Honesty in a Model of Strategic Information Transmission

We consider the following simple model of consumer fraud. Consumers need one of two possible repairs-an expensive repair (denoted E) or an inexpensive repair (denoted I). The exogenous probability that a consumer needs the more costly remedy is r. Each consumer is assumed to know r. An expert can observe with certainty which of the two repairs is needed and can offer to sell either to the consumer. (In a simple extension, we allow for uncertainty with respect to the expert's observation.) We assume that selling the expensive remedy to a consumer who needs only the inexpensive one is more profitable than selling the inexpensive one. Thus, H(EII) > H1(IJI), where H denotes profit and I denotes "conditional on needing." The profit functions are increasing in the price of the respective repairs. (For convenience, we suppress the dependence of H on prices in our notation.) We also assume that a consumer prefers to obtain the appropriate repair to the inappropriate one. Thus, u(EIE) > u(IjE) and u(III) > u(EII), where u is the consumer's payoff. Lastly, we assume that the functions u(EjE) and u(E I) are decreasing in the price of the expensive repair and that u(IIE) and u( III) are decreasing in the price of the inexpensive repair. We also assume that u(IIE) is decreasing in the price of the expensive repair since the consumer ultimately buys the expensive repair in this case. (See below for further assumptions regarding the repairs.) We are interested in the equilibrium levels of honesty in this model. How honest is the expert at the equilibrium? Do consumers always follow the expert's advice? Does the level of honesty increase as the interests of the agents become more similar (in a sense to be defined below)? Our assumptions are similar to those in the abstract strategic information transmission models of Vincent Crawford and Joel Sobel (1982), and Jerry Green and Nancy Stokey (1980). We follow the former paper which asks whether the signals sent become less noisy as the agents' preferences become more similar. Despite the similarity of our model to Crawford-Sobel's, we arrive at some different conclusions with respect to the expert's honesty vis-'a-vis the agents' closeness of interests. We explain this in detail in Section III. In addition we allow that the expert may be uncertain about the true state of the world. We also explicitly model the level of expert honesty, which extends Crawford-Sobel. The outline of the paper is as follows. In Section I we provide the details of the model and show that a unique equilibrium exists. Comparative static results and a simple extension (incorporating the assumption of the incompetent expert) are given in Section II. We draw comparisons with CrawfordSobel in Section III. In Section IV we conclude with a brief summary.