A Short Note on Newcomb’s and Meta-Newcomb’s Paradoxes

[Update: As I suspected, the original computations were false. This has been corrected with a new and more straightforward result!]

For some reasons, I have been thinking about the famous Newcomb’s paradox and I came with a “solution” which I am unable to see if it has been proposed in the vast literature on the topic. The basic idea is that a consistent Bayesian decision-maker should have a subjective belief over the nature of the “Oracle” that, in the original statement of the paradox, is deemed to predict perfectly your choice of taking either one or two boxes. In particular, one has to set a probability regarding the event that the Oracle is truly omniscient, i.e. he is able to foreseen your choice. Another, more philosophical way to state the problem is for the decision-maker to decide over a probability that Determinism is true (i.e. the Oracle is omniscient) or that the Free Will hypothesis is true (i.e. the Oracle cannot predict your choice).

Consider the following table depicting the decision problem corresponding to Newcomb’s paradox:

Matrice

Here, p denotes the probability that the Oracle will guess that you will pick One Box (and thus put 1 000 000$ in the opaque box), under the assumption that the Free Will hypothesis is true. Of course, as it is traditionally stated, the Newcomb’s paradox normally implies that p is a conditional probability (p = 1 if you choose One Box, p = 0 if you choose two boxes), but this is the case only in the event that Determinism is true. If the Free Will hypothesis is true, then p is an unconditional probability as argued by causal decision theorists.

Denote s the probability for the event “Determinism” and 1-s the resulting probability for the event “Free Will”. It is rational for the Bayesian decision-maker to choose One Box if his expected gain for taking one box g(1B) is higher than his expected gain for taking two boxes g(2B), hence if

s > 1/1000.

Interestingly, One Box is the correct choice even if one puts a very small probability on Determinism being the correct hypothesis. Note that is independent of the value of p. If one has observed a sufficient number of trials where the Oracle has made the correct guess, then one has strong reasons to choose One Box, even if he endorses causal decision theory!

Now consider the less-known “Meta-newcomb’s paradox” proposed by philosopher Nick Bostrom. Bostrom introduces the paradox in the following way:

There are two boxes in front of you and you are asked to choose between taking only box B or taking both box A and box B. Box A contains $ 1,000. Box B will contain either nothing or $ 1,000,000. What B will contain is (or will be) determined by Predictor, who has an excellent track record of predicting your choices. There are two possibilities. Either Predictor has already made his move by predicting your choice and putting a million dollars in B iff he predicted that you will take only B (like in the standard Newcomb problem); or else Predictor has not yet made his move but will wait and observe what box you choose and then put a million dollars in B iff you take only B. In cases like this, Predictor makes his move before the subject roughly half of the time. However, there is a Metapredictor, who has an excellent track record of predicting Predictor’s choices as well as your own. You know all this. Metapredictor informs you of the following truth functional: Either you choose A and B, and Predictor will make his move after you make your choice; or else you choose only B, and Predictor has already made his choice. Now, what do you choose?

Bostrom argues that this lead to a conundrum to the causal decision theorist:

If you think you will choose two boxes then you have reason to think that your choice will causally influence what’s in the boxes, and hence that you ought to take only one box. But if you think you will take only one box then you should think that your choice will not affect the contents, and thus you would be led back to the decision to take both boxes; and so on ad infinitum.

The point is that here if you believe the “Meta-oracle”, by choosing Two Boxes you then have good reasons to think that your choice will causally influence the “guess” of the Oracle (he will not put 1000 000$ in the opaque box) and therefore, by causal decision theory, you have to choose One Box. However, if you believe the “Meta-Oracle”, by choosing One Box you have good reasons to think that your choice will not causally influence the guess of the Oracle. In this case, causal decision theory recommends you to choose Two Boxes, as in the standard Newcomb’s paradox.

The above reasoning seems to work also for the Meta-Newcomb paradox even though the computations are slightly more complicated. The following tree represents the decision problem if the Determinism hypothesis is true:

Newcomb

Here, “Before” and “After” denote the events where the Oracle predicts and observes your choice respectively. The green path and the red path in the three correspond to the truth functional stated by the Meta-oracle. The second tree depicts the decision problem if the Free Will hypothesis is true.

Newcomb 2

It is similar to the first one except for small but important differences: in the case the Oracle predicts your choice (he makes his guess before you choose) your payoff depends on the (subjective) probability p that he makes the right guess; moreover, the Oracle is now an authentic player in an imperfect information game with q the decision-maker’s belief over whether the Oracle has already made his choice or not (note that if Determinism is true, q is irrelevant exactly for the same reason than probability p in Newcomb’s paradox). Here, the green and red paths depict the decision-maker best responses.

Assume in the latter case that q = ½ as suggested in Bostrom’s statement of the problem. Denote s the probability that Determinism is true and thus that the Meta-oracle as well as the Oracle are omniscient. I will spare you the computations but (if I have not made mistakes) it can be shown that it is optimal for the Bayesian decision maker to choose One Box whenever s ≥ 0. Without fixing q, we have s > 1-(999/1000q). Therefore, even if you are a causal decision theorist and you believe strongly in Free Will, you should play as if you believe in Determinism!

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Greed, Cooperation and the “Fundamental Theorem of Social Sciences”

An interesting debate has taken place on the website Evonomics over the issue of whether or not economists think greed is socially good. The debate features well-known economists Branko Milanovic, Herb Gintis and Robert Frank as well as the biologist and anthropologist Peter Turchin. Milanovic claims that there is no personal ethics and that morals is embodied into impersonal rules and laws that are built such that it is socially optimal to follow his personal interest as long as one plays along the rule. Actually, Milanovic goes farther than that: it is perfectly right to try to break the rules since if I succeed the responsibility falls on those who have failed to catch me. Such a point of view fits perfectly with the “get the rules right” ideology that dominates microeconomic engineering (market design, mechanism design) and where people’s preferences are taken as given. The point is to set the right rules and incentives mechanisms such as to reach the (second-) best equilibrium.

Not all economists agree with this and Gintis’ and Frank’s answers both qualify some of Milanovic’s claims. Turchin’s answer is also very interesting. At one point, he refers to what he calls the “fundamental theorem of social sciences” (FTSS for short):

In economics and evolution we have a well-defined concept of public goods. Production of public goods is individually costly, while benefits are shared among all. I think you see where I am going. As we all know, selfish agents will never cooperate to produce costly public goods. I think this mathematical result should have the status of “the fundamental theorem of social sciences.”

The FTSS is indeed quite important but formulated this way it is not quite right. Economists (and biologists) have known for long that the so-called “folk theorems” of game theory establish that cooperation is possible in virtually possible in any kind of strategic interactions. To be precise, the folk theorems state that as long as an interaction infinitely repeats with a sufficiently high probability and/or that players have a not too strong preference for the present, then any outcome guaranteeing the players at least their minimax gain in an equilibrium in the corresponding repeated game. This works with all kinds of games, including the prisoner’s dilemma and the related public good game: actually, selfish people will cooperate and produce the public good if they realize that this is in their long term interest to do so (see also Mancur Olson’s “stationary bandits” story for a similar point). So, the true FTSS is rather that “anything goes”: as there are an infinity of equilibria in infinitely repeated games, which one is selected depends on a long list of more or less contingent features (chance, learning/evolutionary dynamics, focal points…). So, contrary to what Turchin claims, the right institutions can in principle incentivize selfish people to cooperate and this prospect may even incentivize selfish people to set up these institutions as a first step!

Does this mean that morality is unnecessary for economic efficiency or that there is no “personal ethics”? Not quite so. First, Turchin’s version of the FTSS becomes more plausible as we recognize that information is imperfect and incomplete. The folk theorems depend on the ability of players to monitor others’ actions and to punish them in case they deviate from the equilibrium. Actually, at the equilibrium we should not observe deviations (except for “trembling hand mistakes”) but this is only because one expects that he will be punished if he defects. It is relatively easy to figure out that imperfect monitoring makes the conditions for universal cooperation to be an equilibrium far more stringent. Of course, how to deal with imperfect and incomplete information is precisely the point of microeconomic engineering (see the “revelation principle”): the right institutions are those that incentivize people to reveal their true preferences. But such mechanisms can be difficult to implement in practice or even to design. The point is that while revelation mechanisms are plausible at some limited scales (say, a corporation) they are far more costly to build and implement at the level of the whole society (if that means anything).

There are reasons here to think that social preferences and morality may play a role to foster cooperation. But there are some confusions regarding the terminology. Social preferences do not imply that one is morally or ethically motivated and the reverse is probably not true altogether. Altruism is a good illustration: animals and insects behave altruistically for reasons that have nothing to do with morals. Basically, they are genetically programmed to cooperate at a cost for themselves because (this is an ultimate cause) it maximizes their inclusive fitness. As a result, these organisms possessed phenotypic characteristics (these are proximate causes) that make them behaving altruistically. Of course, animals and insects are not ethical beings in the standard sense. Systems of morals are quite different. It may be true that morality translates at the choice and preference levels: I may give to a charity not because of an instinctive impulse but because I have a firm moral belief that this is “good” or “right”. For the behaviorism-minded economist, this does not make any difference: whatever the proximate cause that leads you to give some money, the result regarding the allocations of resources is the same. But this can make a difference in terms of institutional design because “moral preferences” (if we can call them like that) may be incommensurable with standard preferences (leading to cases of incompleteness difficult to deal with) or to so-called crowding-out effects when they interact with pecuniary incentives. In any case, moral preferences may make cooperative outcomes easier to achieve, as they lower the monitoring costs.

However, morals is not only embedded at the level of preferences but also at the level of the rules themselves as pointed out by Milanovic: the choice of rules itself may be morally motivated as witnessed by the debates over “repugnant markets” (think of markets for organs). In the vocabulary of social choice theory, morality not only enters into people’s preferences but may also affect the choice of the “collective choice rule” (or social welfare function) that is used to aggregate people preferences. Thus, morality intervenes at these two levels. This point has some affinity with John Rawls’ distinction between two concepts of rules: the summary conception and the practice conception. On the former, a rule corresponds to a behavioral pattern and what justifies the rule under some moral system (say, utilitarianism) is the fact that the corresponding behavior is permissible or mandatory (in the case of utilitarianism, it maximizes the sum of utilities in the population). On the latter, the behavior is justified by the very practice it is constitutive of. Take the institution of promise-keeping: on the practice conception, what justifies the fact that I keep my promises is not that it is “good” or “right” but rather that keeping his promises is constitutive of the institution of promise-keeping. What has to be morally evaluated is not the specific behavior but the whole practice.

So is greed really good? The question is of course already morally-loaded. The answer depends on what we call “good” and on our conception of rules. If by “good” we mean some consequentialist criterion and if we hold the summary conception of rules, the answer will depend on the specifics as indicated in my discussion of the FTSS. But on the practice conception, the answer is clearly “yes, as far as it is constitutive of the practice” and the practice is considered as being good. On this view, while we may agree with Milanovic that to be greedy is good (or at least permissible) as long as it stays within the rules (what Gintis calls “Greed 1” in his answer), it is hard to see how being greedy by transgressing the rules (Gintis’ “Greed 2”) can be good whatsoever… unless we stipulate that the very rules are actually bad! The latter is a possibility of course. In any case, an economic system cannot totally “outsource” morality as what you deem to be good and thus permissible through the choice of rules is already a moral issue.

Working Paper: “Game Theory, Game Situations and Rational Expectations: A Dennettian View”

I have just finished a new working paper entitled “Game Theory, Game Situations and Rational Expectations: A Dennettian View” which I will present at the 16th international conference of the Charles Gide Association for the Study of Economic Thought. The paper is a bit esoteric as it discusses the formalization of rational expectations in a game-theoretic and epistemic framework on the basis of the philosophy of mind and especiallly Daniel Dennett’s intentional-stance functionalism. As usual, comments are welcome.