Journal of Causal Inference 2021; 1 Research Article Open Access Mauricio Gonzalez-Soto*, L. Enrique Sucar, and Hugo Jair Escalante Von Neumann-Morgenstern and Savage Theorems for Causal Decision Making DOI: DOI, Received ..; revised ..; accepted .. Abstract: Causal thinking and decision making under uncertainty are fundamental aspects of intelligent reasoning. Decision making under uncertainty has been well studied when information is considered at the associative (probabilistic) level. The classical Theorems of von Neumann-Morgenstern and Savage provide a formal criterion for rational choice using purely associative information. Causal inference often yields uncertainty about the exact causal structure, so we consider what kinds of decisions are possible in those conditions. In this work, we consider decision problems in which available actions and consequences are causally connected. After recalling a previous causal decision making result, which relies on a known causal model, we consider the case in which the causal mechanism that controls some environment is unknown to a rational decision maker. In this setting we state and prove a causal version of Savage’s Theorem, which we then use to develop a notion of causal games with its respective causal Nash equilibrium. These results highlight the importance of causal models in decision making and the variety of potential applications. Keywords: Causality, Decision Making, Game Theory MSC: 62C99;91A35;62-09 1 Introduction Causal reasoning is a constant element in our lives since we constantly ask why : Why do we get sick? Why does a drug work? Looking for causes is an everyday task and in fact causal reasoning is to be found at the very core of our minds [1–7]. It is even argued that, in a lower level, the brain is a causal inference machine which uses effects to figure out causes [8–11]. Causal thinking also supports statements of the form if...hadn’t occurred then...wouldn’t have happened ; this is known as counterfactual reasoning, which is something humans naturally do since we can easily imagine alternative scenarios [6]. From a planning point of view, causal relations allow us to manipulate our environment and being able to predict effects of a given action [2, 12, 13]. On the other hand, an important aspect of acting in the world is being able to make decisions under uncertain conditions: which route do we use to get to work? could there be traffic? Where to get some lunch? Maybe there is a long waiting line? [5, 14–16]. Von Neumann and Morgenstern gave an answer for how to make choices if rational preferences are assumed, utilities are known, and the decision maker knows the stochastic relation (i.e., probabilities of events) between actions and outcomes: maximize expected utility [17]. If no such relation is known, then Savage showed that a rational decision maker must choose as if she is maximizing expected utility using a subjective probability distribution [18]. Such theorems provide formal criteria for decision making if rationality is assumed. *Corresponding Author: Mauricio Gonzalez-Soto: Coordinación de Ciencias Computacionales, Instituto Nacional de As- trofísica Óptica y Electrónica, Luis Enrique Erro 1, Santa Maria Tonanzintla, México.; E-mail: mauricio@inaoep.mx L. Enrique Sucar: Coordinación de Ciencias Computacionales, Instituto Nacional de Astrofísica Óptica y Electrónica, Luis Enrique Erro 1, Santa Maria Tonanzintla, México.; E-mail: esucar@inaoep.mx Hugo Jair Escalante: Coordinación de Ciencias Computacionales, Instituto Nacional de Astrofísica Óptica y Electrónica, Luis Enrique Erro 1, Santa Maria Tonanzintla, México.; E-mail: hugojair@inaoep.mx arXiv:1907.11752v4 [cs.AI] 5 Apr 2021