Information Revelation Through Signalling

TL;DR

This paper characterizes the minimum number of source symbols a sender can reliably communicate in a signaling game, linking it to a graph-theoretic measure called the vertex clique cover number, and explores implications for equilibrium types.

Contribution

It provides an exact, utility-function-based characterization of the sender's informativeness in signaling games, applicable across all equilibria, and compares it with the receiver's information gain as a leader.

Findings

Informativeness equals the vertex clique cover number of a utility-induced graph.

Informativeness determines the existence of separating, pooling, and semi-separating equilibria.

Informativeness exceeds the receiver's information when the receiver is the leader.

Abstract

This paper studies a Stackelberg game wherein a sender (leader) attempts to shape the information of a less informed receiver (follower) who in turn takes an action that determines the payoff for both players. The sender chooses signals to maximize its own utility function while the receiver aims to ascertain the value of a source that is privately known to the sender. It is well known that such sender-receiver games admit a vast number of equilibria and not all signals from the sender can be relied on as truthful. Our main contribution is an exact characterization of the minimum number of distinct source symbols that can be correctly recovered by a receiver in \textit{any} equilibrium of this game; we call this quantity the \textit{informativeness} of the sender. We show that the informativeness is given by the \textit{vertex clique cover number} of a certain graph induced by the utility function, whereby it can be computed based on the utility function alone without the need to enumerate all equilibria. We find that informativeness characterizes the existence of well-known classes of separating, pooling and semi-separating equilibria. We also compare informativeness with the amount of information obtained by the receiver when it is the leader and show that the informativeness is always greater than the latter, implying that the receiver is better off being a follower. Additionally, we also show that when the players play behavioral strategies, an equilibrium may not exist.