### Article

## Uncertain Rationality, Depth of Reasoning and Robustness in Games with Incomplete Information

Predictions under common knowledge of payoffs may differ from those under arbitrarily, but finitely, many orders of mutual knowledge; Rubinstein's (1989) *Email game* is a seminal example. Weinstein and Yildiz (2007) showed that the discontinuity in the example generalizes: for all types with multiple rationalizable (ICR) actions, there exist similar types with unique rationalizable action. This paper studies how a wide class of departures from common belief in rationality impact Weinstein and Yildiz's discontinuity. We weaken ICR to ICR-lambda, where lambda is a sequence whose n-th term is the probability players attach to (n-1)-th-order belief in rationality. We find that Weinstein and Yildiz's discontinuity remains when lambda_n is above an appropriate threshold for all n, but fails when lambda converges to 0. That is, if players' confidence in mutual rationality persists at high orders, the discontinuity persists, but if confidence vanishes at high orders, the discontinuity vanishes.

We find that, for sufficiently risk-averse agents, strict dominance by *pure or mixed* actions coincides with dominance by *pure* actions in the sense of (Börgers in Econometrica 61(2):423–430, 1993), which, in turn, coincides with the classical notion of strict dominance by pure actions when preferences are asymmetric. Since risk aversion is a *cardinal* feature, all finite single-agent choice problems with *ordinal* preferences admit compatible utility functions which are sufficiently risk averse as to achieve equivalence between pure and mixed dominance. This result extends to some infinite environments.

In everyday economic interactions, it is not clear whether each agent’s sequential choices are visible to other participants or not: agents might be deluded about others’ ability to acquire, interpret or keep track of data. Following this idea, this paper introduces uncertainty about players’ ability to observe each others’ past choices in extensive-form games. In this context, we show that monitoring opponents’ choices does not affect the outcome of the interaction when every player expects their opponents indeed to be monitoring. Specifically, we prove that if players are rational and there is common strong belief in opponents being rational, having perfect information and believing in their own perfect information, then, the backward induction outcome is obtained regardless of which of her opponents’ choices each player observes. The paper examines the constraints on the rationalization process under which reasoning according to Battigalli’s (1996) best rationalization principle yields the same outcome irrespective of whether players observe their opponents’ choices or not. To this respect we find that the obtention of the backward induction outcome crucially depends on tight higher-order restrictions on beliefs about opponents’ perfect information. The analysis provides a new framework for the study of uncertainty about information structures and generalizes the work by Battigalli and Siniscalchi (2002) in this direction.

Estimating how uncertain an AI system is in its predictions is important to improve the safety of such systems. Uncertainty in predictive can result from uncertainty in model parameters, irreducible \emph{data uncertainty} and uncertainty due to distributional mismatch between the test and training data distributions. Different actions might be taken depending on the source of the uncertainty so it is important to be able to distinguish between them. Recently, baseline tasks and metrics have been defined and several practical methods to estimate uncertainty developed. These methods, however, attempt to model uncertainty due to distributional mismatch either implicitly through \emph{model uncertainty} or as \emph{data uncertainty}. This work proposes a new framework for modeling predictive uncertainty called Prior Networks (PNs) which explicitly models \emph{distributional uncertainty}. PNs do this by parameterizing a prior distribution over predictive distributions. This work focuses on uncertainty for classification and evaluates PNs on the tasks of identifying out-of-distribution (OOD) samples and detecting misclassification on the MNIST and CIFAR-10 datasets, where they are found to outperform previous methods. Experiments on synthetic and MNIST and CIFAR-10 data show that unlike previous non-Bayesian methods PNs are able to distinguish between data and distributional uncertainty.

We study an interactive framework that explicitly allows for nonrational behavior. We do not place any restrictions on how players’ behavior deviates from rationality, but rather, on players’ higher-order beliefs about the frequency of such deviations. We assume that there exists a probability *p* such that all players believe, with at least probability *p*, that their opponents play rationally. This, together with the assumption of a common prior, leads to what we call the set of *p*-rational outcomes, which we define and characterize for arbitrary probability *p*. We then show that this set varies continuously in *p* and converges to the set of correlated equilibria as *p* approaches 1, thus establishing robustness of the correlated equilibrium concept to relaxing rationality and common knowledge of rationality. The *p*-rational outcomes are easy to compute, also for games of incomplete information. Importantly, they can be applied to observed frequencies of play for arbitrary normal-form games to derive a measure of rationality *p** that bounds from below the probability with which any given player chooses actions consistent with payoff maximization and common knowledge of payoff maximization.