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Revisiting Common Randomness, No-signaling and Information Structure in Decentralized Control

Apurva Dhingra, Ankur A. Kulkarni

TL;DR

The paper challenges the view that the no-signaling condition exactly captures decentralized information structures by showing that some NS strategies nevertheless require coordination beyond passive common randomness. It demonstrates, through counterexamples and polytope geometry, that active common randomness or one-way communication can realize NS distributions outside the local polytope, implying a mismatch between NS and the strict no-communication requirement. A posterior-distribution perspective links NS to conditional independence properties, offering a complementary way to interpret NS in terms of beliefs about observations. Overall, the work calls for a broader or differently defined information-structure framework and highlights that NS can be strictly larger than what the decentralized control setting would permit under a no-communication assumption.

Abstract

This work revisits the no-signaling condition for decentralized information structures. We produce examples to show that within the no-signaling polytope exist strategies that cannot be achieved by passive common randomness but instead require agents to either share their observations with a mediator or communicate directly with each other. This poses a question mark on whether the no-signaling condition truly captures the decentralized information structure in the strictest sense.

Revisiting Common Randomness, No-signaling and Information Structure in Decentralized Control

TL;DR

The paper challenges the view that the no-signaling condition exactly captures decentralized information structures by showing that some NS strategies nevertheless require coordination beyond passive common randomness. It demonstrates, through counterexamples and polytope geometry, that active common randomness or one-way communication can realize NS distributions outside the local polytope, implying a mismatch between NS and the strict no-communication requirement. A posterior-distribution perspective links NS to conditional independence properties, offering a complementary way to interpret NS in terms of beliefs about observations. Overall, the work calls for a broader or differently defined information-structure framework and highlights that NS can be strictly larger than what the decentralized control setting would permit under a no-communication assumption.

Abstract

This work revisits the no-signaling condition for decentralized information structures. We produce examples to show that within the no-signaling polytope exist strategies that cannot be achieved by passive common randomness but instead require agents to either share their observations with a mediator or communicate directly with each other. This poses a question mark on whether the no-signaling condition truly captures the decentralized information structure in the strictest sense.
Paper Structure (18 sections, 3 theorems, 31 equations, 1 figure, 1 table)

This paper contains 18 sections, 3 theorems, 31 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Background
  3. Notations
  4. Common randomness and decentralized control
  5. Behavioural strategies $\mathscr{B}$:
  6. Local strategies $\mathscr{L}$:
  7. No-signaling strategies $\mathscr{NS}$
  8. No-signaling with passive common randomness
  9. Anantharam and Borkar's conjecture and counterexample
  10. Geometry of no-signaling and Counter-example \ref{['Table : ccex']}
  11. No-signaling with active common randomness
  12. No-signaling with active common randomness
  13. Counterexample of Conjecture \ref{['conj:conj1']}:
  14. Where is the communication?
  15. Strategy within $\mathscr{NS}$ with one-way communication
  16. ...and 3 more sections

Key Result

Proposition 5.1

A strategy $\strat$ satisfies the no-signaling conditions eq:ns if and only if following posterior condition is satisfied

Figures (1)

  • Figure 1: A schematic representation of no-signaling polytope. L and NL label the vertices for local and non-local points. The set bounded by local vertices is $\mathscr{L}$ and overall is the no-signaling polytope $\mathscr{NS}$. Example \ref{['Table : ccex2']} represents a distribution $P(X,Y|A,B) \in \mathscr{NS} \backslash \mathscr{L}$.

Theorems & Definitions (9)

  • Definition 2.1
  • Conjecture 3.1
  • Conjecture 4.1
  • Proposition 5.1
  • proof
  • Lemma 7.1
  • proof
  • Lemma 7.2
  • proof