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. We investigate a signaling game problem where an encoder observes a multi-dimensional source and conveys a message to a decoder, and the quadratic objectives of the encoder and decoder are misaligned due to a bias vector. We first provide a set of geometry conditions that needs to be satisfied in equilibrium considering any multi-dimensional source. Then, we consider independent and identically distributed sources and completely characterize conditions under which an informative linear Nash equilibrium exists. In particular, we show that if the components of the bias vector are not equal in magnitude, then there exists a linear equilibrium if and only if the source distribution is Gaussian. On the other hand, for a linear equilibrium to exist in the case of equal bias components, it is required that the source density is symmetric around its mean. 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 their team theoretic counterpart.

翻译：信号游戏问题 调查由于编码器和解码器使用不同成本功能或前后前后不一致而导致目标不匹配的通信情况 。 我们调查一个信号游戏问题, 编码器观测多维源, 并将信息传递给解码器, 编码器和解码器的二次目标因偏向矢量而出现误差 。 我们首先提供一系列几何条件, 考虑任何多维来源, 平衡时必须满足这些条件 。 然后, 我们考虑独立和相同的分布源, 并完全描述存在信息性线性纳什平衡的条件。 特别是, 我们显示, 如果偏向矢的部件在数量上不相等, 那么只有当源分布为高斯方形时, 才会存在线性平衡 。 另一方面, 如果偏差组成部分相同, 则需要源的密度与其平均值相匹配。 此外, 在高西亚源中, 我们的结果来自可实现的率扭曲感应感应感应感应感测到的组合速度和对等变。