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\begin{document}

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\section*{game theory}

Game theory is the study of strategic interaction between agents. It is often used for mathematical modelling of real-world situations in which agents interact, including in microeconomics, computer science and ecology.

Game theory should not be confused with [[game semantics]], which has some terminology and history in common but has little overlap in practice.

\hypertarget{via_coalgebras}{}\subsection*{{Via coalgebras}}\label{via_coalgebras}

Several authors have approached game theory through coalgebraic systems theory, defining games as elements of the [[terminal coalgebra for an endofunctor|terminal coalgebra]] of an appropriate functor. This puts the concepts of \emph{state} and \emph{state transition} in a game to centre stage. It is closely related to \href{https://en.wikipedia.org/wiki/Extensive-form_game}{extensive form games}, whereby a game is defined by its tree of plays.

Perhaps the first paper to use this approach is \hyperlink{Pavlovic09}{Pavlovic 2009}. Others are \hyperlink{AbramskyWinschel17}{Abramsky and Winschel 2017} and \hyperlink{Ghani18a}{Ghani, Kupke, Lambert and Forsberg 2018}, the latter of which connects coalgebraic game theory with compositional game theory.

\hypertarget{compositional_game_theory}{}\subsection*{{Compositional game theory}}\label{compositional_game_theory}

Open games approach games as open systems, making them morphisms of a [[symmetric monoidal category]] which can be depicted as [[string diagram|string diagrams]], in the spirit of [[applied category theory]]. See \hyperlink{Ghani18b}{Ghani, Hedges, Winschel and Zahn 2018}.

Open games have a close connection to [[lens (in computer science)|lenses]]. This can be used to situate open games inside a symmetric monoidal [[double category]] in a game-theoretically interesting way, see \hyperlink{Hedges18}{Hedges 2018}.

\hypertarget{homotopy_and_game_theory}{}\subsection*{{Homotopy and game theory}}\label{homotopy_and_game_theory}

[[homotopy|Homotopies]] between strategy profiles have been considered as a practical method to compute Nash equilibria (see \hyperlink{Herings10}{Herings and Peeters 2010}). This has never been studied categorically, but it suggests a possible route towards a higher-categorical approach to game theory.

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item $n$Cafe: \href{http://golem.ph.utexas.edu/category/2009/11/combinatorialgame_categories.html}{combinatorial game categories}, \href{http://golem.ph.utexas.edu/category/2012/07/zerodeterminant_strategies_in.html}{zero determinant strategies} in the iterated prisoner's dilemma
\item wikipedia: \href{https://secure.wikimedia.org/wikipedia/en/wiki/Game_theory}{game theory}, \href{http://en.wikipedia.org/wiki/Combinatorial_game_theory}{combinatorial game theory}, \href{http://en.wikipedia.org/wiki/Dominance_%28game_theory%29}{dominance (game theory)}, \href{http://en.wikipedia.org/wiki/Nash_equilibrium}{Nash equilibrium}, \href{http://en.wikipedia.org/wiki/Wild_type}{wild type}
\item William Press, Freeman Dyson, \emph{Iterated prisoner's dilemma contains strategies that dominate any evolutionary opponent}, \href{http://www.pnas.org/content/early/2012/05/16/1206569109.full.pdf+html}{PNAS open access}, March 2012.
\item [[J.R.B. Cockett]], G.S.H. Cruttwell and K. Saff, \emph{Combinatorial game categories}, \href{https://www.mta.ca/uploadedFiles/Community/Bios/Geoff_Cruttwell/CGC.pdf}{pdf}
\item [[André Joyal]], \emph{Remarques sur la th\'e{}orie des jeux \`a{} deux personnes}, Gazette des Sciences Mathematiques du Qu\'e{}bec 1(4):46--52, 1977; Robin Houston's rough translation \href{http://www.ma.man.ac.uk/~rhouston/Joyal-games.ps}{ps}
\item Andr\'e{} Joyal, \emph{Free lattices, communication and money games}, in: Logic and scientific methods. Volume one of the proceedings of the tenth international congress of logic, methodology and philosophy of science, Florence, Italy, Synth. Libr. 259, pages 29--68. Dordrecht: Kluwer Academic Publishers, 1997.
\item [[Dusko Pavlovic]], \emph{A semantical approach to equilibria and rationality}, CALCO 2009. (\href{https://arxiv.org/abs/0905.3548}{arXiv:0905.3548}, \href{http://dx.doi.org/10.1007/978-3-642-03741-2_22}{doi:10.1007/978-3-642-03741-2\_22})
\item [[Samson Abramsky]] and [[Viktor Winschel]], \emph{Coalgebraic analysis of subgame-perfect equilibria in infinite games without discounting}, Mathematical structures in computer science 2017. (\href{https://arxiv.org/abs/1210.4537}{arXiv:1210.4537}, \href{https://doi.org/10.1017/S0960129515000365}{doi:10.1017/S0960129515000365})
\item [[Neil Ghani]], [[Clemens Kupke]], [[Alasdair Lambert]] and [[Fredrik Nordvall Forsberg]], \emph{A compositional treatment of iterated open games}, Theoretical computer science 2018. (\href{https://arxiv.org/abs/1711.07968}{arXiv:1711.07968}, \href{https://doi.org/10.1016/j.tcs.2018.05.026}{doi:10.1016/j.tcs.2018.05.026})
\item [[Neil Ghani]], [[Jules Hedges]], [[Viktor Winschel]] and [[Philipp Zahn]], \emph{Compositional game theory}, LiCS 2018. (\href{https://arxiv.org/abs/1603.04641}{arXiv:1603.04641}, \href{https://doi.org/10.1145/3209108.3209165}{doi:10.1145/3209108.3209165})
\item [[Jules Hedges]], \emph{Morphisms of open games}, MFPS 2018. (\href{https://arxiv.org/abs/1711.07059}{arXiv:1711.07059}, \href{https://doi.org/10.1016/j.entcs.2018.11.008}{doi:10.1016/j.entcs.2018.11.008})
\item P. Jean-Jacques Herings and Ronald Peeters, \emph{Homotopy methods to compute equilibria in game theory}, Economic Theory 2010. (\href{https://doi.org/10.1007/s00199-009-0441-5}{doi:10.1007/s00199-009-0441-5})

\end{itemize}


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