Signaling game problems investigate communication scenarios where encoder(s) and decoder(s) have misaligned objectives due to the fact that they either employ different cost functions or have inconsistent priors. This problem has been studied in the literature for scalar sources under various setups. In this paper, we consider multi-dimensional sources under quadratic criteria in the presence of a bias leading to a mismatch in the criteria, where we show that the generalization from the scalar setup is more than technical. We show that the Nash equilibrium solutions lead to structural richness due to the subtle geometric analysis the problem entails, with consequences in both system design, the presence of linear Nash equilibria, and an information theoretic problem formulation. We first provide a set of geometric conditions that must be satisfied in equilibrium considering any multi-dimensional source. Then, we consider independent and identically distributed sources and characterize necessary and sufficient conditions under which an informative linear Nash equilibrium exists. These conditions involve the bias vector that leads to misaligned costs. Depending on certain conditions related to the bias vector, the existence of linear Nash equilibria requires sources with a Gaussian or a symmetric density. Moreover, in the case of Gaussian sources, our results have a rate-distortion theoretic implication that achievable rates and distortions in the considered game theoretic setup can be obtained from its team theoretic counterpart.

翻译：信号博弈问题研究编码器与解码器因采用不同代价函数或先验不一致而导致目标错位的通信场景。该问题已在文献中针对标量源在多种设置下进行了研究。本文考虑在存在导致准则失配的偏差条件下，基于二次准则的多维源，并证明从标量设置泛化到多维情形不仅仅是技术性的。研究表明，纳什均衡解因问题所涉及的微妙几何分析而展现出结构丰富性，这对系统设计、线性纳什均衡的存在性以及信息论问题形式化均具有意义。我们首先给出任意多维源在均衡状态下必须满足的一组几何条件。随后，考虑独立同分布源，并刻画存在信息性线性纳什均衡的充要条件。这些条件涉及导致代价失配的偏差向量。根据与偏差向量相关的特定条件，线性纳什均衡的存在要求源具有高斯分布或对称密度分布。此外，对于高斯源，我们的结果具有率失真理论方面的意义：在所考虑博弈论设定下可实现速率与失真可从其团队论对应设定中获得。