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Copying Information

Diagram​

Overview​

Copying information duplicates data across parallel game components via the diagonal map Δ: X→X×X. This enables broadcast communication where the same information reaches multiple agents, supporting parallel decision-making with shared context.

Mathematical Structure​

The copy operation Δ: (X,1)→(X×X,1) is the covariant lifting of the diagonal:

  • Δ(x) = (x,x): Duplicates input
  • (Δ,1): (X,1)→(X×X,1): Strategically trivial open game
  • Forms comonoid structure: associative and has identity
  • Preserves information content perfectly

Combined with parallel composition: (G₁⊗G₂)◦Δ broadcasts x to both games.

Key Properties​

  • Non-destructive: Original preserved
  • Perfect fidelity: Copies identical to source
  • Arbitrary fan-out: Create any number of copies
  • Information preservation: No loss in copying

Role in Composition​

Copying enables parallel games to share inputs. The pattern (id⊗Δ)◦G feeds G's output to multiple subsequent games. Essential for broadcast scenarios where multiple agents must receive identical information independently.

Example​

Broadcasting information to multiple agents:

Input: Market update x Copy: Δ(x) = (x,x,...,x) Distribution: Each agent receives identical copy Enables: Parallel independent responses All agents see the same information simultaneously

This models scenarios like auction announcements where all bidders receive identical item descriptions, enabling fair simultaneous bidding.