On the Number of Bins in Equilibria for Signaling Games

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.

Signaling Games for Log-Concave Distributions: Number of Bins and Properties of Equilibria

“…where Γ e and Γ d are the sets of all deterministic (and Borel measurable) functions from M to X and from X to M, respectively. As observed from the definition in (1), under a Nash equilibrium, each individual player chooses an optimal strategy given the strategy chosen by the other player. The quantized nature of Nash equilibria for cheap talk problem [2] motivates the following definition of a scalar quantizer.…”

“…Due to results obtained in [2] and [3], we know that the encoder policy consists of convex bins at a Nash equilibrium 1 . Namely, at a Nash equilibrium, the encoder must employ quantization policies with the cells B i in Definition 1 being intervals.…”

“…, N −1. Due to the definition of Nash equilibrium in (1), these best responses in ( 2) and ( 3) must match each other, and only then can the equilibrium be characterized; i.e., for a given number of bins, the positions of the bin edges are chosen by the encoder, and the centroids are determined by the decoder. Alternatively, the problem can be considered as a quantization game in which the boundaries are determined by the encoder and the reconstruction values are determined by the decoder.…”

Strategic Communication with Decoder Side Information

“…Communication between autonomous devices that have distinct objectives is under study. This problem, referred to as the strategic communication problem, is at the crossroads of different disciplines such as Control Theory [1], [2], Computer Science [3] and Information Theory [4], [7], [8], [9], [10], [11], [12], where it was introduced by Akyol et al in [5], [6].…”

Signaling Games for Log-Concave Distributions: Number of Bins and Properties of Equilibria

“…In these games/problems, the objective functions of the players are not aligned, unlike in classical communication problems. The cheap talk problem was introduced in the economics literature by Crawford and Sobel [2], who obtained the striking result that under some technical conditions on the cost functions, the cheap talk problem only admits equilibria that involve quantized encoding policies, i.e., the observation space is partitioned into intervals and the encoder reveals Part of this work was presented at the 2019 IEEE International Symposium on Information Theory (ISIT), July 7-12, 2019, Paris, France [1]. E. Kazıklı and S. Gezici are with the Department of Electrical and Electronics Engineering, Bilkent University, 06800, Ankara, Turkey, Tel: +90-312-290-3139, Fax: +90-312-266-4192, Emails: {kazikli,gezici}@ee.bilkent.edu.tr.…”