    \begin{algorithm}[H]
    \caption{Min-Max Projected Gradient Descent}\label{algo:minmax}
    \begin{algorithmic}
    \State Initialize \(k=1, \mathbf{w}\sim Uniform(|\Phi|)\)
    \While{\(k<k_{max}\)}
        \State \(\hat{J}=-\infty\)
        \For{\(\sigma_{\text{test}} \in [0,1]\)}
            \If{\(J(\mathbf{w}, \sigma_{\text{test}}) > \hat{J}\)}
                \State \(\sigma \gets \sigma_{\text{test}}\)
                \State \(\hat{J} \gets J(\mathbf{w}, \sigma_{\text{test}})\)
            \EndIf
        \EndFor
        \State Compute gradient \(\nabla_\mathbf{w} J(\mathbf{w},\sigma)\)
        \State \(\mathbf{w} \gets \text{Proj}_{L_2}(\mathbf{w} - \alpha \nabla_\mathbf{w} J(\mathbf{w},\sigma))\)
        \If{\(\left\|\nabla_\mathbf{w} J(\mathbf{w},\sigma) - \frac{\nabla_\mathbf{w} J(\mathbf{w},\sigma)^\top \mathbf{w}}{\|\mathbf{w}\|^2} \mathbf{w}\right\| < \beta\)}
            \State terminate with \(\mathbf{w}\).
        \EndIf
        \State \(k\gets k+1\)
    \EndWhile
    \end{algorithmic}
    \end{algorithm}

\begin{algorithm}
[H]
\caption{Min-Max Projected Gradient Descent} \begin{algorithmic}
\State Initialize \(k=1, \mathbf{w}\sim Uniform(|\Phi|)\)
\While{\(k<k_{max}\)}
\State \(\hat{J}=-\infty\)
\For{\(\sigma_{\text{test}} \in [0,1]\)}
\If{\(J(\mathbf{w}, \sigma_{\text{test}}) > \hat{J}\)}
\State \(\sigma \gets \sigma_{\text{test}}\)
\State \(\hat{J} \gets J(\mathbf{w}, \sigma_{\text{test}})\)
\EndIf
\EndFor
\State Compute gradient \(\nabla_\mathbf{w} J(\mathbf{w},\sigma)\)
\State \(\mathbf{w} \gets \text{Proj}_{L_2}(\mathbf{w} - \alpha \nabla_\mathbf{w} J(\mathbf{w},\sigma))\)
\If{\(\left\|\nabla_\mathbf{w} J(\mathbf{w},\sigma) - \frac{\nabla_\mathbf{w} J(\mathbf{w},\sigma)^\top \mathbf{w}}{\|\mathbf{w}\|^2} \mathbf{w}\right\| < \beta\)}
\State terminate with \(\mathbf{w}\).
\EndIf
\State \(k\gets k+1\)
\EndWhile
\end{algorithmic}
\end{algorithm}
{
    "title": "relationship design for socially-aware behavior in static games",
    "id": "2207.06392",
    "abstract": "autonomous agents can adopt socially-aware behaviors to reduce social costs, mimicking the way animals interact in nature and humans in society. we present a new approach to model socially-aware decision-making that includes two key elements: bounded rationality and inter-agent relationships. we capture the interagent relationships by introducing a novel model called a relationship game and encode agents' bounded rationality using quantal response equilibria. for each relationship game, we define a social cost function and formulate a mechanism design problem to optimize weights for relationships that minimize social cost at the equilibrium. we address the multiplicity of equilibria by presenting the problem in two forms: min-max and min-min, aimed respectively at minimization of the highest and lowest social costs in the equilibria. we compute the quantal response equilibrium by solving a least-squares problem defined with its karush-kuhn-tucker conditions, and propose two projected gradient descent algorithms to solve the mechanism design problems. numerical results, including two-lane congestion and congestion with an ambulance, confirm that these algorithms consistently reach the equilibrium with the intended social costs.",
    "categories": "cs.ma cs.sy eess.sy",
    "doi": "",
    "created": "2022-07-13",
    "updated": "2024-01-25",
    "authors": [
        "shenghui chen",
        "yigit e. bayiz",
        "david fridovich-keil",
        "ufuk topcu"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2207.06392"
}
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