Evidential Decision Theory – EDT – is a branch of decision theory which advises an agent to take actions which, conditional on it happening, maximizes the chances of the desired outcome. As any branch of decision theory, it prescribes taking the action that maximizes utility, that which utility equals or exceeds the utility of every other option. The utility of each action is measured by the expected utility, the averaged by probabilities sum of the utility of each of its possible results. How the actions can influence the probabilities differ between the branches. Causal Decision Theory – CDT – says only through causal process one can influence the chances of the desired outcome 1. EDT, on the other hand, requires no causal connection, the action only have to be a Bayesian evidence for the desired outcome. Some critics say it recommends auspiciousness over causal efficacy2.
One usual example where EDT and CDT are often said to diverge is the Smoking lesion: “Smoking is strongly correlated with lung cancer, but in the world of the Smoker's Lesion this correlation is understood to be the result of a common cause: a genetic lesion that tends to cause both smoking and cancer. Once we fix the presence or absence of the lesion, there is no additional correlation between smoking and cancer. Suppose you prefer smoking without cancer to not smoking without cancer, and prefer smoking with cancer to not smoking with cancer. Should you smoke?” CDT would recommend smoking since there is no causal connection between smoking and cancer. They are both caused by a gene, but have no causal direct connection with each other. Naive EDT, on the other hand, would recommend against smoking, since smoking is an evidence for having the mentioned gene and thus should be avoided. However, a more sophisticated agent following the recommendations of EDT would recognize that if they observe that they have the desire to smoke, then actually smoking or not would provide no more evidence for having cancer; that is, the "tickle" screens off smoking from cancer. (This is known as the tickle defence.)
CDT uses probabilities of conditionals and contrafactual dependence to calculate the expected utility of an action – which track causal relations -, whereas EDT simply uses conditional probabilities. The probability of a conditional is the probability of the whole conditional being true, where the conditional probability is the probability of the consequent given the antecedent. A conditional probability of B given A - P(B|A) -, simply implies the Bayesian probability of the event B happening given we known A happened, it’s used in EDT. The probability of conditionals – P(A > B) - refers to the probability that the conditional 'A implies B' is true, it is the probability of the contrafactual ‘If A, then B’ be the case. Since contrafactual analysis is the key tool used to speak about causality, probability of conditionals are said to mirror causal relations. In most usual cases these two probabilities are the same. However, David Lewis proved 3 its’ impossible to probabilities of conditionals to always track conditional probabilities. Hence evidential relations aren’t the same as causal relations and CDT and EDT will diverge depending on the problem. In some cases, EDT gives a better answers then CDT, such as the Newcomb's problem, whereas in the Smoking lesion problem where CDT seems to give a more reasonable prescription (modulo the tickle defence).