Decision making under uncertain conditions has been well studied when uncertainty can only be considered at the associative level of information. The classical Theorems of von Neumann-Morgenstern and Savage provide a formal criterion for rationally making choices using associative information. We provide here a previous result from Pearl and show that it can be considered as a causal version of the von Neumann-Morgenstern Theorem; furthermore, we consider the case when the true causal mechanism that controls the environment is unknown to the decision maker and propose a causal version of the Savage Theorem. As applications, we argue how previous optimal action learning methods for causal environments fit within the Causal Savage Theorem we present thus showing the utility of our result in the justification and design of learning algorithms; furthermore, we define a Causal Nash Equilibria for a strategic game in a causal environment in terms of the preferences induced by our Causal Decision Making Theorem.
A previous attempt to formalize Decision Theory in the presence of Causal Information is given in @cite_53 , @cite_70 . According to such formulation, a decision maker must choose whatever action is more likely to (causally) produce desired outcomes while keeping any beliefs about causal relations fixed ( @cite_4 ). This is stated by the Stalnaker ( @cite_59 ) equation where @math is to be read as @math @math ( @cite_41 , @cite_11 ). Lewis' and Joyce's work captured the intuition that causal relations may be used to control the environment and to predict what is caused by the actions of a decision maker. In Section we refine the @math operator by an explicit way of calculating the probability of causing an outcome by doing a certain action in terms of Pearl's do-calculus.
@cite_4 @cite_70 @cite_41 @cite_53 @cite_59 @cite_11
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This introduction to decision theory offers comprehensive and accessible discussions of decision-making under ignorance and risk, the foundations of utility theory, the debate over subjective and o ... Preface Introduction: a chance to reconsider 1. Prudential rationality as expected utility maximization 2. Decision problems 3. Savage's theory 4. Evidential decision theory 5. Causal decision theory 6. A general theory of conditional beliefs 7. A representation theorem for causal decision theory 8. Where things stand Notes References. We begin with a rough theory of rational decision-making. In the first place, rational decision-making involves conditional propositions: when a person weighs a major decision, it is rational for him to ask, for each act he considers, what would happen if he performed that act. It is rational, then, for him to consider propositions of the form ‘If I were to do a, then c would happen’. Such a proposition we shall call a counterfactual, and we shall form counterfactuals with a connective ‘☐→' on this pattern: ‘If I were to do a, then c would happen’ is to be written ‘I do a ‘☐→' c happens’. Abstract Newcomb's problem and similar cases show the need to incorporate causal distinctions into the theory of rational decision; the usual noncausal decision theory, though simpler, does not always give the right answers. I give my own version of causal decision theory, compare it with versions offered by several other authors, and suggest that the versions have more in common than meets the eye. A conditional sentence expresses a proposition which is a function of two other propositions, yet not one which is a truth function of those propositions. I may know the truth values of “Willie Mays played in the American League” and “Willie Mays hit four hundred” without knowing whether or not Mays, would have hit four hundred if he had played in the American League. This fact has tended to puzzle, displease, or delight philosophers, and many have felt that it is a fact that calls for some comment or explanation. It has given rise to a number of philosophical problems; I shall discuss three of these. 1. Introduction 2. A brief history of causality 3. Probability, logic and probabilistic temporal logic 4. Defining causality 5. Inferring causality 6. Token causality 7. Case studies 8. Conclusion Appendix A. A little bit of statistics Appendix B. Proofs.