\begin{algorithm}
\caption{Greedy algorithm for the menu selection problem}\label{alg:greedy}
\begin{algorithmic}
\State Initialize $O\gets \emptyset$.\footnote{This can be replaced with any other menu of public goods with no change to the analysis below.} 
\While{$O$ is not $(t,u)$-stable}
\If{$O$ is not $t$-feasible}
\State By definition there exists $j\in O$ such that $|j \succ O \setminus \{ j \} | <t$.
\State Let $j$ be a minimal\newcounter{minimalfootnote}\setcounter{minimalfootnote}{\thefootnote}\footnote{This can be replaced with any other consistent tie-breaking with no change to the analysis below.} such $j$, and update $O\gets O\setminus\{j\}$.
\ElsIf{$O$ is not $u$-uncontestable\footnote{This item can be consistently swapped with the preceding one with no change to the analysis below.}}
\State By definition there exists $j\in G\setminus O$ such that $|j\succ O|\ge u$. 
\State Let $j$ be a minimal\newcounter{savedcurrentfootnote}\setcounter{savedcurrentfootnote}{\thefootnote}\setcounter{footnote}{\theminimalfootnote}\footnotemark\setcounter{footnote}{\thesavedcurrentfootnote} such $j$, and update $O\gets O\cup\{j\}$.
\EndIf
\EndWhile
\end{algorithmic}
\end{algorithm}

\begin{algorithm}
\caption{Greedy algorithm for the menu selection problem}\begin{algorithmic}
\State Initialize $O\gets \emptyset$.\footnote{This can be replaced with any other menu of public goods with no change to the analysis below.}
\While{$O$ is not $(t,u)$-stable}
\If{$O$ is not $t$-feasible}
\State By definition there exists $j\in O$ such that $|j \succ O \setminus \{ j \} | <t$.
\State Let $j$ be a minimal\newcounter{minimalfootnote}\setcounter{minimalfootnote}{\thefootnote}\footnote{This can be replaced with any other consistent tie-breaking with no change to the analysis below.} such $j$, and update $O\gets O\setminus\{j\}$.
\ElsIf{$O$ is not $u$-uncontestable\footnote{This item can be consistently swapped with the preceding one with no change to the analysis below.}}
\State By definition there exists $j\in G\setminus O$ such that $|j\succ O|\ge u$.
\State Let $j$ be a minimal\newcounter{savedcurrentfootnote}\setcounter{savedcurrentfootnote}{\thefootnote}\setcounter{footnote}{\theminimalfootnote}\footnotemark\setcounter{footnote}{\thesavedcurrentfootnote} such $j$, and update $O\gets O\cup\{j\}$.
\EndIf
\EndWhile
\end{algorithmic}
\end{algorithm}
{
    "title": "stable menus of public goods: a matching problem",
    "id": "2402.11370",
    "abstract": "we study a matching problem between agents and public goods, in settings without monetary transfers. since goods are public, they have no capacity constraints. there is no exogenously defined budget of goods to be provided. rather, each provided good must justify its cost, leading to strong complementarities in the \"preferences\" of goods. furthermore, goods that are in high demand given other already-provided goods must also be provided. the question of the existence of a stable solution (a menu of public goods to be provided) exhibits a rich combinatorial structure. we uncover sufficient conditions and necessary conditions for guaranteeing the existence of a stable solution, and derive both positive and negative results for strategyproof stable matching.",
    "categories": "cs.gt econ.th math.co",
    "doi": "",
    "created": "2024-02-17",
    "updated": "",
    "authors": [
        "sara fish",
        "yannai a. gonczarowski",
        "sergiu hart"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2402.11370"
}
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