\begin{algorithm}
\caption{Compute $c^*$}\label{alg:find_cstar}
\begin{algorithmic}
\Require $a_i, B_i, \theta_i$
\\
\State Sort the values $a_i, \theta_i$ according to $\frac{a_{ij}}{\theta_{ij}}$ in a descending order.
If there are goods with $\theta_{ij} = 0$, sort them separately according to $a_{ij}$ and place them as a prefix (lower indices) before the other sorted goods. Equal values are sorted in a lexicographical order. 
\\
\State Set: $a \gets 0,\quad  \theta \gets 0,\quad c_s\gets 0,\quad c^* \gets 0$
\For{$j = 1, \dots, m$}
\State $a \gets a + a_{ij},\ \theta \gets \theta + \theta_{ij}$
\State $c_s \gets \frac{a}{\theta + B_i}$
\State $c^* \gets \max\{c^*, c_s\}$
\EndFor
\State \Return $c^*$
\end{algorithmic}
\end{algorithm}

\begin{algorithm}
\caption{Compute $c^*$}\begin{algorithmic}
\Require $a_i, B_i, \theta_i$
\\
\State Sort the values $a_i, \theta_i$ according to $\frac{a_{ij}}{\theta_{ij}}$ in a descending order.
If there are goods with $\theta_{ij} = 0$, sort them separately according to $a_{ij}$ and place them as a prefix (lower indices) before the other sorted goods. Equal values are sorted in a lexicographical order.
\\
\State Set: $a \gets 0,\quad \theta \gets 0,\quad c_s\gets 0,\quad c^* \gets 0$
\For{$j = 1, \dots, m$}
\State $a \gets a + a_{ij},\ \theta \gets \theta + \theta_{ij}$
\State $c_s \gets \frac{a}{\theta + B_i}$
\State $c^* \gets \max\{c^*, c_s\}$
\EndFor
\State \Return $c^*$
\end{algorithmic}
\end{algorithm}
{
    "title": "asynchronous proportional response dynamics in markets with adversarial   scheduling",
    "id": "2307.04108",
    "abstract": "we study proportional response dynamics (prd) in linear fisher markets where participants act asynchronously. we model this scenario as a sequential process in which in every step, an adversary selects a subset of the players that will update their bids, subject to liveness constraints. we show that if every bidder individually uses the prd update rule whenever they are included in the group of bidders selected by the adversary, then (in the generic case) the entire dynamic converges to a competitive equilibrium of the market. our proof technique uncovers further properties of linear fisher markets, such as the uniqueness of the equilibrium for generic parameters and the convergence of associated best-response dynamics and no-swap regret dynamics under certain conditions.",
    "categories": "cs.gt cs.ma econ.th math.ds",
    "doi": "",
    "created": "2023-07-09",
    "updated": "2024-01-15",
    "authors": [
        "yoav kolumbus",
        "menahem levy",
        "noam nisan"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2307.04108"
}
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