# On the Number of Bins in Equilibria for Signaling Games

## TL;DR

This paper analyzes the number of bins in equilibria for signaling games with misaligned objectives, focusing on exponential and Gaussian sources, and finds conditions for finite or infinite bin equilibria.

## Contribution

It refines previous bounds on the number of bins in equilibria for quadratic cheap talk, specifically for exponential and Gaussian sources, revealing conditions for infinite bin equilibria.

## Key findings

- For exponential sources, a relation between bin count and objective misalignment is established.
- Equilibria with infinitely many bins exist under certain conditions for both exponential and Gaussian sources.
- The upper bound on the number of bins can be infinite depending on source and parameter settings.

## Abstract

We investigate the equilibrium behavior for the decentralized quadratic cheap talk problem in which an encoder and a decoder, viewed as two decision makers, have misaligned objective functions. In prior work, we have shown that the number of bins under any equilibrium has to be at most countable, generalizing a classical result due to Crawford and Sobel who considered sources with density supported on $[0,1]$. In this paper, we refine this result in the context of exponential and Gaussian sources. For exponential sources, a relation between the upper bound on the number of bins and the misalignment in the objective functions is derived, the equilibrium costs are compared, and it is shown that there also exist equilibria with infinitely many bins under certain parametric assumptions. For Gaussian sources, it is shown that there exist equilibria with infinitely many bins.