Biased Revelation.
We consider a two-player game between a sender and a receiver. The game has the following stages: a Nature's move. Nature determines the state of the world w E Q :2 {5, h}. We refer to the two states as 'low' and 'high', respectively. The probability that the state is high is p. The players know the value of p, but they cannot directly observe the true state of the world. 0 Evidence acquisition stage. We assume the sender obtains evidence of the state of the world as follows. If the state of the world is high, the sender obtains evidence with prob- ability qh, whereas if the state is low, the sender obtains the evidence with probability qi. These probabilities are common knowledge. We assume q\" 75 qi. We represent the evidence obtained by the sender 33 6 SS : {emel}, where 35 : .91 when the sender obtains the evidence, and 33 = so otherwise. a Tinnsmission stage. If the sender's obtains 35 = 81, she can reveal the evidence, R, or withhold it, W. If the sender obtains no evidence, 33 : co, she has no choice to make. If the sender reveals, the receiver observes the evidence. If the sender withholds, or if the sender obtained no evidence, 33 = .90, the receiver does not observe the evidence. These assumptions amount to assuming that the sender can pretend she did not obtain evidence (lie by omission), but cannot fabricate evidence (lie by commission). We represent the evidence observed by the receiver as 3" E ST 2 {60,81}, where S'" = 91 when the receiver observes the evidence, and s'" : co otherwise. After the transmission stage, the receiver forms posteriors, par, that the state is it according to her type. That is, n... : 8'" > [0,1]. Note that, typically, both players make a choice, but, in this game it suices to consider the receiver's beliefs, and treat this as her \"choice\". This amounts to assuming a one-toone mapping between the receiver's beliefs and her choice. For instance, suppose that the receiver is considering hiring the sender for a job, as represents the sender's ability, n, represents the receiver's assessment that the sender's ability is high, and that the higher par, the higher the receiver's benets from hiring the sender so the more likely she might be to offer the job or the higher the wage she might oer. We assume that the sender's payoff, 7r, is strictly increasing in the receiver's posterior of the state being high. That is, NW) = 50 + 1#r(8'") for some constants [30 E R and 31 > 0, given that the receiver forms posteriors for an 3" E 8'" according to rEl For instance, in the example where the receiver is considering hiring the sender for a job, the sender benets from the receiver believing her ability is high. The sender's strategy simply species whether she withholds or reveals when she obtains 61, cr 6 {VIC R}. We restrict our analysis to pure strategies
