21  Newcomb’s Paradox

R. Nozick (1969). Newcomb’s Problem and Two Principles of Choice in Essays in Honor of Carl G. Hempel, Nicholas Rescher (ed.), Springer.

There are two boxes in front of us:

1. box \(A\), which contains $1,000;
2. box \(B\), which contains either $1,000,000 or nothing.

You can see inside box \(A\), but not inside box \(B\):

You are offered two choices:

• one-box: choose only box \(B\)
• two-box: choose both box \(A\) and box \(B\)

You can keep whatever is inside any box that is opened, but you do not get to keep what is inside a box that is not opened.

A very powerful being, called the Predictor, who has been invariably accurate in its predictions about your behavior in the past, has already acted in the following way:

1. If the Predictor has predicted that you will open only box \(B\), the being has put $1,000,000 in box \(B\).
2. If the Predictor has predicted that you will open both boxes, the being has put nothing in box \(B\).

Do you choose one-box or two-box?

Warning

You should answer the above question before reading further.

The decision problem described above can be formalized as follows:

• The actions are one-box (selecting only box \(B\)) and two-box (selecting both boxes);
  
• the outcomes are \(1M\) (you receive 1 million dollars), \(1T\) (you receive 1 thousand dollars), \(0\) (you receive nothing), and \(1M+1T\) (you receive one million one thousand dollars); and
• the states are \(pred\_B\) meaning that the Predictor predicted that you would choose only box \(B\), and \(pred\_AB\) meaning that the Predictor predicted that you would choose both boxes.

Then, the decision matrix representing the above decision problem is:

There are two ways to reason about which action you should choose.

1. two-box should be chosen: There are two possible states \(pred\_B\) (the prediction is that you will choose only box \(B\)) and \(pred\_AB\) (the prediction is that you will choose both boxes). If the state is \(pred\_B\), then two-box gives the outcome \(1T\) which is strictly greater than the one-box outcome of \(0\). If the state is \(pred\_AB\), then two-box gives the outcome \(1M+1T\) which is strictly greater than the one-box outcome of \(1M\). In both cases, two-box gives a strictly better outcome than one-box, so two-box should be chosen.

2. one-box should be chosen: Let \(B\) mean that you have chosen the action one-box and \(AB\) mean that you have chosen the action two-box. To calculate the expected utilities of the actions, we consider the following (conditional) probabilities:

   • \(Pr(pred\_B\mid B)\): The probability that the wizard predicted you would choose box \(B\) given that you decided to choose box \(B\).

   • \(Pr(pred\_AB\mid B)\): The probability that the wizard predicted you would choose both boxes given that you decided to choose box \(B\).

   • \(Pr(pred\_B\mid AB)\): The probability that the wizard predicted you would choose box \(B\) given that you decided to choose both boxes.

   • \(Pr(pred\_AB\mid AB)\): The probability that the wizard predicted you would choose both boxes given that you decided to choose both boxes.

   Then, the expected utilities (assuming, for simplicity, that the utility of \(1M\) is \(1,000,000\), the utility of \(0\) is \(0\), the utility of \(1M+1T\) is \(1,001,000\) and the utility of \(1T\) is \(1,000\)) of the actions are:

   • \(EU(\textbf{one-box}) = 1,000,000 * Pr(pred\_B\mid B) + 0 * Pr(pred\_AB\mid B)\)
   • \(EU(\textbf{two-box}) = 1,001,000 * Pr(pred\_B\mid AB) + 1,000 * Pr(pred\_AB\mid AB)\)

   The assumption that the Predictor is invariably accurate about its predictions means that \(Pr(pred\_B\mid B)\) and \(Pr(pred\_AB\mid AB)\) are both close to 1 while \(Pr(pred\_AB\mid B)\) and \(Pr(pred\_B\mid AB)\) are both close to 0. This means that \[1,000,000 * Pr(pred\_B\mid B) + 0 * Pr(pred\_AB\mid B)\] is much greater than \[1,001,000 * Pr(pred\_B\mid AB) + 1,000 * Pr(pred\_AB\mid AB).\] Thus, \[EU(\textbf{one-box}) > EU(\textbf{two-box}),\] and so, one-box should be chosen.

Newcomb’s paradox is interesting because it is a case in which maximizing expected utility seems to recommend an action that is strictly dominated. One response to Newcomb’s paradox is to note that there is something odd about the expected utility calculations. In particular, the expected utility calculations assume that the prediction is probabilistically dependent on your choice.¹ The problem with Newcomb’s paradox is that it sets up a decision problem in which the states are not independent of the actions chosen by the decision maker. A standard assumption in Rational Choice Theory is to rule out such decision problems:

Act-State Independence

In any decision problem, if \(Pr\) is the probability assigned to states, \(X\) is the event that the decision maker chose action \(x\), then for all states \(s\), \(Pr(s)=Pr(s\mid X)\). That is, the probability assigned to a state \(s\) is independent of the action chosen by the decision maker.

See Collins (1999) and Weirich (2020) for further discussion of solutions to Newcomb’s paradox.

Collins, J. 1999. “Newcomb’s Problem.” In International Encyclopedia of Social and Behavorial Sciences. https://philarchive.org/rec/COLNP.

Weirich, Paul. 2020. “Causal Decision Theory.” In The Stanford Encyclopedia of Philosophy, edited by Edward N. Zalta, Winter 2020. Metaphysics Research Lab, Stanford University. https://plato.stanford.edu/archives/win2020/entries/decision-causal/.

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1. There is no assumption that your choice has any causal influence over the prediction.