A True Expert Knows which Question Should be Asked.

We suggest a test for discovering whether a potential expert is informed of the distribution of a stochastic process. In a non-Bayesian non-parametric setting, the expert is asked to make a prediction which is tested against a single realization of the stochastic process. It is shown that by asking the expert to predict a “small” set of sequences, the test will assure that any informed expert can pass the test with probability one with respect to the actual distribution. Moreover, for the uninformed non-expert it is impossible to pass this test, in the sense that for any choice of a “small” set of sequences, only a “small” set of measures will assign a positive probability to the given set. Hence for “most” measures, the non-expert will surely fail the test. We define small as category 1 sets, described in more detail in the paper.

Topologies on Type

We define and analyze a "strategic topology" on types in the Harsanyi-Mertens- Zamir universal type space, where two types are close if their strategic behavior is similar in all strategic situations. For a fixed game and action define the distance be- tween a pair of types as the di¤erence between the smallest " for which the action is " interim correlated rationalizable. We define a strategic topology in which a sequence of types converges if and only if this distance tends to zero for any action and game. Thus a sequence of types converges in the strategic topology if that smallest " does not jump either up or down in the limit. As applied to sequences, the upper-semicontinuity prop- erty is equivalent to convergence in the product topology, but the lower-semicontinuity property is a strictly stronger requirement, as shown by the electronic mail game. In the strategic topology, the set of "finite types" (types describable by finite type spaces) is dense but the set of finite common-prior types is not.rationalizability, incomplete informa- tion, common knowledge, universal type space, strategic topology.

Non-Bayesian Testing of a Stochastic Prediction

We propose a method to test a prediction of the distribution of a stochastic process. In a non-Bayesian non-parametric setting, a predicted distribution is tested using a realization of the stochastic process. A test associates a set of realizations for each predicted distribution, on which the prediction passes. So that there are no type I errors, a prediction assigns probability 1 to its test set. Nevertheless, these sets are small, in the sense that "most" distributions assign it probability 0, and hence there are few type II errors. It is also shown that there exists such a test that cannot be manipulated, in the sense that an uninformed predictor who is pretending to know the true distribution is guaranteed to fail on an uncountable number of realizations, no matter what randomized prediction he employs. The notion of a small set we use is category I, described in more detail in the paper.

Evolution of Preferences

We model, using evolutionary game theory, the implications of endogenous determination of preferences over the outcomes of any given two-player normal form game, G. We consider a large population randomly and repeatedly matched to play G. Each individual has a preference relation over the outcomes of G which may be different than the "true" payoff function in G, and makes optimal choices given her preferences. The evolution of preferences is driven by the payoffs in G that each player obtains. We define stable outcomes (of G) as arising from the stable points of the evolutionary process described above. In our most general model players know the distribution of preferences in the population and observe their opponents' preferences with probability p. They then play a (Bayesian) Nash equilibrium of the resulting game of incomplete information. In the case in which players can perfectly observe their opponents' preferences, i.e., p=1, (where the game is actually one of complete information) an outcome is stable only if it is efficient. Also, an efficient outcome which arises from a strict Nash equilibrium is stable. We also characterize, for 2×2 games, both the stable outcomes and the stable distributions of preferences in the population. When preferences are unobservable, i.e., p=0, we show that stability in our model of evolution of preferences coincides with the notion of neutrally stable strategy (NSS). Finally, we consider robustness of these results. The necessity and sufficiency results are robust to slight changes in p, except for the sufficiency of NSS when p=0: There are in fact (Pareto-inferior) risk-dominant strict equilibria that are not stable for any p>0.Evolution of preferences, observability

Interim Rationalizability

This paper proposes the solution concept of interim rationalizability, and shows that all type spaces that have the same hierarchies of beliefs have the same set of interim rationalizable outcomes. This solution concept characterizes common knowledge of rationality in the universal type space.

Topologies on Types

We define and analyze "strategic topologies" on types, under which two types are close if their strategic behavior will be similar in all strategic situations. To oper- ationalize this idea, we adopt interim rationalizability as our solution concept, and define a metric topology on types in the Harsanyi-Mertens-Zamir universal type space. This topology is the coarsest metric topology generating upper and lower hemiconti- nuity of rationalizable outcomes. While upper strategic convergence is equivalent to convergence in the product topology, lower strategic convergence is a strictly stronger requirement, as shown by the electronic mail game. Nonetheless, we show that the set of "finite types" (types describable by finite type spaces) are dense in the lower strategic topology.