Info-gap theory is a methodology for supporting model-based decisions under severe uncertainty. An info-gap is a disparity between what is known, and what needs to be known in order to make a comprehensive and reliable decision. Info-gaps arise in all areas of activity, and they are resolved only when a surprise occurs, or a new fact is uncovered, or when our knowledge and understanding improve. We know very little about the substance of an info-gap. For instance, we rarely know what unusual event will delay the completion of a task. Even more strongly, we cannot know what is not yet discovered, such as tomorrow’s news, or future scientific theories or technological inventions. The ignorance of these things are info-gaps. An info-gap is a Knightian uncertainty since it is not characterized by a probability distribution.
Info-gap decision theory is based on three elements
The first element is an info-gap model of uncertainty, which is a non-probabilistic quantification of uncertainty. The uncertainty may be in the value of a parameter, such as a drag coefficient or a population size. The uncertainty may be in a vector such as the returns from a portfolio of stocks. An info-gap may in the shape of a utility function or the shape of the tail of the probability distribution function (pdf) of extreme events. An info-gap may be in the size and shape of a set of such entities, such as the set of possible pdf’s or the set of possible utility functions. In all cases an info-gap model is an unbounded family of nested sets of possible realizations. For instance, if the uncertain entity is a function then the info-gap model is an unbounded family of nested sets of realizations of this function. An info-gap model does not posit a worst case or most extreme uncertainty. (Sometimes the family of sets is bounded by virtue of the definition of the uncertain entity. For instance, a probability must be between zero and one, so the family of nested sets of possible probability values is bounded. However, this bound does not derive from knowledge about the event whose probability is uncertain, but only from the mathematical definition of probability. Such an info-gap model is unbounded in the universe of probability values.)
The second element of an info-gap analysis is a model of the system, such as the overall benefit of a specific action or decision. The model expresses our knowledge about the system, and may also depend on uncertain elements whose uncertainty is represented by an info-gap model of uncertainty. The system model also depends on the decisions to be made, and quantifies the outcomes of those decisions given specific realizations of the uncertainties. For instance, the model may express the expected return from a portfolio of stocks.
The third element of an info-gap analysis is a set of performance requirements. These specify values of the outcomes that the decision maker requires or aspires to achieve. These values may constitute success of the decision, or at least minimally acceptable values. For instance, one may require that the average return from the stocks be no less than a specified value. Performance requirements can embody the concept of satisficing: doing good enough or meeting critical requirements. Alternatively, the performance requirements can express windfall aspirations for better-than-anticipated outcomes. Both satisficing and windfalling requirements arise in practice, though satisficing requirements are the most common.
Info-gap decision functions: robustness and opportuneness
These three components – uncertainty model, system model, and performance requirements – are combined in formulating two decision functions that support the choice of a course of action.
The robustness function assesses the greatest tolerable horizon of uncertainty. The robustness function is a quantitative answer to the question: how wrong can we be in our data, models and understanding, and the action we are considering will still lead to an acceptable outcome. The robustness function is based on a satisficing performance requirement. When operating under severe uncertainty, a decision that achieves an acceptable outcome over a large range of uncertain realizations is preferable to a decision that fails to achieve an acceptable outcome even under small error. In this way the robustness function generates preferences on available decisions.
The opportuneness function assesses the lowest horizon of uncertainty that is necessary for better-than-anticipated outcomes to be possible (though not guaranteed). The windfalling decision maker asks: how wrong must we be in order for quite attractive outcomes to be possible? The opportuneness function is based on windfalling rather than satisficing. When operating under severe uncertainty it is possible that best-model anticipations are overly pessimistic; the windfaller seeks to exploit the ambient uncertainty. A decision that would result in a really wonderful outcome if we err only slightly is preferred (by the windfaller) over a decision that requires great deviation in order to enable the same outcome. The opportuneness function thus generates preferences over the available decisions. These preferences may not agree with the preferences generated by the robustness function.
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