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.

We study Hotelling's two-stage model of spatial competition, in which two firms first simultaneously choose locations in the unit interval, then simultaneously choose prices. Under Hotelling's assumptions (uniform distribution of consumers, travel cost proportional to distance, inelastic demand of one unit by each consumer) the price-setting subgames possess equilibria in pure strategies for only a limited set of location pairs. Because of this problem (pointed out independently by Vickrey (1964) and d'Aspremont et al. (1979)), Hotelling's claim that there is an equilibrium of the two-stage game in which the firms locate close to each other is incorrect. A result of Dasgupta and Maskin (1986) guarantees that each price-setting subgame has an equilibrium in mixed strategies. We first study these mixed strategy equilibria. We are unable to provide a complete characterization of them, although we show that for a subset of location pairs all equilibria are of a certain type. We reduce the problem of finding an equilibrium of this type to that of solving three or fewer highly nonlinear equations. At each of a large number of location pairs we have computed approximate solutions to the system of equations. Next, we use our analytical results and computations to study the equilibrium location choices of the firms. There is a unique (up to symmetry) subgame perfect equilibrium in which the location choices of the firms are pure; in it, the firms locate 0.27 from the ends of the market. At this equilibrium, the support of the subgame equilibrium price strategy is the union of two short intervals. Most of the probability weight is in the upper interval, so that this strategy is reminiscent of occasional sales by the firms. We also find a subgame perfect equilibrium in which each firm uses a mixed strategy in locations. In fact, in the class of strategy pairs in which the firms use the same mixed strategy over locations, and this strategy is symmetric about 0.5, there is a single equilibrium. In this equilibrium most of the probability weight of the common strategy is between 0.2 and 0.4, and between 0.6 and 0.8. There is a wide range of pure Nash (as opposed to subgame perfect) equilibrium location pairs: the subgame strategies in which each firm threatens to charge a price of zero in response to a deviation support all but those location pairs in which the firms are very close.