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Closing Game

Diagram​

Overview​

A closing game εX: (X,X)→(1,1) terminates a composition chain by computing final utilities. It takes accumulated game outputs and produces scalar outcomes, converting open game structures into closed forms suitable for equilibrium analysis.

Mathematical Structure​

The counit εX: (X,X)→(1,1) is defined by:

  • Σε = {*}: Single trivial strategy
  • Pε(*,x) = *: Play function maps to unit
  • Cε(*,x,*) = x: Coplay returns the forward value
  • (*,*)∈Bε(x,k) for all contexts: Always in equilibrium

The coplay function identifies the forward value x as the backward utility.

Key Properties​

  • Terminal operation: Marks end of composition chain
  • Utility realization: Computes final payoffs
  • Boundary elimination: Converts (X,X) to scalar (1,1)
  • No further composition: Output is final

Role in Composition​

Closing games provide the counit in the teleological category structure. They enable backward utility flow by identifying forward outputs with backward utilities. Essential for defining Nash equilibria in compositional games—without closing, games remain open-ended.

Example​

Protocol termination computing final payoffs:

Input: Negotiation results, executed transactions Closing: Compute final utilities for all agents Output: Scalar payoff profile No further moves possible The counit "bends" the wire, feeding outputs back as utilities

This represents the moment when a game ends and utilities are realized, enabling equilibrium analysis of the complete interaction.