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.

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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.

Rational Expectations and the Standard Model of Social Ontology

Noah Smith has an interesting post where he refers to an article of Charles Manski about the rational expectations hypothesis (REH). Manski points out that in a stochastic environment it is highly unlikely that expectations are rational in the sense of the REH. However, he ultimately concludes that there are no better alternative. In this post, I want to point out that the REH is actually well in line with what the philosopher Francesco Guala calls in an article the “Standard Model of Social Ontology” (SMOSO), including the fact that it lacks empirical support. This somehow echoes Noah Smith’s conclusion that “rational Expectations can’t be challenged on data grounds”.

Guala characterizes the SMOSO by the following  three elements:

1) Reflexivity: Guala defines this as the fact that “social entities are constituted by beliefs about beliefs” (p. 961). A more general way to characterize reflexivity is that individuals form attitudes (mainly, beliefs) about the systems they are part of and thus attitudes about others’ attitudes. If it is assumed that these attitudes determine people’s actions and in turn, these actions determine the state of the system, then people’s attitudes determine the system. This may lead to the widely discussed phenomenon of self-fulfilling prophecies where the agents’ beliefs about others’ beliefs about the (future) state of the system bring the system to that state.

2) Performativity: it can be defined as the fact that the social reality is literally made by the agents’ attitudes and actions. The classical example is language: performative utterances like “I promise that Y” or “I make you man and wife” not only describe the social reality, they (in the appropriate circumstances) make it by creating a state of affairs that makes the utterance true. Other cases are for instance the fact that some pieces of paper are collectively regarded as money or the fact that raising one’s hand is regarded as a vote in favor of some proposition or candidate.

3) Collective intentionality: attitudes (in particular beliefs) constitutive of the social reality are in some way or another “collective”. Depending on the specific model, collective intentionality can refer to a set of individual attitudes (intentions, beliefs) generally augmented by an epistemic condition (usually, mutual or common knowledge of these attitudes) or a distinct collective attitude of the form “we intend to” or “we believe that”.

The three elements constitutive of the SMOSO are common to almost all the theories and models developed in social ontology and the philosophy of social science for the last thirty years. That does not mean that they fully determine the content of these theories and models: there are several and mutually exclusive accounts of collective intentionality, as well as there are different ways to account for performativity and reflexivity. Now, I want to suggest that many economic models using the REH fall within this general model of social ontology. The REH states that economic agents do not make systematic errors in the prediction of the future value of relevant economic variables. In other words, they make correct predictions on average. Denote X(t) the value of any economic variable you want (price, inflation, …) at time t and X(t+1)^ei the expected value of X at time t+1 according to agent i. Formally, an expectation corresponds to X(t+1)^ei = E[X(t)ΙI(t)^i] with I(t)^i the information available at t for i and E the expectation operator. The REH is the assumption that X(t+1) = X(t+1)^ei + Eu where u is an error term of mean 0. The proximity of the REH with the three elements of the SMOSO is more or less difficult to see but is nevertheless real.

The relationship between the REH and reflexivity is the easiest to state because discussions on rational expectations in the 1950s find their roots in the treatment of the reflexivity issue which itself originates in Oskar Morgenstern’s discussion of the “Holmes-Moriarty paradox”. Morgenstern was concerned with the fact that if the state of affairs that realizes depends on one’s beliefs about others’ beliefs about which state of affairs will realize, then it may be impossible to predict states of affairs. In 1950s, papers by Simon and by Modigliani and Grunberg tackle this problem. Using fixed-point techniques, they show that under some conditions, there is at least one solution x* = F(x*) such that the prediction x* regarding the value of some variable is self-confirmed by the actual value F(x*). In his article on rational expectations, Muth mentions as one of the characteristic of the REH the fact that a public prediction in the sense of Grunberg and Modigliani “will have no substantial effect on the operation of the economic system (unless it is based on inside information)”. So, the point is that a “rational prediction” should not change the state of the system.

The relationship of the REH with performativity and collective intentionality is more difficult to characterize. Things are somewhat clearer however once we realize that the REH implies mutual consistency of the agents’ beliefs and actions (see this old post by economist Rajiv Sethi which makes this point clearly). This is due to the fact that in an economic system, the value X(t+1) of some economic variable at time t+1 will depend on the decisions si made by thousands of agents at t, i.e. X(t+1) = f(s1(t), s2(t), …, sn(t)). Assuming that these agents are rational (i.e. they maximize expected utility), the agent’s decisions depend on their conjectures X(t+1)^ei about the future value of the variable. But then this implies that one’s conjecture X(t+1)^ei is a conjecture about others’ decisions (s1(t), …, si-1(t), si+1(t), …, sn(t)) for any given functional relation f, and thus (assuming that rationality is common knowledge) a conjecture about others’ conjectures (X(t+1)^e1, …, X(t+1)^ei-1, X(t+1)^ei+1, …, X(t+1)^en). Since others’ conjectures are also conjectures about conjectures, we have an infinite chain of iterated conjectures about conjectures. Mutual consistency implies that everyone maximizes his utility given others’ behavior. In general, this will also imply that everyone forms the same, correct conjecture, which is identical to the REH in the special case where all agents have the same information since we have X(t+1) = X(t+1)^ei for all agent i. As Sethi indicates in his post, this is equivalent to what Robert Aumann called the “Harsanyi doctrine” or more simply, the common prior assumption: any disagreement between agents must come from difference in information.

In itself, the relationship between the REH and the common prior assumption is interesting. Notably, if we consider that the common prior assumption is difficult to defend on empirical grounds, this should lead us to consider the REH with suspicion. But it also helps to make the link with the SMOSO. Regarding performativity, we have to give up the assumption (standard in macroeconomics) that the equilibrium is unique, i.e. there are at least two values X(t+1)* and X(t+1)** for which the agents’ plans and conjectures are mutually consistent. Now, any public announcement of the kind “the variable will take value X(t+1)* (resp. X(t+1)**)” is self-confirming. Moreover, this is common knowledge.[1] The public announcement play the role of a “choreographer” (Herbert Gintis’ term) that coordinates the agents’ plans. This makes the link with collective intentionality. It is tempting to interpret the common prior assumption as some kind of “common mind hypothesis”, as if the economic agents were collectively sharing a worldview. Of course, as indicated above, it is also possible to adopt a less controversial interpretation by seeing this assumption as some kind of tacit agreement involving nothing but a set of individual attitudes. The way some macroeconomists defend the REH suggests a third interpretation: economic agents are able to learn about the economic world and this learning generates a common background. In game-theoretic terms, we could also say that agents are learning to play a Nash equilibrium (or a correlated equilibrium).

This last point is interesting when it is put in perspective with Guala’s critique of the SMOSO. Guala criticizes the SMOSO for its lack of empirical grounding. For instance, discussions about collective intentionality are typically conceptual, but almost never build on empirical evidence. Most critics of the REH in economics make a similar point: the REH is made for several reasons (essentially conceptual and theoretical) but has no empirical foundations. The case of learning is particularly interesting: since the 1970s, one of the “empirical” defenses of the REH has been the casual claim that “you can’t fool people systematically”. This is the same as to say that on a more or less short term, people learn how the economy works. This is a pretty weak defense, to say the least. Economists actually do not know how economic agents are learning, what is the rate of the learning process, and so on. Recently, a literature on learning and expectations has been developing, establishing for instance the conditions of convergence to rational expectations. As far as I can tell, this literature is essentially theoretical but is a first step to provide more solid foundations to REH… or to dismiss it. The problem of the empirical foundations for any assumption regarding how agents form expectations is likely to remain though.

Note

[1] Going a little further, it can be shown that if the public announcement is made on the basis of a probabilistic distribution p where each equilibrium is announced with probability p(X(t+1)*), then p also defines a correlated equilibrium in the underlying game, i.e. agents behave as if they were playing a mixed strategy defined by p.