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# Embedding Probabilities, Utilities and Decisions in a Generalization of Abstract Dialectical Frameworks

*Proceedings of the Eleventh International Symposium on Imprecise Probabilities: Theories and Applications*, PMLR 103:246-255, 2019.

#### Abstract

Life is made up of a long list of decisions. In each of them there exists quite a number of choices and most of the decisions are effected by uncertainties and preferences, from choosing a healthy lunch and nice clothes to choosing a profession and a field of study. Uncertainties can be modeled by probabilities and preferences by utilities. A rational decision maker prefers to make a decision with the least regret or the most satisfaction. The principle of maximum expected utility can be helpful in this issue. Expected utility deals with problems in which agents make a decision under conditions in which probabilities of states play a role in the choice, as well as the utilities of outcomes. Argumentation formalisms could be an option to model these problems and to pick one or several alternatives. In this paper, a new argument-based framework, numerical abstract dialectical frameworks (nADFs for short), is introduced to do so. First, the semantics of this formalism, which is a generalization of abstract dialectical frameworks (ADFs for short), based on many-valued interpretations are introduced, including preferred, grounded, complete and model-based semantics. Second, it is shown how nADFs are expressive enough to formalize standard decision problems. It is shown that the different types of semantics of an nADF that is associated with a decision problem all coincide and have the standard meaning. In this way, it is shown how the nADF semantics can be used to choose the best set of decisions.