Newcomb's Paradox Resolved! Paper to appear in Theory and Decision

Newcomb’s Decision Problem. Assume that you are playing a game with Mr. Spock, a superior being from the planet Vulcan, who claims to be able to predict your actions. In this game, there are two boxes, labelled I and II. Box I is transparent and contains one thousand dollars. Box II is opaque and, unbeknownst to you, Mr. Spock puts in it either $1 million or no reward at all. The reward in box I remains untouched. You now have the choice of either taking both boxes or taking only box II. Your reward at the end of this game, consists of the sum of the rewards in the boxes you take. Mr. Spock claims that box II will contain $1 million if and only if you take only box II. What should you do?

Why is Newcomb’s Problem a Paradox? The first significant contribution about this problem, by the Harvard philosopher Robert Nozick, appeared in 1969. In it, Nozick noted that both strategies, namely taking only box II or taking both boxes, can be justified. Taking both boxes (i.e., the “two-box strategy”) is plausible because no matter what box II contains (either a $1 million or nothing), adding the reward of box I ($1,000) increases your payoff. On the other hand, taking only box II (the “one-box strategy”) is plausible because Mr. Spock, by simulating your decision process can anticipate the two-box strategy, which would therefore lead to no reward in box II and thus a payoff of $1,000. This in turn is less than the $1 million you could earn by only taking box II. Over the years, numerous academic papers have appeared discussing the paradox, some arguing for the two-box strategy and some for the one-box strategy, never for both, and never using game theory in a way that would capture the essence of the problem.

A Resolution of the Paradox. A rigorous treatment of the problem must depend on your beliefs about Mr. Spock’s ability to predict your choice. In a game-theoretic setting, this means that with a certain probability p, Mr. Spock can condition his action on your choice, which is as if he were able to change the contents of the opaque box II, just after you made your choice and just before you get to collect your rewards. Depending on whether your belief p exceeds a certain threshold or not, which turns out to be equal to the ratio of the rewards (i.e., 1,000/1000,000 = 0.1%), you will prefer either the one-box strategy or the two-box strategy. If Mr. Spock can observe this belief, then by anticipating your choice he can ensure perfect prediction on the “equilibrium path,” where, perhaps somewhat unexpectedly, an agent (i.e., a possibly different version of you) who thinks ex-post intervention of the being unlikely, ends up with the low reward ($1,000) in the unique Nash equilibrium of the game. If the belief p belongs to your private information, then Mr. Spock can devise—under some technical assumptions—an economic mechanism that leads to full information revelation, guarantees perfect prediction, and implements the (“first-best”) one-box strategy. If the agent is not risk-neutral, his optimal decisions depend generally on two different thresholds. Still, Mr. Spock can solve Newcomb's problem with zero prediction error in equilibrium if the agent is risk-averse. Faced with an agent who is risk-seeking, even a superior being such as Mr. Spock cannot maintain a zero prediction error in equilibrium without relying on noncredible threats.

Applications. The agent’s belief p determines Mr. Spock’s “commitment ability.” For example, if p is high (larger than 0.1% is enough), then Mr. Spock knows that you know that he can react to your choice, and it is best for you to only take box II, which will then also contain $1 million. This is a very happy scenario. By contrast, if p is very low (less than 0.1% in the example), i.e., you do not believe in the being’s ability, then it is best for Mr. Spock to put nothing in box II, because you will end up with two boxes and only $1,000. This is a less happy scenario for you. In any event, a low p means that Mr. Spock’s commitment ability is high, leading to a low payoff for the agent, whereas a high p means that Mr. Spock’s commitment ability is low, leading to a high payoff for the agent. The effect of commitment ability on payoffs for a monopolist was the subject of a well-known (and eventually proved) conjecture by Ronald H. Coase who won the 1991 Nobel Prize in Economics. In a 2014 paper, Prof. Weber examined the effect of commitment as a continuous variable on the strategic interactions between agents, and his research on optimal commitment is ongoing. The present paper on the Newcomb Paradox illustrates that the continuous notion of commitment can be used to explain behavior in situations that might, at least at first sight, seem “paradoxical.” Unsurprisingly, in an invited session, organized by the decision-analysis cluster, at the 2015 INFORMS Annual Meeting in Philadelphia, Pennsylvania (USA), the paper generated an active discussion.

References
• Coase, R.H. (1972) “Durability and Monopoly,” Journal of Law and Economics, Vol. 15, No. 1, pp. 143—149.
• Weber, T.A. (2016) “A Robust Resolution of Newcomb’s Paradox,” Theory and Decision, Forthcoming. [http://dx.doi.org/10.2139/ssrn.2318604]
• Weber, T.A. (2014) “A Continuum of Commitment,” Economics Letters, Vol. 124, No. 1, pp. 67—73.