Skip to main content

Atomic Game

Diagram​

Overview​

An atomic game is an indivisible open game that cannot be decomposed into smaller components. It represents a single decision point where one agent observes context and chooses an action. Atomic games are the primitive building blocks from which all composite games are constructed.

Mathematical Structure​

A decision D: (X,1)→(Y,R) is atomic with:

  • ΣD = X→Y: Strategies are functions from observations to choices
  • PD(σ,x) = σ(x): Play function applies strategy to observation
  • CD(σ,x,r) = *: Coplay returns unit (no coutility generated)
  • (σ,σ')∈BD(x,k) iff σ'(x) ∈ arg max k: Best response maximizes continuation utility

Key Properties​

  • Indivisible: Cannot decompose into sub-games
  • Single agent: One decision maker
  • Function-valued strategies: σ: X→Y maps each observation to a choice
  • Context-dependent optimality: Best response depends on utility function k: Y→R

Role in Composition​

Atomic games compose to form complex interactions. Sequential composition D₁◦D₂ creates multi-stage decisions where the second agent observes the first's action. Parallel composition D₁⊗D₂ models simultaneous choices by independent agents. The atomic property ensures well-defined strategy spaces in compositions.

Example​

Single decision moment that cannot be decomposed:

Context: x (e.g., market signal from past) Play: σ(x) (response chosen, flows to future) Coutility: r (utility from future outcome, flows backward) Continuation: * (no state returned to past, coplay returns unit) Strategy: σ: X→Y mapping context to play Cannot decompose: This represents one indivisible choice

This is the most basic game structure—one agent, one observation, one choice, with the minimal bidirectional flow.