In the classic Bayesian Persuasion model studied by [Kamenica and Gentzkow, 2011], there are two players: the first, called Sender, wishes to persuade the second, called Receiver, to take a desired action. Provided that Sender is ex-post better informed about the underlying state of the world, Sender can leverage their informational advantage by communicating with Receiver via a signal mechanism. However, Sender's ability to effectively manage Receiver's beliefs largely hinges on the assumption that Sender can credibly commit to a signal mechanism. Usually, it is not ex-post optimal for Sender to follow the mechanism, but instead to deviate and send the message that generates the highest payoff. Consequently, Receiver may not have faith in Sender's ability to commit. In this case, all bets are off: persuasion devolves into cheap talk.

Active contour models based on partial differential equations have proved successful in image segmentation, yet the study of their formulation on arbitrary geometric graphs, which place no restrictions in the spatial configuration of samples, is still at an early stage. In this paper, we introduce geometric approximations of gradient and curvature on arbitrary graphs, which enable a straightforward extension of active contour models that are formulated through level sets to such general inputs. We prove convergence in probability of our gradient approximation to the true gradient value and derive an asymptotic upper bound for the error of this approximation for the class of random geometric graphs. Two different approaches for the approximation of curvature are presented, and both are also proved to converge in probability in the case of random geometric graphs. We propose neighborhood-based filtering on graphs to improve the accuracy of the aforementioned approximations and define two variants of Gaussian smoothing on graphs which include normalization in order to adapt to graph nonuniformities. The performance of our active contour framework on graphs is demonstrated in the segmentation of regular images and geographical data defined on arbitrary graphs, using geodesic active contours and active contours without edges as representative models in our experiments.

We consider an infinite collection of agents who make decisions, sequentially, about an unknown underlying binary state of the world. Each agent, prior to making a decision, receives an independent private signal whose distribution depends on the state of the world. Moreover, each agent also observes the decisions of its last K immediate predecessors. We study conditions under which the agent decisions converge to the correct value of the underlying state. We focus on the case where the private signals have bounded information content and investigate whether learning is possible, that is, whether there exist decision rules for the different agents that result in the convergence of their sequence of individual decisions to the correct state of the world. We first consider learning in the almost sure sense and show that it is impossible, for any value of K. We then explore the possibility of convergence in probability of the decisions to the correct state. Here, a distinction arises: if K=1, learning in probability is impossible under any decision rule, while for K ≥ 2, we design a decision rule that achieves it. We finally consider a new model, involving forward looking strategic agents, each of which maximizes the discounted sum (over all agents) of the probabilities of a correct decision. (The case, studied in the previous literature, of myopic agents who maximize the probability of their own decision being correct is an extreme special case.) We show that for any value of K, for any equilibrium of the associated Bayesian game, and under the assumption that each private signal has bounded information content, learning in probability fails to obtain.