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Sequential Game

Diagram​

Overview​

A sequential game G₁◦G₂ connects open games in series. The output of G₁ becomes the input to G₂, creating temporal ordering where later players observe earlier actions. This models perfect information settings where decision-making proceeds in stages.

Mathematical Structure​

For G: (X,S)→(Y,R) and H: (Y,R)→(Z,Q), their composition H◦G: (X,S)→(Z,Q) is defined by:

  • ΣH◦G = ΣG × ΣH: Strategy profiles are tuples
  • PH◦G((σ,τ), x) = PH(τ, PG(σ,x)): Sequential play
  • CH◦G((σ,τ), x, q) = CG(σ, x, CH(τ, PG(σ,x), q)): Composed coplay
  • Best responses decompose: each game optimizes given the other's play

Key Properties​

  • Temporal ordering: Games play in sequence
  • Information flow: Outputs become inputs
  • Observable history: Later players see earlier actions
  • Backward induction: Analysis proceeds from final stage

Role in Composition​

Sequential composition ◦ is the categorical composition operator. It enables multi-stage protocols where each stage depends on previous outcomes. Combines with parallel ⊗ to model complex game trees with both temporal and simultaneous structure.

Example​

Two-stage decision process:

Stage 1: Agent A observes context x, chooses action a Stage 2: Agent B observes (x, a), chooses action b Output: Combined result (a, b) Agent B's choice depends on observing A's prior action

This illustrates perfect information: the second agent sees everything the first agent saw, plus the first agent's choice.