### Abstract

We present a new compositional approach to compositional game theory (CGT) based upon Arrows, a concept originally from functional programming, closely related to Tambara modules, and operators to build new Arrows from old. We model equilibria as a module over an Arrow and define an operator to build a new Arrow from such a module over an existing Arrow. We also model strategies as graded Arrows and define an operator which builds a new Arrow by taking the colimit of a graded Arrow. A final operator builds a graded Arrow from a graded bimodule. We use this compositional approach to CGT to show how known and previously unknown variants of open games can be proven to form symmetric monoidal categories.

Original language | English |
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Number of pages | 27 |

Publication status | Published - 6 Jul 2020 |

Event | Applied Category Theory 2020 - Duration: 6 Jul 2020 → 10 Jul 2020 https://act2020.mit.edu/ |

### Conference

Conference | Applied Category Theory 2020 |
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Abbreviated title | ACT 2020 |

Period | 6/07/20 → 10/07/20 |

Internet address |

### Keywords

- compositional game theory (CGT)
- equilibria
- Arrow
- compositional approach

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## Cite this

Atkey, R., Gavranović, B., Ghani, N., Kupke, C., Ledent, J., & Nordvall Forsberg, F. (2020).

*Compositional Game Theory, compositionally*. Paper presented at Applied Category Theory 2020, .