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Parallel Game

Diagram​

Overview​

A parallel game G₁⊗G₂ executes multiple open games simultaneously. Each game receives the same input independently and produces outputs that are combined. This models simultaneous move scenarios where agents act without observing each other's choices.

Mathematical Structure​

For G₁: (X₁,S₁)→(Y₁,R₁) and G₂: (X₂,S₂)→(Y₂,R₂), the product G₁⊗G₂: (X₁×X₂, S₁×S₂)→(Y₁×Y₂, R₁×R₂) is defined by:

  • ΣG₁⊗G₂ = ΣG₁ × ΣG₂: Strategy profiles are pairs
  • P((σ₁,σ₂), (x₁,x₂)) = (PG₁(σ₁,x₁), PG₂(σ₂,x₂)): Independent play
  • C((σ₁,σ₂), (x₁,x₂), (r₁,r₂)) = (CG₁(σ₁,x₁,r₁), CG₂(σ₂,x₂,r₂)): Parallel coplay
  • Best responses consider equilibrium play of other agents

Key Properties​

  • Simultaneity: All games execute at once
  • Independent play: No direct interaction during execution
  • Shared input: Same context broadcast to all agents
  • Combined output: Results aggregated into tuple

Role in Composition​

Parallel composition ⊗ is the monoidal product operator. Enables simultaneous move games and independent agent decisions. Composes with sequential ◦ to create complex game trees with both temporal and parallel structure.

Example​

Simultaneous auction with multiple bidders:

Input: Item information broadcast to all agents Agents: A₁, A₂, ..., Aₙ each independently decide bid Output: Tuple of bids (b₁, b₂, ..., bₙ) No agent observes others' choices when deciding

This captures the essential feature of simultaneous moves: each agent must commit to a choice before observing what others choose.