Strategic Learning in Teams

This paper analyzes a two-player game of strategic experimentation with three-armed exponential bandits in continuous time. Players face replica bandits, with one arm that is safe in that it generates a known payoff, whereas the likelihood of the risky arms’ yielding a positive payoff is initially unknown. It is common knowledge that the types of the two risky arms are perfectly negatively correlated. I show that the efficient policy is incentive-compatible if, and only if, the stakes are high enough. Moreover, learning will be complete in any Markov perfect equilibrium with continuous value functions if, and only if, the stakes exceed a certain threshold

Maximum of entropy for belief intervals under Evidence Theory

The Dempster-Shafer Theory (DST) or Evidence Theory has been commonly used to deal with uncertainty. It is based on the basic probability assignment concept (BPA). The upper entropy on the credal set associated with a BPA is the only uncertainty measure in DST that verifies all the necessary mathematical properties and behaviors. Nonetheless, its computation is notably complex. For this reason, many alternatives to this measure have been recently proposed, but they do not satisfy most of the mathematical requirements and present some undesirable behaviors. Belief intervals have been frequently employed to quantify uncertainty in DST in the last years, and they can represent the uncertainty-basedinformation better than a BPA. In this research, we develop a new uncertainty measure that consists of the maximum of entropy on the credal set corresponding to belief intervals for singletons. It verifies all the crucial mathematical requirements and presents good behavior, solving most of the shortcomings found in uncertainty measures proposed recently. Moreover, its calculation is notably easier than the upper entropy on the credal set associated with the BPA. Therefore, our proposed uncertainty measure is more suitable to be used in practical applications.Spanish Ministerio de Economia y Competitividad TIN2016-77902-C3-2-PEuropean Union (EU) TEC2015-69496-

Required mathematical properties and behaviors of uncertainty measures on belief intervals

The Dempster–Shafer theory of evidence (DST) has been widely used to handle uncertainty‐based information. It is based on the concept of basic probability assignment (BPA). Belief intervals are easier to manage than a BPA to represent uncertainty‐based information. For this reason, several uncertainty measures for DST recently proposed are based on belief intervals. In this study, we carry out a study about the crucial mathematical properties and behavioral requirements that must be verified by every uncertainty measure on belief intervals. We base on the study previously carried out for uncertainty measures on BPAs. Furthermore, we analyze which of these properties are satisfied by each one of the uncertainty measures on belief intervals proposed so far. Such a comparative analysis shows that, among these measures, the maximum of entropy on the belief intervals is the most suitable one to be employed in practical applications since it is the only one that satisfies all the required mathematical properties and behaviors

Objective and Subjective Rationality in a Multiple Prior Model

A decision maker is characterized by two binary relations. The first reflects decisions that are rational in an “objective” sense: the decision maker can convince others that she is right in making them. The second relation models decisions that are rational in a “subjective” sense: the decision maker cannot be convinced that she is wrong in making them. We impose axioms on these relations that allow a joint representation by a single set of prior probabilities. It is “objectively rational” to choose f in the presence of g if and only if the expected utility of f is at least as high as that of g given each and every prior in the set. It is “subjectively rational” to choose f rather than g if and only if the minimal expected utility of f (relative to all priors in the set) is at least as high as that of g.Rationality, Multiple Priors.

Induced aggregation operators in decision making with the Dempster-Shafer belief structure

We study the induced aggregation operators. The analysis begins with a revision of some basic concepts such as the induced ordered weighted averaging (IOWA) operator and the induced ordered weighted geometric (IOWG) operator. We then analyze the problem of decision making with Dempster-Shafer theory of evidence. We suggest the use of induced aggregation operators in decision making with Dempster-Shafer theory. We focus on the aggregation step and examine some of its main properties, including the distinction between descending and ascending orders and different families of induced operators. Finally, we present an illustrative example in which the results obtained using different types of aggregation operators can be seen.aggregation operators, dempster-shafer belief structure, uncertainty, iowa operator, decision making

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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Models of Subjective Learning

We study a decision maker who faces a dynamic decision problem in which the process of information arrival is subjective. By studying preferences over menus of acts, we derive a sequence of utility representations that captures the decision maker’s uncertainty about the beliefs he will hold when choosing from a menu. In the most general model of second-order beliefs, we characterize a notion of "more preference for flexibility" via a subjective analogue of Blackwell’s (1951, 1953) comparisons of experiments. We proceed to analyze a model in which signals are subsets of the state space. The corresponding representation enables us to compare the behavior of two decision makers who expect to learn differently, even if they do not agree on their prior beliefs. The class of information systems that can support such a representation generalizes the notion of modeling information as a partition of the state space. We apply the model to study a decision maker who anticipates subjective uncertainty to be resolved gradually over time. We derive a representation that uniquely identifies both the filtration, which is the timing of information arrival with the sequence of partitions it induces, and the decision maker’s prior beliefs.Resolution of uncertainty, second-order beliefs, preference for flexibility, valuing binary bets more, generalized partition.