… pour juger de ce que l'on doit faire pour obtenir un bien ou pour éviter un mal,
il ne faut pas seulement considérer le bien & le mal en soi,
mais aussi la probabilité qu'il arrive ou n'arrive pas;
& regarder géometriquement la proportion que toutes ces choses ont ensembles …
- Antoine Arnauld & Pierre Nicole's (1662, IV, 16) La logique, ou l'art de penser in the original French
… to judge what one ought to do to obtain a good or avoid an evil,
one must not only consider the good and the evil in itself,
but also the probability that it will or will not happen;
and view geometrically the proportion that all these things have together …
- Jeffrey's (1981, p. 473) translation
Causal Decision Theory
Causal Decision Theory:
Causal Decision Theory relies on probabilities of conditionals instead of conditional probabilities
Under Causal Decision Theory, good decisions aim to produce (i.e. bring about) good outcomes
Expected utility tracks efficacy rather than auspiciousness
This may be contrasted with Evidential Decision Theory, under which good decisions are indicative of (i.e. provide evidence for) good outcomes
Decision Matrix for Newcomb's Problem
PREDICTION 1 (Put nothing in Box 2)
PREDICTION 2 (Put $1,000,000 in Box 2)
φ_{1} (Take Box 2 only)
$0 (outcome o_{13})
$1,000,000 (outcome o_{15})
φ_{2} (Take Box 1 and Box 2)
$1,000 (outcome o_{24})
$1,001,000 (outcome o_{26})
U(o_{13}) = 0
U(o_{24}) = 1,000
U(o_{15}) = 1,000,000
U(o_{26}) = 1,001,000
Let > denote a subjunctive connective
Let α denote a possible world
Let f denote the selection function operating on a suitable similarity metric of possible worlds
Let f(A, α) denote the selected world that differs minimally from the actual world in which B is evaluated
A > B is true in α if B is true in f(A, α)
A > B is false in α if B is false in f(A, α)
NOTE: Your action of φ_{1}-ing or φ_{2}-ing does not cause the prediction of the daemon predictor (PREDICTION 1 or PREDICTION 2), which happened in the past
According to the dominance principle:
The world has been partitioned into 2 with PREDICTION 1 (Put nothing in Box 2) and PREDICTION 2 (Put $1,000,000 in Box 2)
In the 1^{st} partition (PREDICTION 1), in which the daemon predictor puts nothing in Box 2, two-boxing (yielding outcome o_{24} and $1,000) dominates one-boxing (yielding outcome o_{13} and $0)
In the 2^{nd} partition (PREDICTION 2), in which the daemon predictor puts $1,000,000 in Box 2, two-boxing (yielding outcome o_{26} and $1,001,000) dominates one-boxing (yielding outcome o_{15} and $1,000,000)
The formula for computing the expected utility of an action becomes:
EU(φ_{i}) = ΣP(φ_{i} > o_{ij}) × u(o_{ij})
Let s_{1} denote the state into which the world has been partitioned by PREDICTION 1
Let s_{2} denote the state into which the world has been partitioned by PREDICTION 2