nLab game theory

Idea

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.

Via coalgebras

Several authors have approached game theory through coalgebraic systems theory, defining games as elements of the terminal coalgebra of an appropriate functor. This puts the concepts of state and state transition in a game to centre stage. It is closely related to extensive form games, whereby a game is defined by its tree of plays.

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

Compositional game theory

Open games approach games as open systems, making them morphisms of a symmetric monoidal category which can be depicted as string diagrams, in the spirit of applied category theory. See Ghani, Hedges, Winschel and Zahn 2018.

Open games have a close connection to lenses. This can be used to situate open games inside a symmetric monoidal double category in a game-theoretically interesting way, see Hedges 2018.

Topology and game theory

Deformations between strategy profiles have been considered as a practical method to compute Nash equilibria (see Herings and Peeters 2010).

References

General

• $n$Cafe: combinatorial game categories, zero determinant strategies in the iterated prisoner’s dilemma
• wikipedia: game theory, combinatorial game theory, dominance (game theory), Nash equilibrium, wild type
• William Press, Freeman Dyson, Iterated prisoner’s dilemma contains strategies that dominate any evolutionary opponent, PNAS open access, March 2012.
• J.R.B. Cockett, G.S.H. Cruttwell and K. Saff, Combinatorial game categories, pdf
• André Joyal, Remarques sur la théorie des jeux à deux personnes, Gazette des

  Sciences Mathematiques du Québec 1(4):46–52, 1977; Robin Houston’s rough translation can be found here

• André Joyal, 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.
• Dusko Pavlovic, A semantical approach to equilibria and rationality, CALCO 2009. (arXiv:0905.3548, doi:10.1007/978-3-642-03741-2_22)
• Samson Abramsky and Viktor Winschel?, Coalgebraic analysis of subgame-perfect equilibria in infinite games without discounting, Mathematical structures in computer science 2017. (arXiv:1210.4537, doi:10.1017/S0960129515000365)
• Neil Ghani, Clemens Kupke?, Alasdair Lambert? and Fredrik Nordvall Forsberg, A compositional treatment of iterated open games, Theoretical computer science 2018. (arXiv:1711.07968, doi:10.1016/j.tcs.2018.05.026)
• Neil Ghani, Jules Hedges, Viktor Winschel? and Philipp Zahn?, Compositional game theory, LiCS 2018. (arXiv:1603.04641, doi:10.1145/3209108.3209165)
• Jules Hedges, Morphisms of open games, MFPS 2018. (arXiv:1711.07059, doi:10.1016/j.entcs.2018.11.008)
• P. Jean-Jacques Herings and Ronald Peeters, Homotopy methods to compute equilibria in game theory, Economic Theory 2010. (doi:10.1007/s00199-009-0441-5)

Via modal logic

On epistemic (S5) modal logic in game theory and information science:

• Robert J. Aumann, Interactive epistemology I: Knowledge, International Journal of Game Theory 28 (1999) 263–300 $[$doi:10.1007/s001820050111$]$

• Wiebe van der Hoek, Marc Pauly, Modal logic for games and information, Ch. 20 in: The Handbook of Modal Logic, Studies in Logic and Practical Reasoning 3 (2007) 85-183 $[$doi:10.1016/S1570-2464(07)80023-1, book webpage$]$

• Hans Ditmarsch, Wiebe Hoek, Barteld Kooi, Dynamic Epistemic Logic, Studies In Epistemology, Logic, Methodology, And Philosophy Of Science (SYLI) 337, Springer (2008) [doi:10.1007/978-1-4020-5839-4_2, pdf]