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    … 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

     

     

     

     

    Evidential Decision Theory

    Newcomb's problem
    Image source: http://creatureandcreator.ca/?p=635
    1. Evidential Decision Theory:
    2. Evidential Decision Theory relies on conditional probabilities (or the probabilities of certain outcomes, given the possible actions)
    3. Under Evidential Decision Theory, good decisions are indicative of (i.e. provide evidence for) good outcomes
    4. Expected utility tracks auspiciousness rather than efficacy
    5. This may be contrasted with Causal Decision Theory, under which good decisions aim to produce (i.e. bring about) good outcomes

    Decision Matrix for Newcomb's Problem
    PREDICTION 1
    (Put nothing in Box 2)     		PREDICTION 2
    (Put $1,000,000 in Box 2)
    φ1 (Take Box 2 only) $0 (outcome o13) $1,000,000 (outcome o15)
    φ2 (Take Box 1 and Box 2) $1,000 (outcome o24) $1,001,000 (outcome o26)
    1. Suppose that the daemon predictor is accurate in its predictions 90% of the time
    2. P(o13|φ1) = 0.1
    3. P(o24|φ2) = 0.9
    4. P(o15|φ1) = 0.9
    5. P(o26|φ2) = 0.1


    8. U(o13) = 0
    9. U(o24) = 1,000
    10. U(o15) = 1,000,000
    11. U(o26) = 1,001,000


    14. According to the principle of maximizing expected utility:

    15. EU(φi) = ΣP(oij) × u(oij)



    18. ∴ EU(φ1) = 0.1(0) + 0.9(1,000,000) = 900,000
    19. ∴ EU(φ2) = 0.9(1,000) + 0.1(1,001,000) = 101,000
    20. ∴ EU(φ1) > EU(φ2)


    23. ∴ We arrive at RECOMMENDATION 1 under this version of evidential decision theory:
    24. One-box, φ1, and take Box 2 only
     

     

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