Second order tangent bundles of infinite dimensional manifolds

The second order tangent bundle $T^{2}M$ of a smooth manifold $M$ consists of the equivalent classes of curves on $M$ that agree up to their acceleration. It is known that in the case of a finite $n$-dimensional manifold $M$, $T^{2}M$ becomes a vector bundle over $M$ if and only if $M$ is endowed with a linear connection. Here we extend this result to $M$ modeled on an arbitrarily chosen Banach space and more generally to those Fr\'{e}chet manifolds which can be obtained as projective limits of Banach manifolds. The result may have application in the study of infinite-dimensional dynamical systems.Comment: 8 page

Speculative Trade and the Value of Public Information

In environments with expected utility, it has long been established that speculative trade cannot occur (Milgrom and Stokey [1982]), and that the value of public information is negative in economies with risk-sharing and no aggregate uncertainty (Hirshleifer [1971], Schlee [2001]). We show that these results are still true even if we relax expected utility, so that either Dynamic Consistency (DC) or Consequentialism is violated. We characterise no speculative trade in terms of a weakening of DC and find that Consequentialism is not required. Moreover, we show that a weakening of both DC and Consequentialism is sufficient for the value of public information to be negative. We therefore generalise these important results for convex preferences which contain several classes of ambiguity averse preferences

Admissibility and Event-Rationality

We develop an approach to providing epistemic conditions for admissible behavior in games. Instead of using lexicographic beliefs to capture infinitely less likely conjectures, we postulate that players use tie-breaking sets to help decide among strategies that are outcome-equivalent given their conjectures. A player is event-rational if she best responds to a conjecture and uses a list of subsets of the other players' strategies to break ties among outcome-equivalent strategies. Using type spaces to capture interactive beliefs, we show that common belief of event-rationality (RCBER) implies that players play strategies in S1W, that is, admissible strategies that also survive iterated elimination of dominated strategies (Dekel and Fudenberg (1990)). We strengthen standard belief to validated belief and we show that event-rationality and common validated belief of event-rationality (RCvBER) implies that players play iterated admissible strategies (IA). We show that in complete, continuous and compact type structures, RCBER and RCvBER are nonempty, and hence we obtain epistemic criteria for SinfW and IA.

Theorems and unawareness

This paper provides a set-theoretic model of knowledge and unawareness, in which reasoning through theorems is employed. A new property called Awareness Leads to Knowledge shows that unawareness of theorems not only constrains an agent's knowledge, but also, can impair his reasoning about what other agents know. For example, in contrast to Li (2006), Heifetz, Meier, and Schipper (2006) and the standard model of knowledge, it is possible that two agents disagree on whether another agent knows a particular event

Unawareness of theorems

This paper provides a set-theoretic model of knowledge and unawareness. A new property called Awareness Leads to Knowledge shows that unawareness of theorems not only constrains an agent's knowledge, but also, can impair his reasoning about what other agents know. For example, in contrast to Li (2006), Heifetz et al. (2006a) and the standard model of knowledge, it is possible that two agents disagree on whether another agent knows a particular event. The model follows Aumann (1976) in defining common knowledge and characterizing it in terms of a self-evident event, but departs in showing that no-trade theorems do not hold.

Design Spaces in Visual Analytics Based on Goals: Analytical Behaviour, Exploratory Investigation, Information Design & Perceptual Tasks

This paper considers a number of perspectives on design spaces in visual analytics and proposes a new set of four design spaces, based on user goals. Three of the user goals are derived from the literature and are categorised under the terms exploratory investigation, perceptual tasks, and information design. The fourth goal is categorised as analytical behaviour; a recently defined term referring to the study of decision-making facilitated by visual analytics. This paper contributes to the literature on decision-making in visual analytics with a survey of real-world applications within the analytical behaviour design space and by providing a new perspective on design spaces. Central to our analysis is the introduction of decision concepts and theories from economics into a visual analytics context. Given the recent interest in decision-making we wanted to understand the emerging topic of analytical behaviour as a design space and found it necessary to look at more than just decision-making to make a valuable contribution. The result is an initial framework suitable for use in the analysis or design of analytical behaviour applications

Towards a Theory of Analytical Behaviour: A Model of Decision-Making in Visual Analytics

This paper introduces a descriptive model of the human-computer processes that lead to decision-making in visual analytics. A survey of nine models from the visual analytics and HCI literature are presented to account for different perspectives such as sense-making, reasoning, and low-level human-computer interactions. The survey examines the people and computers (entities) presented in the models, the divisions of labour between entities (both physical and role-based), the behaviour of both people and machines as constrained by their roles and agency, and finally the elements and processes which define the flow of data both within and between entities. The survey informs the identification of four observations that characterise analytical behaviour - defined as decision-making facilitated by visual analytics: bilateral discourse, divisions of labour, mixed-synchronicity information flows, and bounded behaviour. Based on these principles, a descriptive model is presented as a contribution towards a theory of analytical behaviour. The future intention is to apply prospect theory, a economic model of decision-making under uncertainty, to the study of analytical behaviour. It is our assertion that to apply prospect theory first requires a descriptive model of the processes that facilitate decision-making in visual analytics. We conclude it necessary to measure the perception of risk in future work in order to apply prospect theory to the study of analytical behaviour using our proposed model