[Paper Review] Learning and Selfconfirming Equilibria in Network Games

Pierpaolo Battigalli, Fabrizio Panebianco+1 more|arXiv (Cornell University)|Dec 31, 2018

Game Theory and ApplicationsDecision Sciences5 references3 citations

TL;DR

This paper studies learning dynamics in network games where agents have incomplete information about the network structure and only observe their own realized payoffs. It characterizes selfconfirming equilibria as steady states of belief-updating processes, showing that such equilibria can differ from Nash equilibria under imperfect feedback, and establishes conditions under which learning converges to stable, self-confirming action profiles via conjectural best-reply processes.

ABSTRACT

Consider a set of agents who play a network game repeatedly. Agents may not know the network. They may even be unaware that they are interacting with other agents in a network. Possibly, they just understand that their payoffs depend on an unknown state that is, actually, an aggregate of the actions of their neighbors. Each time, every agent chooses an action that maximizes her instantaneous subjective expected payoff and then updates her beliefs according to what she observes. In particular, we assume that each agent only observes her realized payoff. A steady state of the resulting dynamic is a selfconfirming equilibrium given the assumed feedback. We characterize the structure of the set of selfconfirming equilibria in the given class of network games, we relate selfconfirming and Nash equilibria, and we analyze simple conjectural best-reply paths whose limit points are selfconfirming equilibria.

Motivation & Objective

• To analyze how incomplete information about network structure affects learning and equilibrium outcomes in strategic network interactions.
• To characterize the set of selfconfirming equilibria that emerge when agents only observe their own payoffs and update beliefs accordingly.
• To investigate the relationship between selfconfirming equilibria and Nash equilibria under imperfect feedback and limited observability.
• To examine the stability and convergence of learning dynamics via conjectural best-reply processes in network games with local and global externalities.
• To establish conditions under which learning processes converge to selfconfirming equilibria, particularly through contraction mapping arguments on belief update systems.

Proposed method

• Models repeated play in network games where agents update beliefs based solely on observed payoffs, not on others' actions.
• Introduces a belief-updating mechanism where agents form subjective expectations about the payoff-relevant state (e.g., aggregate neighbor actions) and best-respond to them.
• Uses a conjectural variation of best-reply dynamics, where agents adjust actions based on estimated payoff states derived from observed payoffs.
• Applies the Gershgorin circle theorem to the Jacobian of the system to prove stability of rest points, ensuring convergence of learning paths.
• Employs a continuous-time dynamic system defined by equations (15)–(16), with feedback from realized payoffs and belief updates via (17) and (29).
• Analyzes the invertibility and monotonicity of the action profile mapping to the network and payoff parameters, ensuring unique and stable equilibria.

Experimental results

Research questions

• RQ1 Under what conditions do learning processes in network games converge to selfconfirming equilibria rather than Nash equilibria?
• RQ2 How does limited observability—specifically, only observing one’s own payoff—affect the structure and stability of equilibria?
• RQ3 What is the relationship between selfconfirming equilibria and Nash equilibria in network games with incomplete network knowledge?
• RQ4 How do local and global externalities influence the convergence and stability of learning dynamics in such games?
• RQ5 Under what conditions is the learning process a contraction mapping, ensuring convergence to a unique selfconfirming equilibrium?

Key findings

• Selfconfirming equilibria can exist that are not Nash equilibria, especially when agents lack full information about the network or others’ actions.
• The system of equations (16) admits a unique, continuous, and strictly monotonic action profile mapping for any given network and parameter vector, ensuring well-defined equilibrium outcomes.
• The Jacobian of the learning dynamics has row sums bounded between -1 and 1, which, via the Gershgorin circle theorem, guarantees that the system is a contraction and thus converges to a stable fixed point.
• As action levels increase, the row sum of the Jacobian approaches ∑j≠i zij − 1, which is bounded in absolute value by 1 under the model’s assumptions.
• The limit of the row sum as ai → 0 is greater than -1 and approaches -1 only in the limit as ci → 0 and c′i → 0, ensuring stability across the domain.
• The mapping from action profiles to payoff parameters is invertible due to strict monotonicity and continuity, which supports the uniqueness and stability of the equilibrium.