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Open Game

Diagram​

Overview​

An open game G: (X,S)→(Y,R) is the fundamental structure in CGT. It maps observations X to choices Y while computing utilities R and coutilities S. The four components (strategy profiles, play function, coplay function, best response function) enable compositional reasoning about strategic interactions.

Mathematical Structure​

From Definition 3 in the paper, an open game G = (ΣG, PG, CG, BG) consists of:

  • ΣG: Set of strategy profiles
  • PG: ΣG × X → Y: Play function mapping strategies and observations to choices
  • CG: ΣG × X × R → S: Coplay function computing coutility
  • BG: X × (Y→R) → Rel(ΣG): Best response function defining equilibria relative to context

Key Properties​

  • Compositionality: Can combine with other open games via composition operators
  • Bidirectional flow: Forward play (X→Y) and backward coutility (R→S)
  • Context-dependent equilibria: Best responses relative to environment
  • Modular boundaries: Clear input/output interfaces enable reuse

Role in Composition​

Open games compose via ◦ (sequential) and ⊗ (parallel) operators. Sequential composition connects outputs to inputs, creating temporal ordering. Parallel composition combines independent games executing simultaneously. This enables building complex protocols from atomic components.

Example​

A basic decision game D: (X,1)→(Y,R) where an agent observes context and chooses an action:

Context: x (e.g., task specification observed from past) Play: σ(x) (choice made, flows to future) Coutility: r (utility generated by future, flows backward) Continuation: State update (flows back to past) Strategy: σ: X→Y mapping context to play

This represents the simplest non-trivial open game: one agent making one choice with bidirectional information flow.