Causal Decision Theory – CDT – is a branch of decision theory which advises an agent to take actions which maximize the causal consequences on the probability of desired outcomes 1. As any branch of decision theory, it prescribes taking the action that maximizes expected utility, i.e the action which maximizes the sum of the utility obtained in each outcome weighted by the probability of that outcome occurring, given your action. Different decision theories correspond to different ways of construing this dependence between actions and outcomes. CDT focuses on the causal relations between one’s actions and outcomes, whilst Evidential Decision Theory – EDT - concerns itself with what an action indicates about the world (which is operationalized by the conditional probability). That is, according to CDT, a rational agent should track the available causal relations linking his actions to the desired outcome and take the action which will better enhance the chances of the desired outcome.
One usual example where EDT and CDT commonly 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. EDT, on the other hand, would recommend against smoking, since smoking is evidence for having the mentioned gene and thus should be avoided.
The core aspect of CDT is mathematically represented by the fact it uses probabilities of conditionals in place of conditional probabilities 2. 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 cases these two probabilities track each other, and CDT and EDT give the same answers. However, some particular problems have arisen where their predictions for rational action diverge such as the Smoking lesion problem – where CDT seems to give a more reasonable prescription – and Newcomb's problem – where CDT seems unreasonable. David Lewis proved 3 it's impossible for probabilities of conditionals to always track conditional probabilities. Hence, evidential relations aren’t the same as causal relations and CDT and EDT will always diverge in some cases.