Dispersion in retail prices of identical goods is inconsistent with the standard model of price competition among identical firms, which predicts that all prices will be driven down to cost. One common explanation for such dispersion is the use of a loss-leader strategy, in which a firm prices one good below cost in order to attract a higher customer volume for profitable goods. By assuming each consumer is forced to buy all desired goods at a single firm, we create the possibility of an effective loss-leader strategy. We find that such a strategy cannot occur in equilibrium if individual demands are inelastic, or if demands are diversely distributed. We further show that equilibrium loss-leaders can occur (and can result in positive profits) if there are demand complementarities, but only with delicate relationships among the preferences of all consumers.

We show that a simple "reputation-style" test can always identify which of two experts is informed about the true distribution. The test presumes no prior knowledge of the true distribution, achieves any desired degree of precision in some fixed finite time, and does not use "counterfactual" predictions. Our analysis capitalizes on a result of Fudenberg and Levine (1992) on the rate of convergence of supermartingales. We use our setup to shed some light on the apparent paradox that a strategically motivated expert can ignorantly pass any test. We point out that this paradox arises because in the single-expert setting, any mixed strategy for Nature over distributions is reducible to a pure strategy. This eliminates any meaningful sense in which Nature can randomize. Comparative testing reverses the impossibility result because the presence of an expert who knows the realized distribution eliminates the reducibility of Nature's compound lotteries.

Rationalizability is a central solution concept of game theory. Economic models often have many rationalizable outcomes, motivating economists to use refinements of rationalizability, including equilibrium refinements. In this paper we try to achieve a general understanding of when this multiplicity occurs and how one should deal with it. Assuming that the set of possible payoff functions and belief structures is sufficiently rich, we establish a revealing structure of the correspondence of beliefs to sets of rationalizable outcomes. We show that, for any rationalizable action a of any type, we can perturb the beliefs of the type in such a way that a is uniquely rationalizable for the new type. This unique outcome will be robust to further small changes. When multiplicity occurs, then we are in a "knife-edge" case, where the unique rationalizable outcome changes, sandwiched between open sets of types where each of the rationalizable actions is uniquely rationalizable. As an immediate application of this result, we characterize, for any refinement of rationalizability, the predictions that are robust to small misspecifications of interim beliefs. These are only those predictions that are true for all rationalizable strategies, that is, the predictions that could have been made without the refinement.

In some games, the impact of higher-order uncertainty is very large, implying that present economic theories may rely critically on the strong common knowledge assumptions they make. Focusing on normal-form games in which the players' action spaces are compact metric spaces, we show that our key condition, called "global stability under uncertainty," implies that the maximum change in equilibrium actions due to changes in players' beliefs at orders higher than k is exponentially decreasing in k. Therefore, given any need for precision, we can approximate equilibrium actions by specifying only finitely many orders of beliefs.

We exhibit a new equilibrium of the classic Blotto game in which players allocate one unit of resources among three coordinates and try to defeat their opponent in two out of three. It is well known that a mixed strategy will be an equilibrium strategy if the marginal distribution on each coordinate is U[0,(2/3)]. All known examples of such distributions have two-dimensional support. Here we exhibit a distribution which has one-dimensional support and is simpler to describe than previous examples. The construction generalizes to give one-dimensional distributions with the same property in higher-dimensional simplexes as well. As our second note, we give some results on the equilibrium payoffs when the game is modified so that one player has greater available resources. Our results suggest a criterion for equilibrium selection in the original symmetric game, in terms of robustness with respect to a small asymmetry in resources.