I further pursue the proposal of Mankiw to consider fairness or “Just Deserts” as the main guide to tax policy. I suggest very different policy conclusions. In particular, I point out some key assumptions that were left implicit in Mankiw’s analysis, and question whether these hold in practice. I then suggest a different formal approach to determining a fair policy.

We analyze “nice” games (where action spaces are compact intervals, utilities continuous and strictly concave in own action), which are used frequently in classical economic models. Without making any “richness” assumption, we characterize the sensitivity of any given Bayesian Nash equilibrium to higher-order beliefs. That is, for each type, we characterize the set of actions that can be played in equilibrium by some type whose lower-order beliefs are all as in the original type. We show that this set is given by a local version of interim correlated rationalizability. This allows us to characterize the robust predictions of a given model under arbitrary common knowledge restrictions. We apply our framework to a Cournot game with many players. There we show that we can never robustly rule out any production level below the monopoly production of each firm.

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.

Bayesian inference has the advantage of dynamic consistency, but the drawback of rigidity. When a decision-maker's initial model fails a hypothesis test, he may wish to form a new model, violating Bayes' rule. We show that if such "paradigm shifts" are rare, he will be "approximately" dynamically consistent. More specifically, we show that in our setting dynamic consistency is equivalent to the non-existence of Dutch books, and that a decision-maker who is almost always Bayesian will suffer from only "small" Dutch books. This gives the decision-maker some latitude to revise his model while bounding the pain of inconsistency.