Economics, Rational Choice Theory and Utility Maximization

Economist Geoff Hodgson has a short article at Evonomics on the issue of the theoretical and methodological status of the rationality principle in economics. Hodgson sketches an argument that he has more fully developed in his book From Pleasure Machines to Moral Communities. In a nutshell, Hodgson argues that the rationality principle, according to which individuals act such as to maximize their utility, is (i) not a tautology but (ii) is unfalsifiable. Let me take these two points in order.

First, Hodgson argues that the rationality principle is not a tautology because utility maximization “is potentially false”. That is, there may be cases where people’s choices fail to maximize their utility. We may suppose that the non-maximizing behavior may be intentional or not: there are cases where we intend to maximize our utility but we fail, for instance because of weakness of will (a phenomenon known as Akrasia in ancient Greece); there are other cases where reason provides us with good reason to make choices that do not maximize our utility. This latter possibility is at the core of Amartya Sen’s criticism of rational choice theory developed in the 1970’s. Second, Hodgon points out that in spite of the fact that the U-max assumption may be false, we can never know when or at least we can never establish that is false in any specific case. This is due to the fact that it is at least always possible to change the description of some decision problem such as to make the observed behavior compatible with any consistency assumption and thus utility maximization. This is indeed true in all versions of rational choice theory, either under the form of revealed preference theory or of expected utility theory. The basic strategy consists in changing the description of the outcome space of the decision problem such as to make the choice behavior consistent. Unless a behavior is completely random, there should always be in principle a way to rationalize it according to some consistency criterion. U-max only requires transitivity of the underlying preference ordering (plus some reflexivity and continuity conditions). According to Hodgson, these two points make rational choice theory useless as a theory of human behavior. In particular, he rightly note that rational choice theory applies equally well to machines, insects and animals and that as a consequence it cannot tell us anything specific about humans.

I partially agree with Hodgson but his argument requires some qualifications. More specifically, I would argue that (i) actually rational choice theory is a tautology and that (ii) the fact that is unfalsifiable is not necessarily problematic depending on its purpose. Consider the former point first. The only reason Hodgson can validly claim that utility maximization is not a tautology is because he takes utility to be something to be independently measurable. This is of course the way the utility concept was understood by Bentham and by the first marginalists. There is also a bunch of behavioral economists defending a “back to Bentham” paradigmatic move who speak in terms of “experienced utility”, where the latter refers to something akin to happiness or pleasure. Finally, we may also admit that some economists of the Chicago school may have entertained an interpretation of utility as something independently measurable. But all of this is unorthodox. Since the days of Pareto and Samuelson, economic theory (and especially consumer theory) has given up the interpretation of utility as an independently measurable quantity. The ordinalist revolution and Samuelson’s pioneering contribution to revealed preference theory have shown how consumer theory can be formulated without any reference to the utility concept. More exactly, they have established that utility maximization is nothing but a mathematically convenient statement equivalent to the assumption that people make consistent choices and/or have well-ordered preferences. The same is true for expected utility theory, especially Savage’s version which is explicitly behaviorist. Absolutely nothing is assumed regarding what happens “in the head” of the person making some choice. U-max is not an assumption; it is only a descriptive statement of what one is doing. It is a tautology as long as there is always a possibility to rationalize one’s choices in terms of some consistency condition.

Consider now the second point. The fact that the U-max principle is actually a tautology only strengthens Hodgson’s claim that it is unfalsifiable. You cannot falsify a tautology as it is true by definition. Does it make it useless from a scientific perspective? The short answer is clearly “no”. Science is full of useful tautologies, also in economics. Consider only one example coming from biology, one on which Hodgson extensively relies in his work: the Price equation. The Price equation is a highly general mathematical statement of a process of differential replication, i.e. natural selection. The mathematical beauty of the Price equation is that whatever the specificities of the actual process of selection (whether organisms are haploid, haplodiploid, diploid, whether it is cultural or genetic, …), it captures them in a straightforward formula according to which, to simplify matters, the growth rate of some trait in a population can be expressed as the covariance between the trait frequency and the fitness of its bearer. Under the classical meaning of fitness (a measure of reproductive success), Price equation is of course both unfalsifiable and a tautology. But no biologists or behavioral scientists would reject it for this reason. The usefulness of the Price equation comes from its value as a problem-solving device. It gives the scientist a methodological strategy to solve empirical or theoretical problems. As an instance of the latter case, consider for instance how Price equation is useful to derive Hamilton’s rule and to make explicit the assumptions on which the latter rely regarding the property of selection and inclusive fitness.

I would argue that the same is true for rational choice theory in economics. From a Weberian point of view, rational choice theory provides us with a methodological strategy to uncover people’s motivations and reasons for action. Similarly, Don Ross argues it is part of the “intentional stance strategy” through which we are able to understand and predict agents’ behavior. Hodgson is right that rational choice theory is rather weak as an explanatory theory of individual behavior, simply because the theory suffers from an obvious problem of under-determination. But individual behavior is not the right level at which the theory should be applied. It is way more useful for instance to understand how the change in the institutional framework, by modifying people’s incentives and beliefs, may affect their behavior. This strategy is at the core of the applied branch of microeconomics, known as mechanism design. A branch which has enjoyed some empirical successes recently. Of course, there are other reasons to reject the imperialistic claims of the proponents of rational choice theory. I explore some of them in this (version of a) forthcoming paper in the Journal of Economic Methodology.

Working paper: “Bayesianism and the Common Prior Assumption in Game Theory”

I have a new working paper on a slightly esoteric subject, at least for those unfamiliar with decision theory and game theory: “Bayesianism and the Common Prior Assumption in Game Theory: Toward a Theory of Social Interactions“. If everything goes well, I should present this paper at the 3rd Economic Philosophy Conference which will be held in June at Aix-en-Provence. It is organized by Aix-Marseille University and the GREQAM, but one of the co-organizer is my colleague (and office neighbor) from the University of Reims Champagne-Ardenne Jean-Sébastien Gharbi.

This gives me a good excuse for a slight digression: Jean-Sébastien and myself are trying hard to develop economic philosophy at Reims, and though we are currently small in number (hopefully, not for too long!), there has been some momentum shift recently. We have welcome well-known economic methodologist John Davis last year as a visiting professor and we have a significant  numbers  of recent publications in top-journals in economic philosophy and the history of economic thought (Economics and Philosophy, Journal of Economic Methodology, Revue de philosophie économique, European Journal of the History of Economic Thought, Journal of the History of Economic Thought). This is only beginning, or I hope so!

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!

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.

What Are Rational Preferences

Scott Sumner has an interesting post on Econlog about the economists’ use of what can be called the “Max U” framework, i.e. the approach consisting in describing and/or explaining people’s behavior as a utility maximization. As he points out, there are many behaviors (offering gifts at Christmas, voting, buying lottery tickets, smoking) that most economists are ready to deem “irrational” while actually they seem amenable to some kind of rationalization. Sumner then argues that the problem is not with the Max U framework but rather lies in the economists’ “lack of imagination” regarding the ways people can derive utility.

Sumner’s post singles out an issue that lies at the heart of economic theory since the “Marginalist revolution”: what is the nature of utility and of the related concept of preferences? I will not return here on the fascinating history of this issue that passes through Pareto’s ordinalist reinterpretation of the utility concept to Samuelson’s revealed preference account whose purpose was to frame the ordinalist framework in purely behaviorist terms. These debates had also much influence on normative economics as they underlie Robbins’ argument for the rejection of interpersonal comparisons of utility that ultimately led to Arrow’s impossibility theorem and the somewhat premature announcement of the “death” of welfare economics. From a more contemporary point of view, this issue is directly relevant for modern economics and in particular for the fashionable behavioral economics research program, especially as it has now taken a normative direction. Richard Thaler’s reaction to Sumner’s post on Twitter is thus no surprise:

<blockquote class=”twitter-tweet” lang=”fr”><p lang=”en” dir=”ltr”>Yes. This version of economics is unfalsifiable. If people can &quot;prefer&quot; $5 to $10 then what are preferences? <a href=”https://t.co/Cn1XQoIzsh”>https://t.co/Cn1XQoIzsh</a></p>&mdash; Richard H Thaler (@R_Thaler) <a href=”https://twitter.com/R_Thaler/status/680831304175202305″>26 Décembre 2015</a></blockquote>

Thaler’s point is clear: if we are to accept that all the examples given by Sumner are actual cases of utility maximization, then virtually all kinds of behaviors can be seen as utility maximization. Equivalently, any behavior can be explained by an appropriate set of “rational” preferences with the required properties of consistency and continuity. This point if of course far from being new: many scholars have already argued that rational choice theory (either formulated in terms of utility functions [decision theory for certain and uncertain decision problems] or of choice functions [revealed preference theory]) is unfalsifiable: it is virtually always possible to change the description of a decision problem such as to make the observed behavior consistent with some set of axioms. In the context of revealed preference theory, this point is wonderfully made by Bhattacharyya et al. on the basis on Amartya Sen’s long-standing critique of the rationality-as-consistency approach. As they point out, revealed preference theory suffers from an underdetermination problem: for any set of inconsistent choices (according to some consistency axiom), it is in practice impossible to know whether the inconsistency is due to “true” and intrinsic irrationality or is just the result of an improper specification of the decision problem. In the context of expected utility theory, John Broome’s discussion of the Allais paradox clearly shows that reconciliation is in principle possible on the basis of a redefinition of the outcome space.

Therefore, the fact that rational choice theory may be unfalsifiable is widely acknowledged. Is this a problem? Not so much if we recognize that falsification is no longer recognized as the undisputed demarcation criterion for defining science (as physicists are currently discovering). But even if we ignore this philosophy of science feature, the answer to the above question also depends on what we consider to be the relevant purpose of rational choice theory (and more generally of economics) and relatedly, what should the scientific meaning of the utility and preference concepts. In particular, a key issue is whether or not a theory of individual rationality should be part of economics. Three positions seem to be possible: The “Not at all” thesis, the “weakly positive” thesis and the “strongly positive” thesis:

A) Not at all thesis: Economics is not concerned with individual rationality and therefore does not need a theory of individual rationality. Preferences and utility are concepts used to describe choices (actually or counterfactually) made by economic agents through formal (mathematical) statements useful to deal with authentic economic issues (e.g. under what conditions an equilibrium with such and such properties exists?).

B) Weakly positive thesis: Economics builds on a theory of individual rationality but this theory is purely formal. It equates rationality with consistency of choices and/or preferences. Therefore, it does not specify the content of rational preferences but it sets minimal formal conditions that the preference relation or the choice function should satisfy. Preferences and utility are more likely (but not necessarily) to be defined in terms of choices.

C) Strongly positive thesis: Economics builds on a theory of individual rationality and actually parts of economics consist in developing such a theory. The theory is substantive: it should state what are rational preferences, not only define consistency properties for the preference relation. Preferences and in particular utility cannot be defined exclusively in terms of choices, they should refer to inner states of mind (e.g. “experienced utility”) which are accessible in principle through psychological and neurological techniques and methods.

Intuitively, I would say that if asked most economists would entertain something like the (B) view. Interestingly however, this is probably the only view that is completely unsustainable after careful inspection! The problem is the one emphasized by Thaler and others: if rational choice theory is a theory of individual rationality, then it is empirically empty. The only way to circumvent the problem is the following: consider any decision problem Di faced by some agent i. Denote T the theory or model used by the economist to describe this decision problem (T can be either formulated in an expected utility framework or in a revealed preference framework). A theory T specifies, for any Di, what are the permissible implications in terms of behavior (i.e. what i can do given the minimal conditions and constraints defined in T). Denote I the set of such implications and S any subset of these implications. Then, a theory T corresponds to a mapping T: D –> I with D the set of all decision problems or, equivalently, T(Di) = S. Suppose that for a theory T and a decision problem Di we observe a behavior b such that b is not in S. This is not exceptional as any behavioral economist will tell you. What can we do? The first solution is the (naïve) Popperian one: discard T and adopt an alternative theory T’. This is the behavioral economists’ solution when they defend cumulative prospect theory against expected utility theory. The other solution is to stipulate that actually i is not facing decision problem Di but rather decision problem Di’, where T(Di’) = S’ and b ∈ S’. If we adopt this solution, then the only way to make T falsifiable is to limit the range of admissible redefinitions of any decision problem. If theory T is not able to account to some implication b under all the range of admissible descriptions, then it will be falsified. However, it is clear that to define such a range of admissible descriptions necessitates making substantive assumptions over what are rationalizable preferences. Hence, this leads one toward view (C)!

Views (A) and (C) are clearly incompatible. The former has been defended by contemporary proponents of variants of revealed preference theory such as Ken Binmore or Gul and Pesendorfer. Don Ross provides the most sophisticated philosophical defense of this view. View (C) is more likely to be endorsed by behavioral economists and also by some heterodox economists. Both have however a major (and problematic for some scholars) implication once the rationality concept is no longer understood positively (are people rational?) but from an evaluative and normative perspective (what is it to be rational?). Indeed, one virtue of view (B) is that it nicely ties together positive and normative economics. In particular, if it appears that people are sufficiently rational, then the consumer’s sovereignty principle will permit to make welfare judgments on the basis of people’s choices. But this is no longer true under views (A) and (C). Under the former, it is not clear why we should grant any normative significance to the fact that economic agent make consistent choices, in particular because these agents have not to be necessarily flesh-and-bones persons (they can be temporal selves). Welfare judgments can still be formally made but they are not grounded on any theory of rationality. A normative account of agency and personality is likely to be required to make any convincing normative claim. View (C) cannot obviously build on the consumer’s sovereignty principle once it is recognized that people do not always choices in their personal interests. Indeed, this is the very point of the so-called “libertarian paternalism” and more broadly of the normative turn of behavioral economics. It has to face however the difficulty that today positive economics does not offer any theory of “substantively rational preferences”. The latter is rather to be found in moral philosophy and possibly in natural sciences. In any case, economics cannot do the job alone.

Christmas, Economics and the Impossibility of Unexpected Events

surprise

Each year, as Christmas is approaching, economists like to remind everyone that making gifts is socially inefficient. The infamous “Christmas deadweight loss” corresponds to the fact that the resources allocation is suboptimal because people would have chosen to buy different things than the ones they have received as gifts at Christmas if they were given the equivalent value in cash. This is a provocative result but it follows from straightforward (though clearly shortsighted) economic reasoning. I would like here to point out another disturbing result that comes from economic theory. Though it is not specific to the Christmas period it is quite less straightforward, which makes it much more interesting. It is related to the (im)possibility of surprising people.

I will take for granted that one of the points of a Christmas present is to try to surprise the person you’re making the gift to. Of course, many people make wish lists but the point is precisely that 1) one will rarely expect to receive all the items he has indicated on his list and 2) the list may be fairly open or at least give to others an idea of the kind of presents one wish to receive without being too specific. In any case, apart from Christmas, there are several other social institutions whose value is partially derived from the possibility of surprising people (think of April fools). However, on the basis of the standard rationality assumptions made in economics, it is clear that surprising people is simply impossible and even non-sense.

I start with some definitions. An event is a set of states of the world where each person behave in a certain way (e.g. makes some specific gifts to others) and holds some specific conjectures or beliefs about what others are doing and believing. I call an unexpected event an event for which at least one person attributes a null prior probability of realizing. An event is impossible if it is inconsistent with the people’s theory (or model) of the situation they are in. The well-known example of the so-called “surprise exam paradox” gives a great illustration of these definitions. A version of this example is as follows:

The Surprise Exam Paradox: At day D0, the teacher T announces to his students S that he will give them a surprise exam either at D1 or at D2. Denote En the event “the exam is given at day Dn (n = 1, 2)” and assumes that the students S believes the teacher T’s announcement. They also know that T really wants to surprise them and they know that he knows that. Finally, we assume that S and T have common knowledge of their reasoning abilities. On this basis, the students reason the following way:

SR1: If the exam is not given at D1, it will be necessarily given at D2 (i.e. E2 has probability 1 according to S if not E1). Hence, S will not be surprised.
SR2: S knows that T knows SR1.
SR3: Therefore, T will give the exam at D1 (i.e. E1 has probability 1 according to S). Hence, S will not be surprised.
SR4: S knows that T knows SR3.
SR5: S knows that T knows SR1-SR4, hence the initial announcement is impossible.

The final step of S’s reasoning (SR5) indicates that there is no event En that is both unexpected and consistent with S’s theory of the situation as represented by the  assumptions stated in the description of the case. Still, suppose that T gives the exam at D2; then indeed the students will be surprised but in a very different sense than the one we have figured out. The surprise exam paradox is a paradox because whatever T decides to do, this is inconsistent with at least one of the premises constitutive of the theory of the situation. In other words, the students are surprised because they have the wrong theory of the situation, but this is quite “unfair” since the theory is the one the modeler has given to them.

Now, the point is that surprise is similarly impossible in economics under the standard assumption of rational expectation. Actually, this directly follows from how this assumption is stated in macroeconomics: an agent’s expectations are rational if they correspond to the actual state of the world on average. The last clause “on average” means that for any given variable X, the difference between the agent’s expectation of the value of X and the actual value of X is captured by a random error variable of mean 0. This variable is assumed to follow some probabilistic distribution that is known by the agent. Hence, while the agent’s rational expectation may actually be wrong, he will never be surprised whatever the actual value of X. This is due to the fact that he knows the probability distribution of the error term and hence he expects to be wrong according to this probability distribution even though he expects to be right on average.

However, things are more interesting in the strategic case, i.e. when the value of X depends on the behavior of each person in the population, the latter depending itself on one’s expectations about others’ behavior and expectations. Then, the rational expectations hypothesis is akin to assuming some kind of consistency between the persons’ conjectures (see this previous post on this point). At the most general level, we assume that the value of X (deterministically or stochastically) depends on the profile of actions s = (s1, s2, …, sn) of the n agents in the population, i.e. X = f(s). We also assume that there is mutual knowledge that each person is rational: she chooses the action that maximizes her expected utility given her beliefs about others’ actions, hence si = si(bi) for all agents i in the population, with bi agent i’s conjecture about others’ actions. It follows that it is mutual knowledge that X = f(b1, b2, …, bn). An agent i’s conjecture is rational if bi* = (s1*, …, si-1*, si+1*, …, sn*) with sj* the actual behavior of agent j. Denote s* = (s1*(b1*), s2*(b2*), …, sn*(bn*)) the resulting strategy profile. Since there is mutual knowledge of rationality, the fact that one knows s* implies that he knows each bi* (assuming that there is a one-to-one mapping between conjecture and action); hence the profile of rational conjectures b* = (b1*, b2*, …, bn*) is also mutually known. By the same reasoning, a k order of mutual knowledge of rationality entails a k order of mutual knowledge of b* and common knowledge of rationality entails common knowledge of b*. Therefore, everyone correctly predicts X and this is common knowledge.

Another way to put this point is proposed by Robert Aumann and Jacques Dreze in an important paper where they show the formal equivalence between the common prior assumption and the rational expectation hypothesis. Basically, they show that a rational expectation equilibrium is equivalent to a correlated equilibrium, i.e. a (mix-)strategy profile determined by the probabilistic distribution of some random device and where players maximize expected utility. As shown in another important paper by Aumann, two sufficient conditions for obtaining a correlated equilibrium are common knowledge of Bayesian rationality and a common prior over the strategy profiles that can be implemented (the common prior reflects the commonly known probabilistic distribution of the random device). This ultimately leads to another important result proved by Aumann: persons with a common prior and a common knowledge of their ex post conjectures cannot “agree to disagree”. In a world where people have a common prior over some state space and a common knowledge of their rationality or of their ex post conjectures (which here is the same thing), unexpected events are simply impossible. One already knows all that can happen and thus will ascribe a strictly positive probability to any possible event. This is nothing but the rational expectation hypothesis.

Logicians and game theorists who have dealt with Aumann’s theorems have proven that the latter build on a formal structure that is equivalent to the well-known S5 formal system in modal logic. The axioms of this formal system imply, among other things, logical omniscience (an agent knows all logical truths and the logical implications of what he knows) and, more controversially, negative introspection (when one does not know something, he knows it). Added to the fact that everything is captured in terms of knowledge (i.e. true beliefs), it is intuitive that such a system is unable to deal with unexpected events and surprise. From a logical point of view, this problem can be answered simply by changing the axioms of and assumptions of the formal system. Consider the surprise exam story once again. The paradox seems to disappear if we give up the assumption of common knowledge of reasoning abilities. For instance, we may suppose that the teacher knows the reasoning abilities of the students but not that the students knows that he knows that. In this case, steps SR2, SR3 and SR4 cannot occur. Or we may suppose that the teacher knows the reasoning abilities of the students and that the students knows that he knows that, but that the teacher does not know that they know that he knows. In this case, step SR5 in the students’ reasoning cannot occur. In both cases, the announcement is no longer inconsistent with the students’ and teacher’s knowledge. This is not completely satisfactory however for at least two reasons: first, the plausibility of the result depends on epistemic assumptions which are completely ad hoc. Second, the very nature of the formal systems of standard modal logic implies that the agent’s theory of a given situation captures everything that is necessarily true. In the revised version of the surprise exam example above, it is necessarily true that an exam will be given either at day D1 or D2, and thus everyone must know that, and so the exam is not a surprise in the sense of an unexpected event.

The only way to avoid these difficulties is to enter the fascinating but quite complex realm of non-monotonic modal logic and beliefs revision theories. In practice, this consists in giving up the assumption that the agents are logically omniscient in the sense that may not know something that is necessarily true. Faced with an inconsistency, an agent will adopt a belief revision procedure such as to make his belief and knowledge consistent with an unexpected event. In other words, though the agent does not expect to be surprised, it is possible to account for how he deals with unexpected information. As far as I know, there have been very few attempts in economics to build on such kinds of non-monotonic formalization to tackle of expectations formation and revision, in spite of the growing importance of the macroeconomic literature on learning. Game theorists have been more prone to enter into this territory (see this paper of Michael Bacharach for instance) but much remains to be done.