Compositional game theory

We introduce open games as a compositional foundation of economic game theory. A compositional approach potentially allows methods of game theory and theoretical computer science to be applied to large-scale economic models for which standard economic tools are not practical. An open game represents a game played relative to an arbitrary environment and to this end we introduce the concept of coutility, which is the utility generated by an open game and returned to its environment. Open games are the morphisms of a symmetric monoidal category and can therefore be composed by categorical composition into sequential move games and by monoidal products into simultaneous move games. Open games can be represented by string diagrams which provide an intuitive but formal visualisation of the information flows. We show that a variety of games can be faithfully represented as open games in the sense of having the same Nash equilibria and off-equilibrium best responses.Comment: This version submitted to LiCS 201

Compositional Game Theory, compositionally

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

Composing games into complex institutions

Game theory is used by all behavioral sciences, but its development has long centered around tools for relatively simple games and toy systems, such as the economic interpretation of equilibrium outcomes. Our contribution, compositional game theory, permits another approach of equally general appeal: the high-level design of large games for expressing complex architectures and representing real-world institutions faithfully. Compositional game theory, grounded in the mathematics underlying programming languages, and introduced here as a general computational framework, increases the parsimony of game representations with abstraction and modularity, accelerates search and design, and helps theorists across disciplines express real-world institutional complexity in well-defined ways. Relative to existing approaches in game theory, compositional game theory is especially promising for solving game systems with long-range dependencies, for comparing large numbers of structurally related games, and for nesting games into the larger logical or strategic flows typical of real world policy or institutional systems.Comment: ~4000 words, 6 figure

Towards compositional game theory

PhDI gratefully acknowledge that my PhD studies were funded by EPSRC doctoral training grant EP/K50290X/1.I gratefully acknowledge that my PhD studies were funded by EPSRC doctoral training grant EP/K50290X/1.I gratefully acknowledge that my PhD studies were funded by EPSRC doctoral training grant EP/K50290X/1.We introduce a new foundation for game theory based on so-called open games. Unlike existing approaches open games are fully compositional: games are built using algebraic operations from standard components, such as players and outcome functions, with no fundamental distinction being made between the parts and the whole. Open games are intended to be applied at large scales where classical game theory becomes impractical to use, and this thesis therefore covers part of the theoretical foundation of a powerful new tool for economics and other subjects using game theory. Formally we defi ne a symmetric monoidal category whose morphisms are open games, which can therefore be combined either sequentially using categorical composition, or simultaneously using the monoidal product. Using this structure we can also graphically represent open games using string diagrams. We prove that the new de finitions give the same results (both equilibria and o -equilibrium best responses) as classical game theory in several important special cases: normal form games with pure and mixed strategy Nash equilibria, and perfect information games with subgame perfect equilibria. This thesis also includes work on higher order game theory, a related but simpler approach to game theory that uses higher order functions to model players. This has been extensively developed by Martin Escard o and Paulo Oliva for games of perfect information, and we extend it to normal form games. We show that this approach can be used to elegantly model coordination and di fferentiation goals of players. We also argue that a modifi cation of the solution concept used by Escard o and Oliva is more appropriate for such applications.EPSRC doctoral training grant EP/K50290X/1

Morphisms of open games

We define a notion of morphisms between open games, exploiting a surprising connection between lenses in computer science and compositional game theory. This extends the more intuitively obvious definition of globular morphisms as mappings between strategy profiles that preserve best responses, and hence in particular preserve Nash equilibria. We construct a symmetric monoidal double category in which the horizontal 1-cells are open games, vertical 1-morphisms are lenses, and 2-cells are morphisms of open games. States (morphisms out of the monoidal unit) in the vertical category give a flexible solution concept that includes both Nash and subgame perfect equilibria. Products in the vertical category give an external choice operator that is reminiscent of products in game semantics, and is useful in practical examples. We illustrate the above two features with a simple worked example from microeconomics, the market entry game

The game semantics of game theory

We use a reformulation of compositional game theory to reunite game theory with game semantics, by viewing an open game as the System and its choice of contexts as the Environment. Specifically, the system is jointly controlled by $n \geq 0$ noncooperative players, each independently optimising a real-valued payoff. The goal of the system is to play a Nash equilibrium, and the goal of the environment is to prevent it. The key to this is the realisation that lenses (from functional programming) form a dialectica category, which have an existing game-semantic interpretation. In the second half of this paper, we apply these ideas to build a compact closed category of `computable open games' by replacing the underlying dialectica category with a wave-style geometry of interaction category, specifically the Int-construction applied to the cartesian monoidal category of directed-complete partial orders

Composing games into complex institutions

Game theory is used by all behavioral sciences, but its development has long centered around the economic interpretation of equilibrium outcomes in relatively simple games and toy systems. But game theory has another potential use: the high-level design of large game compositions that express complex architectures and represent real-world institutions faithfully. Compositional game theory, grounded in the mathematics underlying programming languages, and introduced here as a general computational framework, increases the parsimony of game representations with abstraction and modularity, accelerates search and design, and helps theorists across disciplines express real-world institutional complexity in well-defined ways. Relative to existing approaches in game theory, compositional game theory is especially promising for solving game systems with long-range dependencies, for comparing large numbers of structurally related games, and for nesting games into the larger logical or strategic flows typical of real world policy or institutional systems