Decision Value and Newcomb's Problem

Nozick's Decision Value (DV) is the sum of three weighted components of utility: causal expected utility (CEU), evidentially expected utility (EEU), and symbolic utility (SU). SU is one of two key subjective elements Nozick has added to decision theory. It assigns a sort of psychological utility to acts that, by their nature, seem to us to symbolize a whole class of similar actions, or to express some belief or value we hold, or to say something about what kind of people we are. The other area of subjectivity introduced is the weighting of the three components by degrees according to an individual's personal preferences.

The values for SU and the weights (W_c, W_e, and W_s) are assigned arbitrarily; one must calculate the other two components. CEU emphasizes the direct causal connection between an action and the outcome in question, while EEU emphasizes the conditional probability of an outcome with respect to an action. This distinction between causal and probabilistic dependence has resulted in wide disagreement as to the relative merits of these two forms of expected utility -- the quintessential example of which is Newcomb's Problem. [The accompanying "Newcomb's Problem" presents for convenience a typical description of the problem, using the terminology of this paper.] Decision value's flexibility not only may appeal to those who find neither the causal nor evidential approaches satisfying, but it may actually better explain how people really do make decisions.

Still, DV is not equipped to solve any problem that is inherent in its components; at best such problems can be minimized through weighting so as to rely less on a flawed component. And I think CEU in particular at least needs an adjustment in the way it is normally applied to Newcomb's Problem. The difficulty, as I see it, lies in presuming that the only relevant causal relationship is between actions taken by the player and the outcomes they produce -- when in fact another causal relationship is implicit in the premises but ignored in the solution. I think the necessary adjustment can be made using something like Skyrms' version of CEU, which explicitly represents factors outside the agent's influence.

When faced with Newcomb's Problem, a player is buffeted by a variety of conflicting motivations. These range from the careful calculation and appreciation of expected utility, through competitive wishes to 'beat' the Predictor, pure greed, pure caution ... all the way down to the most whimsical (perhaps even unconscious) influence I call the "Nah!" Factor: At the very last moment, on impulse, the player says, "Nah!" and does the opposite of what his more carefully considered decision had indicated. Our choices are completely determined by such motivations -- but, since they are 'ours,' we experience such motivations as part of a voluntary decision process, not as subversive, alien influences. The very notion that our decisions might be determined in any way is, for many, both counterintuitive and repugnant. Though I think our decisions really are determined in this way, I needn't argue the point here; it's implicit in 'the rules':

The problem presumes that the Predictor bases his predictions on an extensive understanding of exactly those motivations, right down to the "Nah!" Factor. Every whimsical, contrarian, or even purportedly random choice by a player is governed by a collection of influences that are almost completely transparent to the Predictor. Had he lacked such access, his 99 percent success rate would have been long since destroyed ... or it could have persisted only through blind luck -- something the problem rules out, for one can't rationally base one's choice on a regularity one truly believed to be accidental!

So, while a player's selection is 100 percent determined by some set of motivations, the Predictor's predictions are 99 percent determined by exactly the same set. Because of this common causal influence, the prediction and the choice in each player's case are causally, not just probabilistically, connected.

Though I'm not well qualified to formulate the relevant probabilities, I want to at least try. Here's my approach:

We have no direct knowledge of probabilities concerning our determining motivational forces; all we have is indirect evidence in two forms. First, we have the track record of the Predictor. Secondly, we know that some actual (if undisclosed) fraction f of choices in Newcomb's Problem have been for both boxes, the dominant choice among causal theorists. A third party lacking specific knowledge of the player's internal motivations would have to treat f as the probability that the player will choose both boxes. Though a player himself may be loath to consider his eventual choice to be a matter of probability, he might, approaching the moment of decision, openly declare something like, "I'm about 90 percent set on taking both boxes -- but I'll make up my mind once I'm in there." So even the most independent-minded player might admit that there is some actual probability f that can be assigned to his eventual choice. Let's assign these other symbols:

These relationships emerge:

pr(B) = f
pr(S) = pr(~B) = 1-f
pr(b/B) = pr(s/S) = .99
pr(b/S) = pr(s/B) = pr(b/~B) = .01
pr(b) = pr(b&B) + pr(b&S)
     = [pr(b/B)][pr(B)] + [pr(b/S)][pr(S)]
     = .99f + .01(1-f) = .99f + .01 -      = .98f + .01
pr(s) = .01f + .99(1-f) = .01f + .99 - .99f
    = .99 - .98f

Because in this case our expected utility depends on the value of the fraction f of double-boxers, we might call it Proportionate Expected Utility (PEU). My formula is derived from Skyrms, who would identify two cases K_s and K_b -- each a full description of one relevant way the world might be that is outside the player's influence. (In K_s, the Predictor forecasts S, and in K_b he forecasts B.) I have eliminated impossible cases (e.g., s without $M) and redundancies [pr(s) = pr(K_s = pr($M is in the box)]:

PEU(S) = pr(s)[pr(s/S) x $M] + pr(b)[pr(b/S) x $0]
     = (.99 - .98f)[.99 x $M] + (.98f +      = (.99 - .98f)($990,000)

PEU(B) = pr(s)[pr(s/B) x $MT] + pr(b)[pr(b/B) x $T]
     = (.99 - .98f)[.01 x $MT] + (.98f +      = (.99 - .98f)($10,010) + (.98f +

CEU, calculated on the presumption that the relationship between B and b is probabilistically but not causally linked, finds CEU(B) = CEU(S) + $1,000; in Skyrms' formulation, CEU(S) = pr(s) x $1,000,000.

It is interesting to see how PEU adjusts when f is manipulated. When f approaches 0 (virtually everyone is a single- boxer), PEU(S) approaches $980,100 and PEU(B) approaches $9,909.90 + $9.90 = $9,919.80. This compares to EEUs of $990,000 and $11,000, respectively. At the other extreme, as f approaches 1, PEU(S) nears $9,900 and PEU(B) approaches $100.10 + $980.10 = $1,080.20.

For PEU, then, as for EEU, the single-box choice is always more valuable.

Bayes' Theorem gives us the connection under PEU between B and b -- one that, because it is so counterintuitive, may be worth examining:

	pr(b/B)pr(B)
pr(B/b) =	-----------------------------
	pr(b/B)pr(B) + pr(b/~B)pr(~B)
	(.99)(f)
=	-------------------------
	(.99)(f) + (.01)(1-f)

Special cases of interest:	f [= pr(B)]	pr(B/b)
	1	1
	.5	.99
	.01	.5
	0	0

It might be possible to formulate probabilities based directly on some Skyrmsian set K_M of motivations that determine both player choice and the Predictor's forecast; if so, it's beyond my limited knowledge of probability. I can only guess that the most likely approach in this regard would be to carry K_M all the way through the calculations as an unknown (as I did above with f), revealing the way it affects the range of possible values for other factors.

I suspect there is an element of SU in the way CEU is applied to Newcomb's Problem: the utility of believing that one's choices are not predetermined, even though they actually are. In the absence of such a belief (or, in its place, some belief to the effect that even though one's actions are determined it is important to behave as though they aren't), one is likely to suffer a diminished motivation (such beliefs are themselves motivating factors) for action in open-ended choice situations (unlike the forced-choice case we have been studying) and a consequent loss of potential utility that could have been earned through action.

I should mention that, if I'm right about any substantial portion of this, the consequences aren't nearly as profound for other decision problems as for Newcomb's Problem. The Prisoner's Dilemma, for example -- even though it is claimed by some to be essentially the same problem -- doesn't necessarily imply a common causal influence for both prisoners. They must certainly wonder whether they think sufficiently alike to make cooperation probable. But, while I think that both their decisions are determined by a set of motivations, we have no way of knowing to what degree those sets overlap.

Finally, I think that DV in general suffers from the same "directional" problem I have been criticizing. Too much attention is paid to the subjective ("independence" of choice, SU, personalized weighting of DV factors) rather than the objective (unrecognized causative factors, consequences, principles, and values). But that's a topic for another essay....

Campbell, Richmond, "Background for the Uninitiated," in R. Campbell & L. Sowden, eds., Paradoxes of Rationality and Cooperation. University of British Columbia Press, 1985.

Eells, Ellery, Rational Decision and Causality 1982

Nozick, Robert, "Newcomb's Problem and Two Principles of Choice," in R. Campbell & L. Sowden, eds., Paradoxes of Rationality and Cooperation. University of British Columbia Press, 1985.

Nozick, Robert, The Nature of Rationality. Princeton, NJ: Princeton University Press, 1993.

Skyrms, Brian, Causal Necessity. Yale University Press, 1980.