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Unattainability of Common Knowledge in Asymmetric Games with Imperfect Information

Fabian Farestam, Dilian Gurov

TL;DR

The paper addresses whether common knowledge, formalized as $C\,Off$, can be attained in highly asymmetric, imperfect-information games. It introduces MAGIIAN, a minimal two-agent scenario where a human acts and an AI observes without acting, and uses an iterative Kripke-structure update to analyze evolving knowledge states. The main contribution is a formal proof that, for action sequences such as $i r^n$, the knowledge state never reaches $C\,Off$, meaning common knowledge of the alarm being off is unattainable. This result highlights fundamental limits on coordination in decentralized systems with uneven information access and informs the design of robust, communication-restricted decision processes.

Abstract

In this paper, we present a conceptual model game to examine the dynamics of asymmetric interactions in games with imperfect information. The game involves two agents with starkly contrasting capabilities: one agent can take actions but has no information of the state of the game, whereas the other agent has perfect information of the state but cannot act or observe the other agent's actions. This duality manifests an extreme form of asymmetry, and how differing abilities influence the possibility of attaining common knowledge. Using Kripke structures and epistemic logic we demonstrate that, under these conditions, common knowledge of the current game state becomes unattainable. Our findings advance the discussion on the strategic limitations of knowledge in environments where information and action are unevenly distributed.

Unattainability of Common Knowledge in Asymmetric Games with Imperfect Information

TL;DR

The paper addresses whether common knowledge, formalized as , can be attained in highly asymmetric, imperfect-information games. It introduces MAGIIAN, a minimal two-agent scenario where a human acts and an AI observes without acting, and uses an iterative Kripke-structure update to analyze evolving knowledge states. The main contribution is a formal proof that, for action sequences such as , the knowledge state never reaches , meaning common knowledge of the alarm being off is unattainable. This result highlights fundamental limits on coordination in decentralized systems with uneven information access and informs the design of robust, communication-restricted decision processes.

Abstract

In this paper, we present a conceptual model game to examine the dynamics of asymmetric interactions in games with imperfect information. The game involves two agents with starkly contrasting capabilities: one agent can take actions but has no information of the state of the game, whereas the other agent has perfect information of the state but cannot act or observe the other agent's actions. This duality manifests an extreme form of asymmetry, and how differing abilities influence the possibility of attaining common knowledge. Using Kripke structures and epistemic logic we demonstrate that, under these conditions, common knowledge of the current game state becomes unattainable. Our findings advance the discussion on the strategic limitations of knowledge in environments where information and action are unevenly distributed.
Paper Structure (9 sections, 3 theorems, 5 equations, 5 figures)

This paper contains 9 sections, 3 theorems, 5 equations, 5 figures.

Table of Contents

  1. Introduction
  2. A two-agent game with imperfect information
  3. Background
  4. Games with imperfect information
  5. Epistemic logic
  6. An iterative model update construction
  7. Unattainability of common knowledge
  8. Related Work
  9. Conclusion

Key Result

Lemma 4.1

In our game, the equivalence relations on the histories satisfy the following inductive properties: where $h_1, h_2 \in H_G$ and $a, b \in Act$.

Figures (5)

  • Figure 1: The scenario presented as a game graph.
  • Figure 2: the resulting sequences and the relations between them.
  • Figure 3: Model update, applied to $\mathcal{M}_2$ (step 1), resulting in $\mathcal{M}_3$ (step 4).
  • Figure 4: $\mathcal{M}_{1}$ with $h_1=\text{Start i Off r Off}$$\leadsto \mathcal{M}_{2}$ with $h_1=\text{Start i Off r Off r Off}$$\leadsto \mathcal{M}_{3}$ with $h_1=\text{Start i Off r Off r Off r Off}$
  • Figure 5: Visualisation of why only two jumps are required after the update.

Theorems & Definitions (11)

  • Definition 3.1: MAGIIAN
  • Definition 3.2: Full play
  • Definition 3.3: Full history
  • Definition 3.4: KT45$^n$ syntax
  • Definition 3.5: Kripke structure
  • Definition 3.6: KT45$^n$ semantics
  • Definition 3.7: Induced model
  • Lemma 4.1
  • Lemma 4.2: Model update
  • Theorem 5.1
  • ...and 1 more