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The Curse of Shared Knowledge: Recursive Belief Reasoning in a Coordination Game with Imperfect Information

Thomas Bolander, Robin Engelhardt, Thomas S. Nicolet

TL;DR

Common knowledge enables safe coordination but is often unattainable in real settings. The authors design a novel recursive coordination game (the Canteen Dilemma) with imperfect information to compare common knowledge with nth-order shared knowledge, using 802 participants across MTurk and classroom experiments. Across results, participants behave as if they possess common knowledge even when only shallow $n$th-order shared knowledge exists, incurring nontrivial penalties for miscoordination. The study introduces the 'curse of shared knowledge' and highlights important implications for understanding Theory of Mind, coordination under uncertainty, and the design of human-AI and multi-agent systems.

Abstract

Common knowledge is crucial for safe group coordination. In its absence, humans must rely on shared knowledge, which is inherently limited in depth and therefore prone to coordination failures, because any finite-order knowledge attribution allows for an even higher order attribution that may change what is known by whom. In three separate experiments involving 802 participants, we investigate the extent to which humans can differentiate between common knowledge and nth-order shared knowledge. We designed a two-person coordination game with imperfect information to simplify the recursive game structure and higher-order uncertainties into a relatable everyday scenario. In this game, coordination for the highest payoff requires a specific fact to be common knowledge between players. However, this fact cannot become common knowledge in the game. The fact can at most be nth-order shared knowledge for some n. Our findings reveal that even at quite shallow depths of shared knowledge (low values of n), players behave as though they possess common knowledge, and claim similar levels of certainty in their actions, despite incurring significant penalties when falsely assuming guaranteed coordination. We term this phenomenon 'the curse of shared knowledge'. It arises either from the players' inability to distinguish between higher-order shared knowledge and common knowledge, or from their implicit assumption that their co-player cannot make this distinction.

The Curse of Shared Knowledge: Recursive Belief Reasoning in a Coordination Game with Imperfect Information

TL;DR

Common knowledge enables safe coordination but is often unattainable in real settings. The authors design a novel recursive coordination game (the Canteen Dilemma) with imperfect information to compare common knowledge with nth-order shared knowledge, using 802 participants across MTurk and classroom experiments. Across results, participants behave as if they possess common knowledge even when only shallow th-order shared knowledge exists, incurring nontrivial penalties for miscoordination. The study introduces the 'curse of shared knowledge' and highlights important implications for understanding Theory of Mind, coordination under uncertainty, and the design of human-AI and multi-agent systems.

Abstract

Common knowledge is crucial for safe group coordination. In its absence, humans must rely on shared knowledge, which is inherently limited in depth and therefore prone to coordination failures, because any finite-order knowledge attribution allows for an even higher order attribution that may change what is known by whom. In three separate experiments involving 802 participants, we investigate the extent to which humans can differentiate between common knowledge and nth-order shared knowledge. We designed a two-person coordination game with imperfect information to simplify the recursive game structure and higher-order uncertainties into a relatable everyday scenario. In this game, coordination for the highest payoff requires a specific fact to be common knowledge between players. However, this fact cannot become common knowledge in the game. The fact can at most be nth-order shared knowledge for some n. Our findings reveal that even at quite shallow depths of shared knowledge (low values of n), players behave as though they possess common knowledge, and claim similar levels of certainty in their actions, despite incurring significant penalties when falsely assuming guaranteed coordination. We term this phenomenon 'the curse of shared knowledge'. It arises either from the players' inability to distinguish between higher-order shared knowledge and common knowledge, or from their implicit assumption that their co-player cannot make this distinction.

Paper Structure

This paper contains 24 sections, 4 theorems, 2 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. The curse of shared knowledge
  3. Experimental Design: Materials and Methods
  4. Game Strategies
  5. Results
  6. Discussion
  7. Conclusion and future work
  8. Experimental design and data collection
  9. Mixed effect logistic regression
  10. Payoffs and Penalties
  11. Formal Analysis
  12. Materials and Methods: Further Details
  13. Mechanical Turk (MTurk) Experiment, Including Walkthrough and Screenshots
  14. DTU Experiments
  15. MTurk Settings
  16. ...and 9 more sections

Key Result

Lemma 1

If a strategy $s$ is Pareto optimal then $s(t) =o$ for all $t \geq 9{:}00$.

Figures (15)

  • Figure 1: Percentage of canteen choices as a function of arrival times. The colored lines are logistic regression lines with 95% confidence intervals shown as translucent bands. Fitted parameters show significant differences in the slope and intercept between MTurk and DTU experiments ($p < .0001$).
  • Figure 2: Violin plots of certainty estimates. In each round, participants were asked how certain they were of successful coordination with their colleague. Blue areas show the results from MTurk ($\text{n}=4260$) and orange areas show the results from DTU1 and DTU2 combined ($\text{n}=3172$). We predefined a five point likert scale of certainty estimates as: 'very uncertain', 'slightly certain, 'somewhat certain', 'quite certain', and 'very certain', and translated them into the numerical values of probability estimates used in the payoff calculations (see Appendix \ref{['appendix:payoffs']}). The white dots correspond to the median certainty estimate.
  • Figure 3: Percentage of coordinations and miscoordinations as a function of arrival times. Green means coordinating into the canteen, purple means coordinating into the office, and red means miscoordination. We use the notation 8:00/8:10 to denote the union of the arrival pairs $(8{:}00,8{:}10)$ and $(8{:}10,8{:}00)$, i.e., the arrival time combinations where one of the players arrive at $8{:}00$ and the other at $8{:}10$. Miscoordinations approach $50\%$ at 8:40/8:50 and 8:50/9:00.
  • Figure 4: Mean frequencies of canteen choices for all possible arrival times as a function of the number of rounds played. The three fitted horizontal lines, corresponding to arrival times $8{:}40$, $8{:}50$, and $9{:}00$ respectively, represent weighted linear squares (WLS) with the weights chosen to be the square root of the number of data points constituting the mean frequencies for each round, also shown by dot size.
  • Figure 5: Logistic regression of the relationship between card number and choice of a white stone in the qualitative experiment at DIS. The resulting curve shows a very similar overall trend as in the main experiments shown in Fig. \ref{['fig:1']}.
  • ...and 10 more figures

Theorems & Definitions (9)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Definition 4
  • Theorem 5
  • proof