A Topology for Information Structures Based on Approximate Common Knowledge

Concetti Chiave

This paper introduces the "almost common knowledge topology" on information structures, demonstrating it as the coarsest topology that ensures the continuity of equilibrium outcomes in games with incomplete information.

Sintesi

• Bibliographic Information: Bergemann, D., Morris, S., & Veiel, R. (2024). A Strategic Topology on Information Structures. arXiv preprint arXiv:2411.09149.
• Research Objective: This paper aims to characterize the coarsest topology on information structures that guarantees the continuity of equilibrium outcomes in games of incomplete information.
• Methodology: The authors develop a new topology called the "almost common knowledge topology" based on the concept of approximate common knowledge of interim beliefs. They then analyze the properties of this topology and its relationship to equilibrium outcomes in games.
• Key Findings: The paper's central finding is that the almost common knowledge topology is indeed the coarsest topology that generates continuity of equilibrium outcomes. This means that any other topology that also ensures continuity must be at least as fine as the almost common knowledge topology. Additionally, the authors prove that simple information structures, where each player has finitely many types with distinct first-order beliefs, form a dense subset within this topology.
• Main Conclusions: The almost common knowledge topology provides a robust framework for studying information structures in games. The density of simple information structures simplifies the analysis of information design problems, as focusing on these simpler structures is sufficient for many purposes.
• Significance: This research significantly contributes to the field of game theory by providing a new topological tool for analyzing information structures. The almost common knowledge topology offers a deeper understanding of how information affects strategic interactions and equilibrium outcomes.
• Limitations and Future Research: The paper primarily focuses on belief-invariant Bayes correlated equilibria. Exploring the implications of the almost common knowledge topology for other equilibrium concepts and solution concepts could be a fruitful avenue for future research. Additionally, investigating the topology's applications in specific economic or game-theoretic models would be of interest.

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A Strategic Topology on Information Structures

by Dirk Bergema... alle arxiv.org 11-15-2024

https://arxiv.org/pdf/2411.09149.pdf

Domande più approfondite

How might the almost common knowledge topology be applied to analyze information design problems in specific economic contexts, such as auctions or signaling games?

The almost common knowledge (ACK) topology provides a powerful tool for analyzing information design problems due to its ability to characterize the robustness of equilibrium outcomes to perturbations in the information structure. Here's how it can be applied to auctions and signaling games:
Auctions:

Optimal Information Disclosure: In auction design, a central question is how much information the seller should reveal to maximize revenue. The ACK topology can help determine how sensitive the optimal information disclosure policy is to small changes in the bidders' beliefs. For instance, if a slight change in the information structure leads to a significant shift in equilibrium bidding behavior, the seller might prefer a more robust disclosure policy.
Collusion: The ACK topology can be used to analyze the robustness of collusive agreements between bidders. If a collusive equilibrium is highly sensitive to small changes in the information structure, it is less likely to be sustainable. This is because slight misperceptions or uncertainties about the signals received by other bidders could lead to a breakdown of the collusive agreement.
Interdependent Values: In auctions with interdependent values, bidders' valuations depend on both their private information and the information held by other bidders. The ACK topology can be used to study how the degree of information interdependence affects equilibrium outcomes and the seller's revenue.
Signaling Games:

Robustness of Signaling Equilibria: Signaling games often exhibit multiple equilibria. The ACK topology can be used to assess the robustness of these equilibria to perturbations in the information structure. For example, if a small change in the sender's signaling strategy leads to a large change in the receiver's beliefs and actions, the original equilibrium might be considered less plausible.
Information Design by the Sender: The sender in a signaling game might have an incentive to manipulate the information structure to achieve a more favorable outcome. The ACK topology can be used to analyze the sender's optimal information design problem and determine how much the sender can benefit from strategically disclosing or withholding information.
Cheap Talk: The ACK topology can be applied to cheap talk games, where the sender's messages have no direct bearing on payoffs. It can help determine the conditions under which informative communication can arise in equilibrium and how robust such communication is to small changes in the players' beliefs.
In both auctions and signaling games, the ACK topology allows us to move beyond simply identifying equilibrium outcomes and delve into the strategic implications of different information structures. By understanding how sensitive equilibrium outcomes are to perturbations in beliefs, we can gain valuable insights into the design of robust mechanisms and the strategic behavior of players in these settings.

Could there be alternative notions of "closeness" between information structures that capture other relevant aspects of strategic behavior beyond equilibrium outcomes?

Yes, absolutely. While the almost common knowledge topology elegantly captures the robustness of equilibrium outcomes, alternative notions of "closeness" can be conceived to capture other crucial aspects of strategic behavior. Here are a few possibilities:

Complexity-Based Closeness: Instead of focusing solely on equilibrium outcomes, we could define closeness based on the complexity of strategic reasoning required by the players. Two information structures could be considered close if they induce similar levels of strategic complexity, even if their equilibrium outcomes differ. This notion of closeness would be particularly relevant in settings where bounded rationality or cognitive limitations play a significant role.
Learning-Based Closeness: We could define closeness based on the speed and efficiency of learning under different information structures. Two information structures could be considered close if they lead to similar learning dynamics and convergence rates to equilibrium. This notion of closeness would be particularly relevant in dynamic games where players learn about the underlying state or the actions of others over time.
Welfare-Based Closeness: Instead of focusing solely on individual payoffs, we could define closeness based on the overall welfare implications of different information structures. Two information structures could be considered close if they lead to similar levels of social welfare, even if the distribution of payoffs among players differs. This notion of closeness would be particularly relevant in mechanism design problems where the designer aims to maximize social welfare or achieve a particular social objective.
Behavioral Closeness: Instead of assuming perfectly rational players, we could define closeness based on the similarity of predicted behavior under different information structures using behavioral models. Two information structures could be considered close if they lead to similar deviations from rational play or similar patterns of biases and heuristics. This notion of closeness would be particularly relevant in experimental economics and behavioral game theory.
These are just a few examples, and the most appropriate notion of closeness will depend on the specific economic context and the research question at hand. By exploring alternative notions of closeness, we can gain a more nuanced understanding of the relationship between information and strategic behavior beyond the traditional equilibrium framework.

If we relax the assumption of a common prior, how would the definition of the almost common knowledge topology and its properties change?

Relaxing the common prior assumption significantly impacts the definition and properties of the almost common knowledge (ACK) topology. Here's a breakdown of the key changes and challenges:
Definition:

Multiple Priors: Instead of a single common prior, each player i would have their own prior belief μi ∈ Δ(Θ) over the state space.
Belief Disagreements: The definition of the ACK topology would need to account for the possibility of persistent belief disagreements among players, even after observing their signals. This is because, without a common prior, players might interpret the same information differently and arrive at different posterior beliefs.
Higher-Order Beliefs: The notion of "closeness" between information structures would need to be extended to encompass not only first-order beliefs about the state but also higher-order beliefs about other players' beliefs. This is because, in the absence of a common prior, higher-order beliefs can significantly impact strategic behavior.
Challenges and Modifications:

No Universal Type Space: Without a common prior, a universal type space that captures all possible hierarchies of beliefs might not exist. This is because players might have different and incompatible views about the possible beliefs of others.
Measurability Issues: Defining common knowledge and approximate common knowledge becomes more challenging without a common prior. The events used to define these concepts might not be measurable with respect to all players' beliefs.
Equilibrium Existence: The existence of equilibrium, even weaker notions like correlated equilibrium, is not guaranteed without a common prior. This is because belief disagreements can lead to strategic uncertainty and prevent players from coordinating their actions.
Potential Approaches:

Common Supporting Beliefs: One approach is to focus on information structures where players have a common supporting belief, meaning there exists a single prior from which all players' beliefs can be derived by conditioning on their private information. This approach retains some of the tractability of the common prior setting while allowing for some degree of belief heterogeneity.
Type Spaces with Heterogeneous Priors: Another approach is to work with type spaces that explicitly model heterogeneous priors. This approach is more general but also more complex, as it requires specifying a prior for each player and a belief updating rule that governs how players revise their beliefs upon receiving new information.
Robustness to Perturbations: Instead of focusing on exact equilibrium notions, one could analyze the robustness of strategic behavior to small perturbations in players' beliefs. This approach relaxes the requirement of equilibrium and focuses on identifying strategies that perform well across a range of plausible beliefs.
Relaxing the common prior assumption significantly complicates the analysis of information structures and strategic behavior. However, it also opens up exciting avenues for research by allowing us to study the impact of belief disagreements and higher-order uncertainty on economic outcomes.