wouldn't you know it? The new board voted down the Be-Less-Cruel policy.
On Monday, the new board — minus two original members, including its former chairman — passed the unamended policy banning assisted dying at St. Boniface.
That's religion for you. It doesn't matter what people want. It doesn't matter how awful or misguided your thinking is. It's Catholic doctrine and therefore it's set in stone… until someone further up the chain decides to change it.
Patients will suffer because Catholics refuse to have compassion for the people who need help the most. Keep in mind that very few people meet the qualifications to even request this procedure. It's not like the original policy meant anyone could just demand death and doctors would have to go along with it. And realize that most people don't get to choose which hospital they go to in the event of an emergency.
There is no rational justification to overturn the policy. And using a loophole to shoot it down sends a strong message to the doctors and administrators at that hospital that their expertise can be ignored and their wisdom doesn't matter because the Vatican knows better than everyone.


Christened 10 Jun 1839 St James, Church James's Street.
Ellen has her Son-in-Law living with her and his children, But he is a Widow.
She is now On her own.
[S21] 1911 Census, (Got from Archive Library.), 1911 Census. Got from Library. (Reliability: 4), 12 Oct 2007.


text:
E-books: Book Series
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Karen O'Brien
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The Humanities Research Centre of the University of Warwick in collaboration with Ashgate has re-launched its book series. Warwick Studies in the Humanities aims to bring together innovative work of a high academic standard which crosses disciplinary borders in the Arts and Humanities. It provides a forum for volumes exploring new dimensions of cultural history from the early modern period to the present, and for works that investigate aspects of contemporary cultural production within and across national boundaries. The series reflects the breadth of the interdisciplinary work carried out at Warwick's Humanities Research Centre, and includes work of both European and extra-European scope.
German Culture and the Uncomfortable Past
Representations of National Socialism in Contemporary Germanic Literature, 1st Edition
Edited by Helmut Schmitz
Beginning with the question of the role of the past in the shaping of a contemporary identity, this volumes spans three generations of German and Austrian writers and explores changes and shifts in the aesthetics of Vergangenheitsbewältigung (coming to terms with the past). The purpose of the book…
Staging Dario Fo and Franca Rame
Anglo-American Approaches to Political Theatre, 1st Edition
By Stefania Taviano
This is the first extended treatment of the English translations, stagings, and reception of the political plays of Dario Fo and Franca Rame. Focusing on the United Kingdom and the United States, Stefania Taviano offers a critique of the cultural stereotyping and political conservatism that have…
Ignazio Silone in Exile
Writing and Antifascism in Switzerland 1929–1944, 1st Edition
By Deborah Holmes
Italian writer and political activist Ignazio Silone spent fifteen years from 1929 to 1944 as a political exile in Switzerland. Focusing on this period, this book throws new light on Silone's complex biography and shows how his literary production influenced and was influenced by fellow antifascist…
Illuminating Eco
On the Boundaries of Interpretation, 1st Edition
By Charlotte Ross
Edited by Rochelle Sibley
Illuminating Eco covers the range of British scholarship on the prolific literary and theoretical work of Umberto Eco. With essays by scholars such as Michael Caesar and David Robey, the volume provides an overview of current research being carried out by a new generation of academics. In addition,…
Epistolary Selves
Letters and Letter-Writers, 1600–1945, 1st Edition
Edited by Rebecca Earle
This volume of ten essays discusses the pivotal role that letters have played in social, economic and political history from the seventeenth to the twentieth century. The recent scholarly interest in the history of reading has as yet yielded few studies which consider letters as a category of…
George Eliot and Europe
Edited by John Rignall
This book is based on a conference held in Warwick in July 1995. It is a collection of essays which explore various aspects of George Eliot's relation to the literature and culture of Continental Europe. The essays range widely over the novelist's life and work, examining her Journals and…


Discussion in 'NOD32 version 2 Forum' started by Mister Natural, Sep 2, 2008.
I received a popup on my system running 2.7.32 with RA console and mirror offering a component update. I was wondering if this component is NODv3 being offered or a 2.7 update?
The latest version of the 2.70.x series is v2.70.39, so you were most likely being offered the opportunity to update to that newer release.
ESET is not performing Program Component Updates for going from NOD32 v2.70 to the v3.0 series, as customers may wish to deploy ESET NOD32 Antivirus v3.0 with different configurations than used with NOD32 v2.70, or be deploying ESET Smart Security in place of NOD32 v2.70.


text:
Q: How can I conform to ShapeStyle protocol in SwiftUI? I am looking to conform to ShapeStyle, the goal is just conforming to ShapeStyle(Nothing more and nothing less).
Here is my code and the issue:
struct MyContentView: View {
    var body: some View {
        Circle()
            .fill(RedView())
    }
}

struct RedView: View {
    var body: some View {
        Color.red
    }
}

So I know I could use Color.red directly instead of a view, but as I said, the goal of this question is conformation to ShapeStyle. For those going to share apple developer link about ShapeStyle, I have to say I was there before.
Error:

Instance method 'fill(_:style:)' requires that 'RedView' conform to 'ShapeStyle'

A: Disclaimer: below is only for demo, because all used API are private (by convention, not by design), so reaction of Apple if you submit app with such code to AppStore is unpredictable.
Note: as used interfaces are private they can be changed with any new version.
Main part:
  public func _apply(to shape: inout SwiftUI._ShapeStyle_Shape) {
    Color.red._apply(to: &shape)  // << here !!
  }

Which gives on Xcode 13.3 / iOS 15.4

Complete findings and code is here

A: If you want any kind of View to conform to ShapeStyle, then @Asperi provided here the answer.
If you want a Color to conform to ShapeStyle like in your example, it already adopts that protocol, so you just need a little tweak to make it work, like this:
struct RedView {
    var render: some ShapeStyle {
        Color.red
    }
}

struct MyContentView: View {
    var body: some View {
        Circle()
            .fill(RedView().render)
    }
}

A: From the documentation for ShapeStyle:

You don't use the ShapeStyle protocol directly. Instead, use one of the concrete styles that SwiftUI defines. To indicate a specific color or pattern, you can use Color or the style returned by image(_:sourceRect:scale:), or one of the gradient types, like the one returned by radialGradient(_:center:startRadius:endRadius:). To set a color that's appropriate for a given context on a given platform, use one of the semantic styles, like background or primary.

I believe what you're trying to do can be achieved with:
struct MyContentView: View {
    var body: some View {
        Circle()
            .fill(.red)
    }
}

redpajama_set_name: RedPajamaStackExchange


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4 Ghost: Okay, granted that's a decent start.
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Last Modified: Saturday, 2 January 2010; 02:11:01 PST.