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1. What is rational choice?

1 What is rational choice? *

1.1 Introduction *

1.2 The role of rationality and uncertainty in economics in general *

1.2.1 Rationality *

1.2.2 Uncertainty *

1.2.3 Sub-conclusion *

1.3 Rational Choice *

1.4 Rational Choice in Economics: Expected Utility Theory *

1.5 Which concept of rationality is "correct?" *

1.5.1 Sub-conclusion *

1.6 Rational Choice about Information in Economics: Search theory *

1.6.1 Stigler *

1.6.2 Rotschild *

1.6.3 Hirschleifer and Riley *

1.7 Indeterminacy of Rational Choice in General *

1. Introduction

If we disagree on the definition of rationality we may also give different answer to the question of whether it is possible to make a rational decision about how much information to collect. To avoid misunderstandings of this sort, it is necessary to discuss the concept of rationality in general. This is the first aim of this section.

The second aim is to relate Elster’s discussion of rationality to standard economic thinking both about rationality and the collection of information. This includes a presentation of von Neumann-Morgenstern’s expected utility hypothesis and theories of search in economics.

The third aim is to locate my question within the larger body of questions in economics and rational choice theory. There are many possible reasons why it may be impossible to make a rational choice, and the focus in this paper is only on one particular argument for the indeterminacy of rational choice. Moreover, locating the question in the academic map also requires a discussion of the role of rationality and uncertainty/information more generally in economics.

 



 

 

2. The role of rationality and uncertainty in economics in general

 

1. Rationality

What is the role of the rationality assumption in economics? For some economists rational maximization is one of a few core defining features of economics. As Gary Becker writes:

"The combined assumptions of maximizing behavior, market equilibrium, and stable preferences, used relentlessly and unflinchingly, form the heart of the economic approach." (Quoted in Hirschleifer 1985, 301. Originally in Becker 1976, p. 4)

In this approach economics is by definition the analysis of the implications of rational choice.

On the other side there are economists who emphasise the limitations of rationality. As Kenneth J. Arrow (1987, p. 25) writes:

It is most plausible under very ideal assumptions. When these conditions cease to hold, the rationality assumptions become strained and possibly even self-contradictory. They certainly imply an ability at information processing and calculation that is far beyond the feasible and that cannot well be justified as the result of learning and adaptation.

Moreover, "there is no general principle that prevents the creation of an economic theory based on hypotheses other than rationality" (Arrow 1987, p. 25). For instance a theory of consumption based on a psychological theory of habit-formation.

The quotations clearly reflect a disagreement about the role of the assumption of rational behaviour in economics. Although I shall make a few comments on the debate in a later chapter, I have doubts about the fruitfulness of trying to answer the general question of "for or against the use of the assumption of rationality." In any case the question is too large for me and I have chose to focus on an critical evaluation of only two arguments for the interdetminacy of rational choice in one field (the collection of information).

 

2. Uncertainty

The meaning and role of uncertainty in economics is controversial. For some uncertainty is a situation in which it is impossible to attach probabilities to the outcomes. Others deny this impossibility and argue that for all practical purposes uncertainty and risk are synonyms. Risk is here defined as a situation in which you do not know which outcome ("state of the world") will obtain, but you do have beliefs that can be expressed as numerical probabilities.

Depending on the two different meanings of uncertainty economists have tended to take different positions on its importance. Those who believe radical uncertainty exists (like the Post Keynesians), also believe the concept is extremely important especially in demonstrating that the economy is not self-regulating. On the other hand there are those who try to show that the assumption of certainty is not vital to many of the core results in economics (such as the existence of general equilibrium, stability, and its pareto optimality).

Let me make the above general comments slightly more precise. General equilibrium theory requires a complete set of markets; there is a price such that for each commodity demand is equal to supply. Without claiming to understand the proofs, I must accept that economists have rigorously demonstrated both the existence of this general equilibrium and that it is Pareto optimal. This proof requires certain assumptions and one of these is certainty. In what way does uncertainty make the problem more difficult?

A commodity may be defined by at least four features: (1) Its physical description; (2) The place of delivery; (3) The time of delivery; And (4) The state of nature obtaining. In short, an ice-cream at a beach in 1997 in sunny weather is a different commodity than an ice-cream in a forest in 1994 when it is raining. This definition of a commodity drastically increases the number of markets that have to exist to have a complete set of markets. Moreover, uncertainty seems to create large problems for (3) and (4) since this requires a complete set of future contingent markets. For instance, it is possible to buy "options" (a right to buy something in the future at a specified price), but empirically speaking not all commodities have existing options markets as. The same goes for (4); it is possible to buy insurance for some states of nature (car-crash), but uncertainty/risk may to destroy the existence of some of these markets (unemployment insurance) (see Barr, but also my criticism). In this way uncertainty makes it more likely that a complete set of markets does not exist and that the Pareto optimality of a decentralized self-regulating economy no longer holds. This is one of the key theoretical significance of uncertainty.

 

3. Sub-conclusion

In sum, whatever your opinion on the meaning of uncertainty and rationality it is clear that the debate about its meaning and implications are central to economics. The general significance may derive from what some consider to be the ultimate economics question: "Does the free-market produce a stable and efficient economy?" The question may be far, far in the background in this paper, but ultimately questions of the possibility of rational behaviour in an uncertain world are relevant to this question since the existing proofs assume both rationality and (often) certainty.

 

3. Rational Choice

What does it mean to make a rational choice and what does the different the different theories assume is necessary before it is possible to make a rational choice? Here is one possible answer to the first question:

Ideally, a fully satisfactory rational-choice explanation of an action would have the following structure. It would show that the action is the (unique) best way of satisfying the full set of the agent's desires, given the (uniquely) best beliefs the agent could form, relatively to the (uniquely determined) optimal amount of evidence. We may refer to this as the optimality part of the explanation. In addition the explanation would show that the action was caused (in the right way) by the desires and beliefs, and the beliefs caused (in the right way) by consideration of the evidence. We may refer to this as the causal part of the explanation. These two parts together yield a first-best rational-choice explanation of the action. The optimality part by itself yields a second-best explanation, which, however, for practical purposes may have to suffice, given the difficulty of access to the psychic causality of the agent. (Elster 1985, p. 71)

According to this view there are two general demands that have to be met before we can use rational choice to explain an action: First, the demands of optimality. Second the demands of causality. The demands of optimality can be divided into three requirements: optimality in the choice of action from the feasible set, optimality of beliefs for a given set of information, and optimality in the collection of information. The two causal demands require that action and beliefs be caused "in the right way" given preferences, beliefs and evidence. For instance, assume it is rational for me to press a green button (not the red), and I do so. We would not call this a rational action is the reason I pressed the green button was that somebody pushed me and I accidentally hit it. The same goes for beliefs. I may, for example, make two errors when I calculate probabilities, but these two errors may cancel each other out so the final belief is optimal. This is an example of evidence causing the beliefs in the wrong (non-rational) way.

The quotation also suggests that we might classify different definitions of rationality according to how many of the three optimality demands they require. Traditionally economics is often said to employ a "thin" notion of rationality in which an action is rational as long as it has the highest expected utility for given beliefs and given information i.e. the last two requirements need not be satisfied. More recently, however, the rational expectations revolution in economics has broadened the standard economic concept of rationality to include rational beliefs: an action is not viewed as rational even if it was the best given the beliefs when the beliefs were not rational given the evidence. Beyond this, however the quotation is of little help as a classificatory device:

First, the third demand may be considered redundant if one considers "collecting information" as a feasible action along with the other possible actions. Then the demand that the collection of information is taken care of by the assumption that we should always choose the optimal act (which would be collect information if this has a higher expected utility than the other alternatives).

Second, one might add even stronger requirements to the list. Having argued that economists use a "thinner" notion of rationality than Elster, we could then go to the other extreme to consider the very broad notion used by many philosophers. In this tradition rationality is often defined as being or acting "reasonable." Robert Nozick (1993) in his book The Nature of Rationality, argues that it is not reasonable to consider acting on all kinds of preferences as rational. For instance, he (1993, p. 144) claims that it is irrational for an agent "to have desires that she knows are impossible to fulfill (sic)" and he devotes a sub-chapter to discuss the many demands he want to make on preferences before they should be called rational. These discussions are often said to be about substantive rationality, as opposed to the economists more instrumental concept of rationality (i.e. they are about what we should want, not only what we should do to get something we want). This substantive demand goes beyond Elster's definition since he (at least in the article above) does not want to make this a part of the standard definition of rationality.

Third, there are concepts of rationality that do not fit into the quotation at all. Consider Shaun Hargreaves Heap distinction between instrumental, procedural and expressive rationality. The first two are reasonably well known in the literature, but expressive rationality -"making sense of self through creative choice" - is not easily related to Elster’s three demands.

I could go on to discuss the almost endless varieties of rationality, but within the context of this paper only a few issues are relevant. First of all I need to give a more detailed picture of the standard economic view of rational choice. Second, it is important to discuss to what extent we should make the optimal formation of beliefs and collection of information a part of their definition of rationality. This is directly relevant to the argument about infinite regress and problem of estimation, unlike the more philosophical view that certain preferences are non-rational. The answer to the last question does not affect the argument in chapter three.

 

4. Rational Choice in Economics: Expected Utility Theory

Imagine that you have to select one action (x_i) from a set of feasible actions (X). Assume, moreover, that you are in a situation of uncertainty. Which action should you choose?

In its prescriptive variant expected utility theory says that we should choose that action which maximizes expected utility. As Schoemaker (1982) points out, the theory can also be used descriptively ("this is how people choose"), predictively ("I expect him to chose x since it is the act that maximizes expected utility") and postdictively i.e. it is used as a non-falsifiable assumption that guides research. Anomalies do not destroy the theory, but stimulates search for the "unknown" variable that makes the behaviour conform to the theory. My interest is here mainly in the normative aspects of the theory i.e. whether it can tell us what to do. In the following I shall thus present the basics of the theory with special emphasis on exactly what is required before the theory can be put to use.

How do we calculate the expected utility? To answer this we first specify our uncertainty as a list of possible "states of the world" (each state is denoted s_i and is a member of the set of possible states S). We then have a list of possible actions and possible states that together form the set of possible consequences (c_xs). A simple example is the following: You have to chose whether to bring an umbrella or not when you go for a walk (x₁= bring umbrella, x₂= not bring umbrella). There are two possible "states of the world": s₁= it will rain, s₂= it will not rain. Crosstabulating this we have the following four possible consequences:

 

Table 1: Calculation of Expected Utility

		Possible states (S)
		s1 = Rain (probability p1)	s2 = No rain (probability p2)
Possible actions (X)	x1 (Bring umbrella)	c11 (it rains and you have an umbrella)	c12 (you brought the umbrella, but is does not rain)
	x2 (Do not bring umbrella)	c21 (it rains and you did not bring an umbrella)	c22 (you did not bring the umbrella and it did not rain)

 

The expected utility of an action is calculated by multiplying the utility of each possible consequence of an action with the (subjective or given) probability that the consequence will occur. Formally in our example:

EU (x₁) = v(c₁₁) p₁ + v(c₁₂) p₂

EU (x₁) = v(c₂₁) p₁ + v(c₂₂) p₂

Or, more generally:

EU (x) = S v(c_xs) p_s

Maximization of expected utility then simply means that you choose that alternative which has the highest expected utility when it is calculated in the way described above.

So far all I have done is to describe exactly how one calculates the expected utility from an action. How can this procedure be justified as the rational way of making a choice? The answer is that the decision rule "maximize expected utility" (MEU) follows from what some people think are appealing axioms. More specifically, we must assume that the preference function v(.) must (or should) satisfy the following:

1. Completeness (all possible consequences are ranked)

2. Continuity (increasing a probability will eventually make you switch from one alternative to another)

3.Transitivity (if you prefer x₁ to x₂ and x₂ to x₃, the you must also prefer x₁ to x₃)

4. Independence (my choice between x₁ and x₂ is not changed if each alternative is changed by the same amount, say x₃ is added to both alternative) (this is comparable to Savage’s sure-thing principle).

von Neumann and Morenstern showed that if we accepted these assumptions, then it follows that we should follow the MEU rule (described above) to determine which action we should select. The intuitive idea may have been around for a long time (at least since Bernoulli’s solution to the St. Petersburg Paradox, and certainly since Savage and Ramsey), but it was von Neuman and Morgenstern who rigorously proved that the MEU rule followed from certain appealing axioms. In the words of Hirschleifer and Riley (1992: 15):

The great contribution of Neumann and Morgenstern was to show that, given plausible assumptions about individual preferences, it is possible to construct a v(c) function - "cardinal" in that only positive linear transformations thereof are permissible - whose joint use with the Expected Utility Rule ... will lead to the correct ordering of actions.

They made the connection between the ranking of outcomes (v(.)) and the choice of action and proved that as long as certain restrictions on the preference function was accepted, the maximization of expected utility would necessarily follow.

The Expected Utility Hypothesis has been extensively discussed, and especially the fourth assumption (independence) has been questioned. The purpose of this section, however, is not to present a detailed review of the debates around the hypothesis. Instead I simply wanted to describe the basics of the theory and make its assumptions explicit.

 

5. Which concept of rationality is "correct?"

von Neumann and Morgenstern are silent about the rationality of the probabilities in their theory. As mentioned Elster and modern economists criticise this silence, and argue that a decision should not be labeled rational if it is not based on optimal beliefs and an optimal amount of evidence. To examine this issue I have chosen to discuss some of Russell Hardin’s arguments from his book One for All: The Logic of Group Conflict. Some may believe it is a bit on the side to discuss the rationality of individual action in ethnic conflicts in a paper on economics, but I believe there are good reasons to focus on Hardin: He is a well respected academic (so I am not attacking a soft target), he knows the general topic well (he is an authority on game theory, collective action and rationality), and he discusses the specific question head on (should we require beliefs and the collection of information to be rational before we label the decision rational?). The fact that the context is ethnic violence and not, say, investment, makes little difference to the principles involved.

The aim of Hardin’s book, expressed in his own words, is "to go as far as possible with a rational choice account of reputedly primordial, moral, and irrational phenomena of ethnic identification and action" (1995, p. 16). A short summary of his theory of ethnic violence goes as follows. It is rational to identify with a group (since it provides both security, material benefits and satisfies a psychic need to belong somewhere). Being a member of a group affects your beliefs since it tends to reduce awareness of alternative ways of doing things, as well as inducing the belief that what we do is the right thing to do (is-ought fallacy). Given these beliefs, it becomes rational for people who want power to play on people's ignorance and the belief that we are "better" that the other. Finally, group violence happens when the leaders find that this is their best way of maintaining power (for instance to distract people from economic failure). Using nationalist propaganda, they create a belief that it is in peoples' self interest to engage in a pre-emptive attack against the other group. Once violence starts there is a spiral that only increases violence, since it creates hate as well as an even stronger belief that we must destroy the other side before they kill us (and there is no way the parties can credibly promise not to discriminate the other).

Although this to some extent is a plausible story, we have to ask whether it is intuitive to label it rational. More specifically, is the formation of beliefs behind nationalism and ethnic violence rational? Hardin (1995, p. 62, emphasis removed) admits that beliefs used to explain group conflict are "not convincing, even patently not so in the sense that it would not stand serious scrutiny..." (adding that this "does not entail that people cannot believe it" but this is not relevant to my argument). But how can it be rational to act on beliefs that are obviously wrong? Hardin (1995, p. 62-63) answer is worth quoting in at length:

One might say that the supposed knowledge of ethnic or national superiority is corrupt at its foundation. Unfortunately this is true also of other knowledge, perhaps of almost all knowledge of factual matters. [...] Hence, at their foundations there is little to be distinguish supposed knowledge of normative from that of factual matters [...] Should we say that anyone who acts on such knowledge is irrational? We could, but then we would be saying that virtually everyone's actions are always irrational. It seems more reasonable to say that one's beliefs may have corrupt foundations but that, given those beliefs, it is reasonable to act in certain ways rather than others is one wishes to achieve particular goals.

[...]

Someone who carries through on an ethnic commitment on the claim that her ethnic group is in fact superior, even normatively superior, to others, may not be any more irrational than I am in following my geographic knowledge. She merely follows the aggregated wisdom of her ethnic group.

In short, because all knowlege is corrupted at its base it "would be odd [...] to conclude that the action was irrational when taken if it was fully rational given the available knowledge" (Hardin 1995, p. 16, my emphasis).

Who is correct, Hardin or Elster? First of all, there are several internal inconsistencies in Hardin's argument. For instance, even if we agree that rationality demands only optimality for given information, it is difficult to see how people can believe that the individuals in their ethnic group descend from one "original" group Eve. This is a common belief among nationalist (see Connor 1994). Hence, "patently false beliefs" do not require us to collect information to be falsified, they may be irrational even for a given knowledge. Secondly, at one point he also claims that "a full account of rational behaviour must include the rationality of the construction of one's knowledge set. Costs and benefits of gaining particular bits and categories of knowledge may be related to one's circumstances...". Maybe, I failed to understand this (and it is preceded by the term "On this view ..."), but as far as I can see it is in stark contradiction to the rest of the argument. Finally, there is an inconsistency in his arguments since he later (1995, p. 192) attacks communitarianism using the following two arguments:

The chief epistemological problem with particularistic communitarianism is that it violates the dictum of the epigraph of this chapter: The important thing is not to stop questioning [...] To question such beliefs is to increase the chance of bettering them. (p. 192)

"Commonsense epistemology allows for variations in our confidence of our knowledge. My belief that concern for human welfare dominates concern for various community values or even community survival is radically different from my belief that certain rough physical laws hold sway over us." (p. 210)

He cannot at the same time demand that the construction of beliefs should not be rational and claim that communitarians are wrong in not questioning these belief. Indeed, if it is true what he says that all knowledge is corrupt at its foundation, then we should put little faith in the preposition that we can increase the reliability of our beliefs and values by questioning them.

Second, we might question the argument that all knowledge is equally corrupt at its foundation. As I shall discuss later, Jon Elster and Leif Johansen have made similar claims, but they in no way go this far. Is it really true that all our knowledge is so weak that none of the differences are worth seeking out or acting on?

Third, and perhaps most important, is the suggestion that to demand that we should construct the set of knowledge in a rational way must lead us to conclude that "virtually everyone's actions are always irrational." My immediate response would be that his argument leads to an equally odd conclusion: To reject the demand for rational construction of beliefs makes almost all behaviour rational. A better approach, I think, would be to say that rational/irrational/non-rational are not labels that either apply or do not apply. It is a question of degrees. An action can be more or less rational depending on variables such as the rationality of the construction of the beliefs behind the action. Hence, the information demand does not committ me to the position that almost all action is irrational, as Hardin claims. Moreover, his argument makes it far too easy (and uninteresting) to prove that a phenomenon is caused by individually "rational" action.

In the end it boils down to our intuition. Is it intuitive to buy a house or a car without first collecting some information about its quality? If the answer is no, then you accept some kind of demands on the collection of information before you label the decision rational. Would you say that a man who went out today to walk on water because he believed he was God, is rational? If no, then you accept that the beliefs have to be in some proportion to the evidence before the action is defined as rational.

In addition to the arguments about the intuitive appeal of including belief formation and information collection in the definition of rationality, we may add a methodological and historical argument. With the rational expectations revolution in economics there has been a tendency to widen the scope of rationality even in economics. As described by Roger E. Backhouse (1995, p. 118):

In the post-war period economic theory has been dominated by the attempt to explain economic phenomena in terms of rational behaviour. In macroeconomic this has taken the form of providing a microeconomic foundation for macroeconomic theories: deriving macroeconomic relationships as the outcome of individuals' optimizing subject to the constraints imposed by their endowments, markets and technology. There has been an aversion to what Lucas has termed 'free parameters': parameters describing individual or market behaviour that are not derived from the assumption of utility or profit maximization.

One of these free parameters was the assumption of either rigid or only backwards looking adaptive expectations, so in the 1970s Lucas, in the words of Backhouse (1995, p. 123) argued that optimizing behaviour "should be systematically applied to all aspects of macroeconomic models, including the formation of expectations." This demonstrates the tendency for economics to go beyond the thin notion of rationality and it identifies this trend as a key to progress in economic science.

Why should economist take this step? As mentioned in the quotation, the main argument is not that we can explain more doing so, but that the explanation is better grounded. The underlying notion is one that conceives of progress in a discipline as explaining much using little (reductionism and parsimony). But, if this is the underlying justification, the logical step is to include Elster's third requirement: If we demand that beliefs be rational for a given set of information parsimony also demands that we should demand that the collection of information be rational. As the discussion below demonstrates, not everybody will agree to this.

 

1. Sub-conclusion

I have argued in favor of Elster's definition of rationality and against Hardin. The argument had two main aspects. First there is the theoretical presuppostition based on parsimony that if we assume maximizing behavior in the choice of action for given beliefs, then we should also assume it when people form belief and when they collect information. Second, when faced with some concrete examples it sounded intuitively wrong to exclude optimal beliefs and optimal collection of information from the definition of rationality.

 

6. Rational Choice about Information in Economics: Search theory

Expected utility theory is about choice in general, while this paper is about one particular type of choice viz. the choice of how much information to collect. This may not be the favourite topic of economists, but since Stigler (1960) the question has received some attention. In this sub-section I want to trace some of this development as well as presenting what I consider to be the most developed model of how much information to collect (presented in chapter 5 of Hirschleifer and Riley, 1992).

 

1. Stigler

Stigler (1960) starts by noting that price dispersion is an empirical fact. This, in turn, makes it profitable to search for the lowest price before you buy a product. However searching is costly, so the question is whether there is an optimal amount of search. Stigler shows that if we assume the distribution of prices is known, then it is possible to conduct an optimal search. Assume, for instance, that we know that half the stores charge $1 and the other half charge $2 for the same product. This implies that the expected price if we visit one store is $1.50. If we sample two stores, the probability of finding one with the lowest price is one minus the probability of sampling two stores with high prices:

Probability of finding at least one store with the minimum price = 1 - ( ½ * ½)= 0.75

Thus, the expected price if we try two stores is $1.25 [(1 * 0.75) + (2 * 0.25)]

and the gain visiting one more store (going from one to two) is $0.25. In the same way we can calculate the expected price after sampling n stores and the gain from going from n-1 to n:

 

Table 2 Expected price after sampling n stores

Number of stores visited (n)	Probability of finding at least one store with the lowest price	Probability of n stores with highest price	Expected price	Expected gain from increasing sample from n-1 to n
1	0.5	0.5	1.5
2	0.75	0.25	1.25	0.25
3	0.875	0.125	1.125	0.125
4	0.9375	0.0625	1.0625	0.0625

Based on this one might believe that the choice of an optimal amount of information is simple. The collection of information is optimal when the expected gain is equal (or as close as possible) to the expected cost of further collection. Assuming that the cost of going to one more store (or taking one more phone call or whatever it is that brings information) is known this is a simple exercise. In our case, if the cost of increasing the sample size by one is 0.11 (and constant for al n), the optimal search is n=3.

Unfortunately things are not that simple. There are two main problems. First, to make the example work we had to make some quite strong assumptions, such as the one about knowing the distribution of prices. Second - and more damaging in this case - the claimed optimal search rule is not optimal at all! I shall deal with the second problem first.

Assume that you find the lowest price in the first store. It is then pointless to search more stores even if the "optimal rule" told you to sample three stores. One might try to avoid this problem by specifying the following rule: "Collect n price quotations, but stop searching before n if you find the lowest price." This, however, does not solve all our problems. Imagine, for instance, that you decide to collect three price quotations (which was the optimal size of sample), but you happen to draw three stores with the highest price. Should you then resign and buy the commodity at the highest price? In fact, after conducting three searches and having found three high prices, it is still optimal to conduct new searches. Stigler’s "fixed sample rule" is not "credible" since it is not optimal to stop after collecting the "optimal" number of price quotations if these quotations are all high. The expected gain of collecting one more after collecting n high prices is always 0.5 which is higher than the cost of searching. This is a general problem with "fixed sample" search rules, and to be fair Stigler is aware of the problem, but "leave it to other" to specify the optimal sequential search procedure.

The second problem worth mentioning is the rather stringent assumptions needed to make the example work. Take, for instance, the assumption that the individual knows the probability distribution of prices. First, it seems strange to assume that you know the distribution but do not know anything about which stores are most likely to have the lowest price. Second, often we do not have a good idea of the true distribution so we have to ask whether it is possible to conduct a search when the distribution is unknown. The first of these is not really a problem since it is at least logically possible to know the general shape of the distribution without having specific knowledge of where to find the lowest price. I shall return to this later. The second problem has been partially answered by Rotschild (1974).

 

2. Rotschild

Unlike Stigler who specified fixed sample search rules for a known distribution, Rotschild (1974) tries to construct and examine the optimal sequential search rule for an unknown distribution. The main purpose of the article is to compare the following five properties of different optimal search-rules (different in the sense that some assume a known distribution and others do not):

1. Does it (the search rule) imply a well-behaved demand function (demand is non-increasing in prices)?

2. Is search behaviour a function of the cost of search and the distribution of prices?

3. Does the amount of search decrease when the cost of search increase?

4. Does total cost decrease when prices become more dispersed?

5. Does increased price dispersion also increase search?

I can make no claim to fully understand the proof of the theorems behind his conclusion that "enough has been said to establish that the properties of search rules are not different." I did, however, understand that both the existence of an optimal search rule and its properties in his article rested on some very strict assumptions. The general idea is that an agent starts with some initial beliefs and updates these according to Bayes rule as he receives new information. Based on these beliefs, in turn, the agent decides whether it pays to go on searching. To make this procedure work, Rotschild must assume (and he is very honest about this) that the prior beliefs have a particular distribution (a Dirilicht distribution - the multinominal equivalent of the Beta distribution), that the agent knows all the possible outcomes (but not their probabilities), and that the learning process is localized in the sense that observing a price of 10 does not affect my probability of observing a price of 11 or 1000. This is strange because one might believe that observing several prices in one neighborhood also increases the probability of the other prices in that neighborhood (and reduced the probability of prices far away). On the other hand, one cannot conclude from the statement that "Rotschild proofs need these assumptions" to "when these assumptions are not met no optimal search rule exist and/or it does not have the same properties as search rules for known distributions." The fact that he did not manage to establish a more general proof, does not mean that such a proof does not exist and he makes some comments to the effect that he considers it likely that the results are more general.

 

3. Hirschleifer and Riley

To end this section on search theory in economics, I want to examine what I found to be the most general and useful frame. The choice about whether to collect information or not can be viewed as any other choice: We should try to collect more information when the expected value of this alternative is higher than the expected value of the other possible actions. But how do we work out the expected value of more information? And, what are the relevant variables?

To illustrate their general answers these question, Hirschleifer and Riley (1992: 173) use the example of an agent who believes there might be oil in a field. In this situation the agent has to decide whether to drill a test well or go ahead with a major investment without collecting more information. The structure of the prior beliefs is given by the agent’s beliefs about the geological structure of the soil and there are three such structures (favorable geological structure: 0.9 probability of hitting oil, moderate: probability 0.3 of oil, hopeless: impossible to find oil). In the terminology of expected utility theory there are three "states of the world." Before drilling the agent believes that the probability of favourable geological structure is 0.1, the probability of a moderate structure is 0.5 and the probability of a hopeless structure is 0.4. Finally, they assume that the result of the test drill is not conclusive i.e. the result is only "wet" or "dry." Whether the result is "wet" or "dry" depends on the geological structure, and the probability of "wet" if the true state is "favourable geological structure" is 0.9, compared to 0.3 probability of wet for "moderate" and 0 probability of wet if the true state is "hopeless." Given all this rather condensed information, we now ask three questions. First, how should you estimate the probability of oil given the result from a test-drill. Second, what is the value of doing the test-drill (i.e. gather information).

To impose some order on the information, Hirschleifer and Riley use three different matrix: the likelihood matrix (L), the joint probability matrix (J), and the posterior matrix (O). The likelihood matrix specifies the probability of each message given the state of the world, P(m/s); The joint probability matrix gives the probability of each combination of states and messages P(sm); Lastly, the posterior gives the probability of a state of the world give a message. Filling in the information above we have:

 

Table 3: The likelihood, joint probability and posterior matrix

	The likelihood matrix				The joint probability matrix					The posterior matrix
		Message
		Wet (m1)	Dry (m2)			Wet (m1)	Dry (m2)	Prior beliefs			Wet (m1)	Dry (m2)
States of the world (Geological structure)	Favorable (s1)	0.9	0.1		Favorable (s1)	0.09	0.01	0.1		Favorable (s1)	0.375	0.013
	Moderate (s2)	0.3	0.7		Moderate (s2)	0.15	0.35	0.5		Moderate (s2)	0.625	0.461
	Hopeless (s3)	0	1		Hopeless (s3)	0	0.40	0.4		Hopeless (s3)	0	0.526
						0.24	0.76

Combining the likelihood matrix with the prior beliefs about the geological structure, we find the joint probability matrix. The posterior matrix is found by combining the information from the likelihood matrix and the joint probability matrix.

At this point a short summary of the intuition in these calculations may be in order. After the test drill you have two pieces of information relevant to the estimation of the probability of the various geological structures. First, the result of the test drill. Second, the prior beliefs about the geological structure. The final rational estimate is a combination of the two. For instance, initially the agent believed that the probability of a favourable geological structure was 0.1, but after receiving a message of "wet" the new probability estimate becomes 0.375. The method used to derive this result is called Bayesian updating and can be stated in words as follows: "The posterior probability that an individual should attach to state s, after receiving a message m, is equal to the prior probability pis multiplied by the likelihood qms of message m and then divided by a normalizing factor which is the overall probability of receiving the message" (p. 175)

Having considered the answer to the first question in some detail, it remains to answer the question of how much the information is worth. Hirschleifer and Riley (1992: 180) first define this as the difference between the expected utility of doing you will receive when choosing an action based on current information vs. the expected utility of choosing an action after receiving information. However, the answer is really slightly more complicated since they also assume that people can only buy an "information service" and not one piece of information. Hence, in the oil-case we could buy a test, but we could not buy the result "wet" - the result of the test may be both wet and dry! The expected difference between the expected utility of doing you will receive when choosing an action based on current information vs. the expected utility of choosing an action after receiving information (the sum of each expected difference multiplied by its probability).

w_n=U(x_m; p_s.m) - U(x₀; p_s.m)

W( m) = E w_n = S q_m U(x_m; p_.m) - U(x₀; p_.m)

W( m) = S q_m S p_s.m v(c*_s.m) - S S p_s.m q_m v(c*_s0)

W( m) = S S p_s.m q_m v(c*_s.m) - S p_s v(c*_s0)

"The value of the message service is just the difference between expected utility with and without the service." (p. 180)

To work out the precise answer in terms of oil, we have to make some assumptions about the costs and gains. Assume the following payoffs:

Drilling and wet: 1 000 000

Drilling and dry: - 400 000

Not drilling: - 50 000 (relocation costs)

If we also assume that the agent is risk-neutral, the preference scaling function is linear in income, v(c) = c.

Hirschleifer and Riley then makes the example unnecessarily complicated by asking how much one would pay for a geological analysis before the test drill (as opposed to how much one would pay to do the test drill). In any case, if we follow their example, they assume that the likelihood matrix of the geological analysis is as follows

 

Table 4: The likelihood and posterior matrix (of the geological analysis)

	The likelihood matrix				The posterior matrix
		Message				Message
		Wet (m1)	Dry (m2)			Wet (m1)	Dry (m2)
States of the world (Geological structure)	wet	0.6	0.4		wet	0.486	0.136
	dry	0.2	0.8		dry	0.514	0.864

 

The expected gain from drilling (x1) is -$64,000 [(0.24 * 1 000 000) + (0.76 * 400 000)] . This is less than the expected loss of not frilling (-$50,000). Hence, the optimal action before receiving information is "not drilling" with an expected payoff of -$50,000.

The next step is to calculate the expected utility of the optimal action after receiving information. To do so we need the posterior probability matrix.

If the message is "dry", the expected payoff from drilling (x1) is

If dry - best action (least worst) is not drilling

Payoff, drilling: [(0.136 * 1 000 000) + (0.854 * 400 000)]

Payoff, not drilling: -$50,000

If wet - best action is drilling:

Payoff, drilling: [(0.486* 1 000 000) + (0.136 * 400 000)] = $140,400

Payoff, not drilling: -$50,000

"So the expected value of the information is $140,400 + $50,000 = $190,400. This is the value of the message service" (p. 183).

More generally, the maximum a person should be willing to pay for an information service is the x that solves the following:

S S p_s.m q_m v(c*_{s.m -} x) = S p_s v(c*_s0)

The value of information can also be visualized in a diagram. (p. 181-182 in Hirschleifer and Riley).

...

This concludes my treatment of the standard theory of rational choice in economics and the standard frame for determining how much information to collect.

 

7. Indeterminacy of Rational Choice in General