Assume, strictly for the time being, that the chief executive has a posterior belief that the signaled value of z is true only when the messages sent by both signalers agree. Assume, also strictly for the time being, that when the messages disagree he believes that z lies between ϖ − 2xs1 and ϖ + 2xs1. (At the end of this section, I demonstrate that these beliefs are consistent with the senders' strategies.)

Given the above posterior beliefs, the chief executive will find it optimal to choose k = z when the messages agree. This is the case because, given that x = k − z, k = z will yield x = 0, which is his ideal point. Both senders will prefer to signal the true value of z, if the chief executive's choice when the messages do not agree will yield a value of x further from both their ideal points than xc. Because ω is uniformly distributed, and given that when the messages disagree the chief executive believes that it lies between ϖ − 2xs1 and ϖ + 2xs1, the chief executive's choice of k when the messages disagree is ϖ. (This choice maximizes his expected utility when z lies between these values, because it minimizes the expected distance of x from his ideal point.) Given that x = k − z, the chief executive's choice of k = ϖ when the messages disagree yields outcomes that are to the left of −2xs1 when z < ϖ + 2xs1 and to the right of 2xs1 when z < ϖ − 2xs1.