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\input{sections/macros}

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\title{Finite-horizon Equilibria for Neuro-symbolic Concurrent Stochastic Games}

% The standard author block has changed for UAI 2022 to provide
% more space for long author lists and allow for complex affiliations
%
% All author information is authomatically removed by the class for the
% anonymous submission version of your paper, so you can already add your
% information below.
%
% Add authors
\author[1]{Rui~Yan \thanks{Equal Contributions.}}
\author[1]{Gabriel~Santos\hspace{2.5pt}$^*$}
\author[2]{Xiaoming~Duan}
\author[3]{David~Parker}
\author[1]{Marta~Kwiatkowska}
% Add affiliations after the authors
\affil[1]{%
    Department of Computer Science\\
    University of Oxford\\
    Oxford, UK
}
\affil[2]{%
     Department of Automation\\
     Shanghai Jiao Tong University\\
     Shanghai, China
}
\affil[3]{%
    School of Computer Science\\
    University of Birmingham\\
    Birmingham, UK
  }
  \begin{document}
\maketitle
\begin{abstract}
We present novel techniques
for neuro-symbolic concurrent stochastic games,
a recently proposed modelling formalism to represent a set of probabilistic agents
operating in a continuous-space environment
using a combination of neural network based perception mechanisms
and traditional symbolic methods.
To date, only zero-sum variants of the model were studied,
which is too restrictive when agents have distinct objectives.
%
We formalise notions of equilibria for these models
and present algorithms to synthesise them.
Focusing on the finite-horizon setting,
and (global) social welfare subgame-perfect optimality,
we consider two distinct types: Nash equilibria and correlated equilibria.
% undiscounted
We first show that an exact solution based on backward induction
% based on local optimality for NS-CSGs,
may yield arbitrarily bad equilibria.
We then propose an approximation algorithm
called frozen subgame improvement,
which proceeds through iterative solution of nonlinear programs.
% encodes the computation of SPEs as a nonlinear program.
% The algorithm proceeds by solving a smaller nonlinear program iteratively to ease the
% scalability issues and can monotonically improve the social welfare.
We develop a prototype implementation and demonstrate the
benefits of our approach on two case studies: an automated
car-parking system and an aircraft collision avoidance system.
% the VerticalCAS (VCAS) aircraft system. 
\end{abstract}

\input{sections/introduction}

\input{sections/background}

\input{sections/algorithms}

\input{sections/experiments}

\section{Conclusions}

We have considered finite-horizon equilibria computation for %the formalism of
CSGs whose agents are equipped with NN-based perception mechanisms.
We developed an approximate algorithm that improves on social welfare equilibria values and strategies, for both SPNE and SPCE, compared to backward induction, which can only reason about local optimality. A prototype implementation showcased its applicability and advantages on two case studies. Future work will focus on infinite-horizon properties (incorporating finite-horizon equilibria with receding horizon synthesis \citep{VR-MF-TW-RMM:15}) and temporal logic specifications.

% \rui{There are two formats for references (full and abbreviated names). Which one should we use?}\marta{abbreviated takes less space, but check with the Call fro Papers} \rui{The template adopts the full case. I will follow that. (two extra pages for references, and it should be fine)}


% \begin{contributions} % will be removed in pdf for initial submission,
%                       % so you can already fill it to test with the
%                       % ‘accepted’ class option
%     Briefly list author contributions.
%     This is a nice way of making clear who did what and to give proper credit.

%     H.~Q.~Bovik conceived the idea and wrote the paper.
%     Coauthor One created the code.
%     Coauthor Two created the figures.
% \end{contributions}

\begin{acknowledgements} % will be removed in pdf for initial submission,
                         % so you can already fill it to test with the
                         % ‘accepted’ class option
This project was funded by the ERC under the European Union’s Horizon 2020 research and innovation programme (\href{http://www.fun2model.org}{FUN2MODEL}, grant agreement No.~834115).
\end{acknowledgements}

\bibliography{references}

% \clearpage

% \input{appendix}

\end{document}
