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AbstractWe 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. JEL Classi…cation: K9
We study a one-sided offers bargaining game in which the buyer has private information about the value of the object and the seller has private information about his beliefs about the buyer. We show that this uncertainty about uncertainties dramatically changes the set of possible outcomes when compared to two-sided private information. In particular, higher order beliefs can lead to a delay in reaching agreement even when the seller makes frequent offers, while in the case of two-sided first order private information, agreement is reached almost instantly. Furthermore, we show that not all types of higher order beliefs lead to a delay: the crucial condition is that when uncertain about uncertainties, one assigns positive probability to certainty.
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