We introduce open games as a compositional foundation of economic game theory. An open game is played relative to an arbitrary, open environment. We show that a variety of games can be faithfully modelled as open games in the sense of having the same Nash equilibria and off-equilibrium best responses. Open games can be represented by string diagrams which provide an intuitive but formal visualisation of the information flows. A key feature of compositional game theory is its tight link to functional programming. Game theoretic models so can be seamlessly translated into computer code making it relevant for software systems based on economic designs such as auctions.
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