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In “Technical Note—Ranking Distributions When Only Means and Variances Are Known,” Müller, Scarsini, Tsetlin, and Winkler address the question of ranking distributions when only the first two moments—that is, means and variances—are known. This is important in decision making under uncertainty, with potential applications in economics, finance, statistics, and other areas. Previous results require some assumptions about the shape of the distributions, while this paper’s approach is to impose bounds on how much marginal utility can change, thus constraining risk preferences. Such a ranking is consistent with almost stochastic dominance and provides a new connection between the Sharpe and Omega ratios from finance.

In finite games mixed Nash equilibria always exist, but pure equilibria may fail to exist. To assess the relevance of this nonexistence, we consider games where the payoffs are drawn at random. In particular, we focus on games where a large number of players can each choose one of two possible strategies, and the payoffs are i.i.d. with the possibility of ties. We provide asymptotic results about the random number of pure Nash equilibria, such as fast growth and a central limit theorem, with bounds for the approximation error. Moreover, by using a new link between percolation models and game theory, we describe in detail the geometry of Nash equilibria and show that, when the probability of ties is small, a best-response dynamics reaches a Nash equilibrium with a probability that quickly approaches one as the number of players grows. We show that a multitude of phase transitions depend only on a single parameter of the model, that is, the probability of having ties., 29 pages, 7 figures

Optimal Policies for Online Platforms When Social Learning Occurs Before buying products online, consumers read the reviews written by the previous customers. If they buy the product, they write a review themselves. When the product is of unknown quality, consumers learn it over time; that is, social learning occurs. If consumers have various purchase options of similar products of different brands, the platform that they use may affect this social learning by choosing the order in which the products appear on its website. In “Product Ranking in the Presence of Social Learning,” Maglaras, Scarsini, Shin, and Vaccari compare various policies that the platform may adopt, with the goal of maximizing its revenue collected from commission fees for sold items. The criterion to compare the policies is the worst-case regret with respect to a fully informed platform benchmark.