Open Games: A Compositional Game Theory Overview

• Open games are compositional game theory models defined as open systems with explicit forward (play) and backward (coplay) interfaces that facilitate the modeling of strategic interactions.
• They employ categorical tools such as lenses and symmetric monoidal categories to enable sequential and parallel composition, preserving equilibrium through context-dependent best responses.
• Recent extensions incorporate mixed strategies, Bayesian reasoning, and learning embeddings, allowing applications to extensive-form, iterated, and stochastic game frameworks.

Searching arXiv for recent and foundational papers on open games and related variants. Open games are a compositional foundation of game theory in which a game fragment is modeled as an open system interacting with an environment rather than as a closed normal-form or extensive-form object. In the foundational formulation, an open game $\mathcal G:(X,S)\to(Y,R)$ is specified by a set of strategy profiles, a play function, a coplay function, and a context-indexed best-response relation; the environment is summarized by an observation $x\in X$ and a continuation $k:Y\to R$, while backward-flowing coutility $S$ makes sequential and parallel composition possible (Ghani et al., 2016). Subsequent work recast this architecture in terms of lenses and symmetric monoidal categories, introduced morphisms and state-based solution concepts, extended the framework to mixed and Bayesian settings, related it to learning and backpropagation, and developed compositional accounts of extensive-form and infinitely iterated games (Hedges, 2017).

1. Foundational definition and semantic ingredients

In the original compositional formulation, an open game is a 4-tuple

$$\mathcal G = (\Sigma_{\mathcal G}, \mathbf P_{\mathcal G}, \mathbf C_{\mathcal G}, \mathbf B_{\mathcal G})$$

of type

$\mathcal G : (X,S) \to (Y,R),$

where $\Sigma_{\mathcal G}$ is a set of strategy profiles, $$\mathbf P_{\mathcal G} : \Sigma_{\mathcal G} \times X \to Y$$ is the play function, $$\mathbf C_{\mathcal G} : \Sigma_{\mathcal G} \times X \times R \to S$$ is the coplay function, and $$\mathbf B_{\mathcal G} : X \times (Y \to R) \to \operatorname{Rel}(\Sigma_{\mathcal G})$$ is the best-response function (Ghani et al., 2016). The pair $x\in X$0 is the input interface and $x\in X$1 the output interface. Forward-flowing data are observations and moves; backward-flowing data are utility-like values and coutilities.

A context is a pair $x\in X$2 with $x\in X$3 and $x\in X$4. The continuation $x\in X$5 summarizes “the future”: if the game outputs $x\in X$6, then $x\in X$7 is the value returned by the surrounding environment. Equilibrium is therefore relative to context. In the deterministic recasting reviewed by Bolt, Hedges, and Zahn, equilibrium in context is the fixed-point condition

$x\in X$8

(Bolt et al., 2019).

The additional backward type $x\in X$9 is the paper’s notion of coutility. It is the utility-like value generated by an open game and returned to its environment, and it is the main device that allows local game fragments to be composed without flattening them into a single global game (Ghani et al., 2016). This distinguishes open games from ordinary extensive-form or normal-form representations, where no separate backward interface is exposed.

The basic strategic atom is a decision node. For utility maximization, the paper defines a decision

$k:Y\to R$0

with strategy space $k:Y\to R$1, play given by evaluation, trivial coplay, and best response determined by $k:Y\to R$2 on the continuation (Ghani et al., 2016). More generally, the same paper defines $k:Y\to R$3 from a multi-valued selection function $k:Y\to R$4, which makes the framework compatible with higher-order game formulations beyond ordinary utility maximization.

2. Categorical and diagrammatic structure

Open games are the morphisms of a symmetric monoidal category. Sequential composition corresponds to categorical composition; simultaneous play corresponds to the monoidal product. On objects,

$k:Y\to R$5

and the monoidal unit is $k:Y\to R$6 (Ghani et al., 2016). Sequential composition transforms contexts by threading utility backward through coplay, while tensor product constructs unilateral-deviation continuations for each component from a joint continuation.

A major conceptual refinement is the identification of the non-strategic part of open games with lenses. In Hedges’ formulation, a lens

$k:Y\to R$7

is a pair $k:Y\to R$8 with

$k:Y\to R$9

and strategically trivial open games are essentially lenses (Hedges, 2017). This observation explains why open games have a bidirectional structure: play is the forward map and coplay is the backward update.

The same paper isolates the categorical structure of open games and lenses as a teleological category. Teleological categories are symmetric monoidal categories equipped with a wide symmetric monoidal subcategory of dualisable morphisms, an involutive symmetric monoidal functor

$S$0

and counits

$S$1

but no corresponding units in general (Hedges, 2017). This is the source of the oft-cited description “counits without units.” It explains why string diagrams for open games look reminiscent of compact closed categories while still allowing wire-bending only in one direction.

Diagrammatically, open games are represented with forward-flowing wires for observations and moves and backward-flowing wires for utility and coutility. Hedges proves a coherence theorem for this graphical language by constructing a free teleological category of diagrams and showing that diagrammatic equivalence coincides with semantic equality in every teleological category, including $S$2 (Hedges, 2017). That result formalizes a diagrammatic practice that had previously been used informally in compositional game theory.

3. Morphisms, states, and explicit agency

A second phase of the theory adds morphisms between open games. Hedges defines a notion of morphism $S$3 consisting of lenses

$S$4

together with a strategy map $S$5, subject to commutation and best-response preservation conditions (Hedges, 2017). With these morphisms, open games form a symmetric monoidal pseudo double category whose horizontal 1-cells are open games, whose vertical 1-morphisms are lenses, and whose 2-cells are morphisms of open games.

This added structure yields a flexible solution concept: states. A state of $S$6 is a strategy together with a continuation that is stable for all histories, and Hedges proves

$S$7

so states are morphisms out of the monoidal unit in the vertical category (Hedges, 2017). In that framework, ordinary Nash equilibria appear as states of suitable scalar games, while $S$8-separable states correspond to subgame perfect equilibria in sequential settings. The same paper shows that products in the vertical category act as an external choice operator and illustrates the construction on the market entry game.

A complementary development appears in “Translating Extensive Form Games to Open Games with Agency,” which argues that earlier formulations lacked an explicit notion of player and therefore of agency (Capucci et al., 2021). The revised framework separates an open game into an arena and a selection function. For a player set $S$9, strategy and reward spaces are indexed as

$$\mathcal G = (\Sigma_{\mathcal G}, \mathbf P_{\mathcal G}, \mathbf C_{\mathcal G}, \mathbf B_{\mathcal G})$$0

and the global selection function factors as a Nash product

$$\mathcal G = (\Sigma_{\mathcal G}, \mathbf P_{\mathcal G}, \mathbf C_{\mathcal G}, \mathbf B_{\mathcal G})$$1

This makes playerhood explicit and supports reparametrisation, regrouping, cloning, and internal and external choice operators.

Those operators are not ancillary. They are what makes it possible to translate extensive-form games with both perfect and imperfect information into open games while preserving equilibrium behavior (Capucci et al., 2021). In the imperfect-information case, a cloning lens forces all occurrences of an information set to share a single strategic choice. This translation result is one of the clearest demonstrations that open games can cover classical tree-based game theory while retaining compositional structure.

4. Probabilistic and Bayesian generalizations

Two major extensions relax the original deterministic, complete-information setting. The first is the mixed-strategy extension of compositional game theory. In “Compositional Game Theory with Mixed Strategies: Probabilistic Open Games Using a Distributive Law,” probabilistic open games keep the same basic interface structure, but the equilibrium function ranges over distributions: $$\mathcal G = (\Sigma_{\mathcal G}, \mathbf P_{\mathcal G}, \mathbf C_{\mathcal G}, \mathbf B_{\mathcal G})$$2 where $$\mathcal G = (\Sigma_{\mathcal G}, \mathbf P_{\mathcal G}, \mathbf C_{\mathcal G}, \mathbf B_{\mathcal G})$$3 is the discrete probability distribution monad (Ghani et al., 2020). Parallel and sequential composition are redefined using expected-utility semantics, the monad structure of $$\mathcal G = (\Sigma_{\mathcal G}, \mathbf P_{\mathcal G}, \mathbf C_{\mathcal G}, \mathbf B_{\mathcal G})$$4, and a natural transformation

$$\mathcal G = (\Sigma_{\mathcal G}, \mathbf P_{\mathcal G}, \mathbf C_{\mathcal G}, \mathbf B_{\mathcal G})$$5

used to lift equilibrium predicates through mixed state distributions. The resulting probabilistic open games again form a symmetric monoidal category. Matching Pennies is recovered with the usual mixed equilibrium $$\mathcal G = (\Sigma_{\mathcal G}, \mathbf P_{\mathcal G}, \mathbf C_{\mathcal G}, \mathbf B_{\mathcal G})$$6, and sequential settings recover backward-induction behavior in conditioned games (Ghani et al., 2020).

The second extension, “Bayesian open games” by Bolt, Hedges, and Zahn, reconstructs the framework over the Kleisli category $$\mathcal G = (\Sigma_{\mathcal G}, \mathbf P_{\mathcal G}, \mathbf C_{\mathcal G}, \mathbf B_{\mathcal G})$$7 of the finite-support distribution monad, not by naively replacing deterministic maps with stochastic ones, but by replacing ordinary lenses with coend lenses (Bolt et al., 2019). A Bayesian open game is a morphism in $$\mathcal G = (\Sigma_{\mathcal G}, \mathbf P_{\mathcal G}, \mathbf C_{\mathcal G}, \mathbf B_{\mathcal G})$$8, and a context can be represented concretely as $$\mathcal G = (\Sigma_{\mathcal G}, \mathbf P_{\mathcal G}, \mathbf C_{\mathcal G}, \mathbf B_{\mathcal G})$$9, where $\mathcal G : (X,S) \to (Y,R),$0 is a hidden state or type space, $\mathcal G : (X,S) \to (Y,R),$1 is a prior over hidden state and observation, and $\mathcal G : (X,S) \to (Y,R),$2 is a continuation. Bayesian updating is built in through

$\mathcal G : (X,S) \to (Y,R),$3

This extension preserves the central compositional slogan—games are morphisms and equilibrium is compositional—while capturing chance moves, stochastic strategies, incomplete information, and Bayesian Nash equilibrium (Bolt et al., 2019). The paper shows that standard Bayesian games, first-price sealed-bid auctions, and sequential asymmetric-information examples such as markets for lemons fit into the framework. A plausible implication is that the original open-games program survives probabilistic generalization only after substantial changes to its semantic infrastructure, especially in the treatment of contexts and hidden correlations.

5. Learning, iteration, and diegetic feedback

One line of work links open games to learning theory. In “From open learners to open games,” Hedges proves a faithful symmetric monoidal functor

$\mathcal G : (X,S) \to (Y,R),$4

from open learners to open games (Hedges, 2019). Under this embedding, parameter sets become strategy sets, implementation becomes play, request becomes coplay, and the best-response relation is defined by the learner’s update rule: $\mathcal G : (X,S) \to (Y,R),$5 The paper interprets a sufficiently simple supervised neural network as an open game in which each parameter is treated as if controlled by a different player, and the game’s best response relation encodes the dynamics of gradient descent (Hedges, 2019).

A second line concerns repetition and infinite play. “A Compositional Treatment of Iterated Open Games” introduces a conditioning operator $\mathcal G : (X,S) \to (Y,R),$6 that enforces optimality in every contingent subgame and then constructs an infinitely repeated game $\mathcal G : (X,S) \to (Y,R),$7 as the final coalgebra of the functor

$\mathcal G : (X,S) \to (Y,R),$8

(Ghani et al., 2017). In the iterated game,

$\mathcal G : (X,S) \to (Y,R),$9

so strategies are history-dependent, and plays live in $\Sigma_{\mathcal G}$0. Equilibrium is defined as the greatest fixpoint of a monotone operator on predicates over repeated-game strategies. This gives a coalgebraic semantics for infinite repetition and provides a compositional route to subgame perfection (Ghani et al., 2017).

A third line internalizes equilibrium reasoning itself. “Diegetic Representation of Feedback in Open Games” replaces extra-diegetic equilibrium predicates by internal backward propagation of payoff functions (Capucci, 2022). Instead of arbitrary utility and coutility types, the paper fixes a payoff object $\Sigma_{\mathcal G}$1 and uses

$\Sigma_{\mathcal G}$2

Backward propagation is functorial via

$\Sigma_{\mathcal G}$3

and the crucial lax monoidal structure is the Nashator

$\Sigma_{\mathcal G}$4

This turns a joint payoff function into unilateral deviation payoff functions. Players are then modeled as explicit subsystems over the arena, and the paper proves that their fixpoint behaviours are Nash equilibria (Capucci, 2022). The same work emphasizes a close analogy with backpropagation in learning systems, while stressing that games differ because the relevant feedback is unilateral rather than fully cooperative.

6. Distinct literatures sharing the term “open”

The phrase “open games” is not unique to compositional game theory. Several adjacent literatures use similar terminology for substantively different objects.

Notion	Core object	Relation to compositional open games
Open-source games	Transparent program-submission games	Distinct game-theoretic setting with program equilibria (Sistla et al., 29 Nov 2025)
Open-ended games	Growing policy populations and meta-games	Not the categorical open-games framework (Balduzzi et al., 2019, Nieves et al., 2021)
Open parity games	Parity games with graph interfaces	Interface-based, but explicitly distinguished from standard open games (Watanabe et al., 2021)
Open-open games / open determinacy	Topological or class-game determinacy notions	Unrelated use of “open” [(Kalemba et al., 2012); (Gitman et al., 2015)]

In the 2025 paper “Evaluating LLMs in Open-Source Games,” an open-source game is a setting in which each player submits a program $\Sigma_{\mathcal G}$5 that can inspect the opponent’s source code before returning an action, and a program equilibrium is a Nash equilibrium of the resulting game of submitted code (Sistla et al., 29 Nov 2025). That literature is concerned with transparent code, program inspection, and empirical approximate program equilibria under replicator dynamics, not with lenses, coutility, or compositional best-response semantics.

Likewise, the open-ended game literature studies growing policy populations, gamescapes, fictitious play, and PSRO-style solvers in non-transitive environments (Balduzzi et al., 2019, Nieves et al., 2021). The relevant objects are meta-games over expanding policy sets, not open systems $\Sigma_{\mathcal G}$6. “Open parity games,” by contrast, are interface-based parity games on graphs whose semantics are winning regions rather than equilibria; the paper introducing them explicitly notes that their interfaces are graph-theoretic open ends rather than the choice-and-utility interfaces of compositional game theory (Watanabe et al., 2021).

A common misconception is therefore that any interface-based or non-closed game model counts as an open game in the Ghani–Hedges–Winschel–Zahn sense. The literature does not support that identification. What the various usages share is a broad compositional intuition—games with exposed boundaries or extensible strategy spaces—but not a common formalism.

The standard compositional notion remains the one introduced as a foundation of economic game theory: open games are morphisms in a symmetric monoidal category, built from play, coplay, strategy profiles, and context-relative best response, with later work extending this core to teleological diagrams, morphisms and states, mixed and Bayesian reasoning, explicit agency, learning-theoretic embeddings, diegetic feedback, and infinite iteration (Ghani et al., 2016).