Utility Maximization A Tautology?

Economists proved over half a century ago that certain stories are unfounded in the theory. For example, one might think that if some workers are involuntarily unemployed, a drop in real wages would lead to a tendency for the labor market to clear. The Cambridge Capital Controversy revealed some difficulties. In response, some economists turned to the Arrow-Debrue-McKenzie model of intertemporal equilibria in which it is not clear that one could even talk about such concepts. The Mantel-Sonnenschein-Debreu theorem shows that this model lacks empirical content. Utility theory provides a closure for some models. Formally, one can demonstrate the existence of equilibria under certain assumptions. But existence does not get one very far.

My purpose of this post is to note that some saw utility theory as a useless tautology at the time of the marginal revolution:

"It is interesting, in this connection, that the earliest critics saw in the theory of marginal utility what we have called a behaviourist theory of choice ... and used exactly the same arguments against it which will be used below against this latter version. Thus [John] Cairnes wrote about Jevon's theory: 'What does it really amount to? In my apprehension to this, and no more - that value depends upon utility, and that utility is whatever affects value. In other words, the name "utility" is given to the aggregate of unknown conditions which determine the phenomenon, and then the phenomenon is stated to depend upon what this name stands for.' Jevon's theory was believed to say no more than this: 'that value was determined by the conditions which determine it - an announcement, the importance of which, even though presented under the form of abstruse mathematical symbols, I must own myself unable to discern'. Some Leading Principles of Political Economy, 1874, p. 15.

[John] Ingram took the same view in A History of Political Economy, 1888, ed. by Ely, 1915, p. 228 and passim. Cairnes, Ingram, and other early critics of marginal utility had, however, directed their criticism also against the mathematical method generally, and the discussion went soon into other channels. The marginalists met the criticism by claiming to be proponents of logical and mathematical method and their tautological psychology thus escaped its well-deserved criticism." -- Gunnar Myrdal (1953) The Political Element in the Development of Economic Theory (trans. by Paul Streeten, Routledge & Kegan Paul, p. 231.

Obviously, Cairnes and Ingram could not have known about results demonstrated a century later. Utility theory manages simultaneously to not say anything about market phenomena, to not be good armchair theorizing, and to be empirically false at the level of the individual.

Election Paradoxes And Faustian Agents

I have been trying to reread Donald Saari on election paradoxes. I have previously considered a few parallels between the Condorcet paradox and models of agents as composed of multiple selves. It seems to me that one could draw more analogies here. I do not plan to pursue the research agenda outlined here - I'm not sure how plausible its results would be. Anyways, Saari provides a comprehensive analysis of a range of voting procedures. Could a fuller range of such procedures - as opposed to pairwise majority rule - be applied to models of multiple selves?

For example, consider a model of a person as having multiple selves, where each one of those selves has a set of preferences over commodities. And suppose the individual, in making choices, resolves those selves with a procedure analogous to an election procedure (e.g., plurality vote, antiplurality vote, Borda Count). Suppose which procedure is used is context-dependent. Can an outside agent modify the context somehow such that the individual follows a different procedure, with consequent effects on the individual's choice?

Or consider two people each composed of the same number of multiple selves, with the preferences of those selves the same across these two people. But suppose each person resolves those selves with a different voting procedure. Maybe these two different voting procedures yield the same "best" choice for one specific menu of choices, but order the non-best choices differently. So if a new menu was created with the best choice removed, these two people - who have identical preferences, in some sense - would make different choices.

I suppose if you follow research along these lines, it would be theoretical research. I do not know how an experiment could elicit the required information to determine the preferences of the multiple selves and the election procedure. I guess the challenge would be to come up with an account consistent with some behavioral anomaly arising in economics experiments. Even better might be to suggest a new experiment and to implement it.

References

• Donald G. Saari (2001). Chaotic Elections! A Mathematician Looks at Voting, AMS.

Humans And Other Animals

Figure 1: Chapuchin Monkeys, Our Cousins

What do we think about generalizations, validated partly with experiments with non-human animals, for economics?

Nicholas Georgescu-Roegen is an economist widely admired by heterodox economists. He quit the American Economic Association in response to their flagship publication, the American Economic Review, publishing articles on, if I recall correctly, pigeons. Researchers were trying to demonstrate that properly trained pigeons had downward-sloping demand curves. I gather they wanted to show income effects and substitution effects, as well, with these laboratory experiments.

On the other hand, are we not supportive of behavioral economists undermining utility theory? I am thinking of controlled experiments that demonstrate people do not conform to the axioms of preference theory. And some of these experiments, as illustrated in the YouTube video linked above, extend beyond humans.

I have a suggestion to resolve such a tension. One might want to treat investigations of humans as a naturalistic enterprise. If so, one would not want to impose an a priori boundary on the different constituents of minds. Whether some species of animals has some sense of self, expectations of the future, primitive languages, or what not should be found by empirical investigation. On the other hand, activities that depend on the existence of social institutions cannot be expected to be found in animals not embedded in any society. And demand curves, if they were to exist, would only arise in specific market institutions.

Reference

• Philip Mirowski (1994). The realms of the Natural, in Natural Images in Economic Thought (ed. by P. Mirowski), Cambridge University Press.

Thoughts On Davis' Individuals and Identity in Economics

I have previously gone on about multiple selves, also known as Faustian agents. I had not considered how an individual manages these selves in making plans and decisions. My point was to apply Arrow's impossibility theorem at the level of the single agent, thereby demonstrating the necessity of some argument for characterizing an individual by a utility function.

Consider many individuals interacting in a market, each being composed of multiple selves. What, in the analysis, groups together sets of these multiple selves to identify individuals? This problem, and similar problems with many other decision theory analyses, is the theme of John D. Davis' 2011 book, Individuals and Identity in Economics.

By the way, an interesting issue arises with multiple selves interacting through time. One might justify hyperbolic discounting by thinking of an individual as composed of a different self at each moment in time. Why should these selves make consistent plans? Might one self start an action based on a plan for future actions, only to have a future self revise or reject that plan? This is the third or fourth time I have started reading Davis' book. Anyways, on pages 41 and 42, Davis writes:

"...[Herbert] Simon's recommendation to abandon the standard utility function framework was not influential in economics, but Lichtenstein and Slovic's demonstration of preference reversals was. Most economists initially dismissed it on a priori grounds, but David Grether and Charles Plott believed that they could go farther and demonstrate that preference reversals could not possibly exist. They identified thirteen potential errors in psychologists' preference reversal experimental methodology and accordingly set out to show that preference reversals were only an artifact of experimental design. Nonetheless, they ended up confirming their existence as well as Simon's judgement of utility functions:

'Taken at face value the data are simply inconsistent with preference theory and have broad implications about research priorities in economics. The inconsistency is deeper than mere lack of transitivity or even stochastic transitivity. It suggests that no optimization principles of any sort lie behind the simplest of human choices and that uniformities in human choice behavior which lie behind market behavior may result from principles which are of a completely different sort from those generally accepted.'(Grether and Plott 1979, 623; emphasis added)

Published in the American Economic Review, this was a momentous admission for economists. However, for many psychologists the debate was already long over, and research had moved on to which theories best explained preference construction. James Bettman published what is regarded as the first theory of preference construction in the same year Grether and Plott's paper appeared (Bettman 1979), a major review of preference construction theories appeared in 1992 (Payne and Bettman 1992), and Lichtenstein and Slovic's retrospective volume appeared in 2006 (Lichtenstein and Slovic 2006). As Slovic put it in 1995, 'It is now generally recognized among psychologists that utility maximization provides only limited insight into the processes by which decisions are made' (Slovic 1995, 365). Grether and Plott, interestingly, extended their own critique of standard rationality to Kahneman and Tverky's proposed prospect theory replacement, implicitly highlighting the difference between the two currents in Edwards' B[ehavioral] D[ecision] R[esearch] program.

'We need to emphasize that the phenomenon causes problems for preference theory in general, and not for just the expected utility theory. Prospect theory as a special type of preference theory cannot account for the results.' (Grether and Plott 1982; 575)

So, given the data and what economists have said years ago about it in the most prominent and most prestigious economics journal in America, one can expect mainstream economists today to have rejected utility theory, revealed preference theory, prospect theory, and the usual old textbook derivation of market demand curves and factor supply curves. Right?

Mainstream And Non-Mainstream Economics: Research Areas Transgressing The Boundaries

1.0 Introduction

Mainstream and non-mainstream economics can be read as sociological categories, defined by what conferences economists attend, in which journals they publish, and through patterns of referencing. One might expect the intellectual content of the theories put forth by mainstream and non-mainstream economists to cluster, too. In some sense, non-mainstream economists are also automatically heterodox, where heterodoxy refers to the content of theories. For example, heterodox economists tend to prefer theories in which agents are socially embedded and constituted, in some sense, by society (instead of being pre-existing, asocial monads).

The point of this post, though, is to illustrate that the boundary between mainstream and non-mainstream economists is not hard and fast, at least as far as ideas go. I point out two-and-a-half areas where both categories of economists are developing similar ideas.

2.0 Complex Economic Dynamics

Economic models are available which exhibit complex, non-linear dynamics, including chaos. Richard Goodwin, Steve Keen, and Paul Ormerod are some self-consciously non-mainstream heterodox economists who have developed such models. Jess Benhabib and John Geanakoplos are some authoritative mainstream economists on certain models of this type. I also want to mention some researchers who I do not feel comfortable putting in either category. As I understand it, J. Barkley Rosser, Jr. makes an effort to talk to both mainstream and non-mainstream economists. I do not know enough about, for example, Anna Agliari to say what she would say about these categories. And Donald G. Saari is a mathematician interested in social science; so I am not sure how these categories would apply to him, if at all.

3.0 Multiple Selves

I have previously commented on theories of multiple selves, also known as Faustian agents. I particularly like the conclusion, from the Arrow impossibility theorem, that an agent's preferences cannot necessarily be characterized by a utility function, given a theory of modular minds.

I do not think I know enough about these theories to talk authoritatively on this subject. Specifically, I have some dim awareness that a large literature exists here about time (in)consistency of decisions. But I am aware that this is a topic of research among both non-mainstream and mainstream economists. I cite John B. Davis, Ian Steedman, and Ulrich Krause as non-mainstream, heterodox economists with literature in this area. And I cite E. Glen Weyl as a mainstream economist also with literature here.

4.0 Choice Under Uncertainty

Keynes distinguished between risk and uncertainty. Post Keynesian economists have famously developed this theme. Works seen as part of mainstream economics in their time also distinguish between risk and uncertainty, for example:

"...Let us suppose that a choice must be made between two actions. We shall say that we are in the realm of decision making under:

• Certainty if each action is known to lead invariably to a specific outcome...
• Risk if each action leads to one of a set of possible outcomes, each outcome occurring with a known probability. The probabilities are assumed known to the decision maker...
• Uncertainty if either action or both has as its consequences a set of possible specific outcomes, but where the probabilities of these outcomes are completely unknown or are not even meaningful."

-- R. Duncan Luce and Howard Raiffa, Games and Decisions, Harvard University (1957): p. 13.

I only feel entitled to count this as half an example. I find that other literature on the foundations of decision theory is also clear on assumptions about known outcomes and probabilities necessary to characterize a situation of risk. But I do not know of contemporary mainstream economists researching choice under uncertainty (as opposed to risk). I think elements of Chapter 13 of Luce and Raiffa, on decision making under uncertainty, has entered the teaching of business schools targeted towards, for example, corporate managers.

5.0 Reflections

I do not think that this post has demonstrated an openness in mainstream economics. Further work would need to show an awareness among mainstream researchers of parallel work by non-mainstream economists, a willingness to critically engage that work, and a willingness to cite it in mainstream literature. Furthermore, one would like to show that the implications of such work is transitioning into the teaching of economists at all levels. I have seen some economists verbally affirm that economies are complex dynamic systems and then ignore the implications of such a claim. Some economists - for example, Yanis Varoufakis - have expressed skepticism that cutting edge mainstream economics research, in which unique deterministic outcomes do not obtain, can be successfully transitioned. Nevertheless, I find the parallel research noted above to be intriguing.

Against "Science", "Reality", And "Free Will"

1.0 Introduction

You may have noticed. I am not overly fond of neoclassical economics. But today I thought I would talk about criticisms you might find in the blogosphere that I find unpersuasive. That is, I do not like certain one-line assertions, without additional elaborations. I make no attempt to demonstrate here that some make these assertions.

2.0 "Economics Is Not Science"

If you are not arguing about the history, philosophy, or sociology of science, why would you care if a particular field is a science? Should you not be more concerned if the arguments in a field tend to be persuasive, if the norms in the field lead to such arguments? I can see a role here for classifying types of assumptions. One can argue about whether economists put forth supposedly substantial theories that cannot be falsified by any logical or empirical findings. Likewise, perhaps some communities of economics are not as quick as they should be to discard empirically falsified theories. Or one could ask if whatever laws are supposed to be embedded in economic models are restricted to certain institutional and historical instances of capitalism. I hope in putting forth criticisms along these lines, I try to provide concrete examples, not just abstract claims.

3.0 "Economic Theory Does Not Correspond To Reality"

What does use of the word "reality" add to an argument about the persuasiveness or non-persuasiveness of a certain set of doctrines? (I do not mean here to downgrade Tony Lawson's research into ontology and economics. In particular, I do not have a problem with the idea that economic systems are invariably open systems. I guess this idea is in tension with my simultaneous interest in natural experiments.)

4.0 "People Have Free-Will; Thus, Economies Cannot Be Modeled With Mathematics"

I am also not fond of the claim that, since people have free will, one cannot apply mathematics to economics. First, I think at least some applications of mathematics in economics are about algorithms and accounting conventions. I do not see how ideas about consumer choice are relevant to much of this work. Second, I tend to think of the distinction between free will versus determinism as one of those tired dualisms that the linguistic turn in philosophy should have dissolved. I usually cite work drawing on Ludwig Wittgenstein for this sort of point. But let me mention J. L. Austin's "A Plea For Excuses" as being directly relevant for an analysis of when an action is voluntary and of when an agent is responsible or blameworthy for what they do. Austin argues for distinctions that you might not initially see. Third, I agree that describing agents as if they calculate how to obtain a maximum utility curve, given preferences and constraints does not leave room for genuine individual choice. One might try to problematize individual choice and seek more sophisticated models. Such an approach does not necessitate the rejection of mathematics.

Amartya Sen And The Second Phase Of The Classical Revival

Adam Smith and David Ricardo exemplify classical economics. Economists lost and forgot many good ideas in classical economics with the advent of marginalism.

The first phase of the classical revival clarified the classical theory of value and distribution. Scholars achieved this clarification by building on Piero Sraffa's work. Amartya Sen, a student of Maurice Dobb, Piero Sraffa, and Joan Robinson, has commented (e.g., Sen 1974, Sen 2003) a couple of times on Sraffa. He's even had something appreciative to say of the labor theory of value.

Ricardo and Smith have different writing styles, with Ricardo more formal and Smith treating a wider range of material. Furthermore, the theory of value and distribution is only a component, albeit an essential one, of classical economics. In Smith, individuals are not exclusively self-interested utility-maximizers. They exhibit a variety of types of motivations, including sympathy and a regard to social conventions. Furthermore, Smith's statements cannot be partitioned into statements of facts and statements of values. Smith's judgements on what is desirable are multidimensional, not based solely on increasing, for example, income per head.

I have been reading Sen's Development as Freedom and related literature. Like Smith, Sen treats a wider ranges of ideas than are dealt with in the classical theory of value. Sen argues that economic development must be considered along many dimensions, not just income per head. Some of the goals he considers, such as the empowerment of women, are good in themselves. But his consideration of the interconnected instrumental roles of many of these goods provides a rich empirical research program. As examples, I cite the role of decreased income inequality in raising life expectancy, the role of democracy of preventing famine even when crops fail, and the role of increased female literacy in decreasing infant mortality for girls.

Vivian Walsh, while drawing on Hilary Putnam, points out that facts, values, and conventions are, as in Smith, impossible to separate in Sen's work. He looks to Sen as exemplifying the "second phase" of a revival of classical economics. This second phase is less formal than the first, and it treats a wider range of issues.

Perhaps a historian of ideas would find it of interest to research Wittgenstein's role in all of this. Putnam drew on the later Wittgenstein in developing his pragmatic anti-dualism. And, of course, Wittgenstein acknowledges Sraffa's influence. Perhaps a study along these lines would strengthen claims of the complementary nature of Sraffa's and Sen's economics.

(I want to recall that I have commented before on a limited amount of Sen's early work on choice. I also want to recall Daniel Little's post on the second phase of the classical revival.)

(It seems to me that Sen's work already answers the concerns raised in some recent discussions of propertarian confusions about freedom.)

References

• Nuno Martins (2011). "The Revival of Political Economy and the Cambridge Tradition: From Scarcity Theory to Surplus Theory" Review of Political Economy, V. 23, N. 1 (January): pp. 111-131
• Hilary Putnam (2004). The Collapse of the Fact/Value Dichotomy and Other Essays, Harvard University Press. [To read.]
• Hilary Putnam and Vivian Walsh (2009). "Entanglement through Economic Science: The End of a Separate Welfare Economics", Review of Political Economy, V. 21, N. 2 (April): pp. 291-297.
• Hilary Putnam and Vivian Walsh (editors) (2011). The End of Value-Free Economics, Routledge. [To read; a collection that includes some of my other references.]
• Amartya Sen (1974). "On Some Debates in Capital Theory", Economica, V. 41, Iss. 163 (August): pp. 328-335.
• Amartya Sen (1997a). "Maximization and the Act of Choice", Econometrica, V. 65, N. 4 (July): pp. 745-779.
• Amartya Sen (1997.) "Dobb, Maurice Herbert", in The New Palgrave: A Dictionary of Economics (ed. by J. Eatwell, M. Milgate, and P. Newman), Macmillan Press.
• Amartya Sen (1999). Development as Freedom, Alfred A. Knopf.
• Amartya Sen (2003). "Sraffa, Wittgenstein, and Gramsci", Journal of Economic Literature, V. 41 (December): pp. 1240-1255.
• Amartya Sen (2005). "Walsh on Sen after Putnam", Review of Political Economy, V. 17, N. 1 (January): pp. 107-113
• Amartya Sen (2010). "Adam Smith and the Contemporary World", Erasmus Journal for Philosophy and Economics, V. 3, Iss. 1 (Spring): pp. 50-67.
• Vivian Walsh (2000). "Smith After Sen", Review of Political Economy, V. 12, No. 1.
• Vivian Walsh (2003). "Sen After Putnam", Review of Political Economy, V. 15, No. 3 (July).
• Vivian Walsh (2008). "Freedom, Values and Sen: Towards a Morally Enriched Classical Economic Theory", Review of Political Economy, V. 20, N. 2 (April): pp. 199-232.

On The Lack Of Persuasiveness Of Austrian-School Economists

Mattheus von Guttenberg exemplifies what I think are defects in many fanboys of Austrian school economics. Among these defects is an uncritical acceptance of Ludwig von Mises' characterization of his own theories. And another defect is uncritical acceptance, likewise, of what Mises, or even worse, Murray Rothbard, had to say about the mainstream economics of their day. And a third defect is to apply these characterizations to mainstream economics of our day, while remaining quite ignorant of relevant trends in contemporary economics. Without more widespread correction of such defects, advocates of the Austrian school should not be able to persuade many economists, both orthodox and heterodox, of the worth of their views.

For this post, I focus on the theory of choice.

Here are examples of arguably a weak understanding of both the Austrian school and of mainstream economic theory:

"...we're not rejecting cardinal utility functions because it's hip and counter-culture. There's a distinct reason utility functions are impossible and unrealistic, and that's because utility cannot be known or measured... The degree to which we draw swooping utility functions overlaying cost curves is a unacceptable practice borrowed from coordinate geometry. Utility, again - is ordinal, it is intrinsically subjective, and it cannot be made known by other people." -- Mattheus von Guttenberg

"The concept of diminishing marginal utility is implicit in the logic of action, the Austrians just draw it to the fore." -- Mattheus von Guttenberg

The claim that utility reaches an interval-level measurement scale is a conclusion formally drawn from the Von Neumann and Morgenstern axioms (which can be considered independently of game theory). Most introductory economic textbooks claim that utility only reaches an ordinal-level measurement scale, anyways. The introductory textbooks have a different set of axioms, where choice among a set of goods with specified probability is not formally modeled. And they assert that the utility obtained is not interpersonally comparable. Mattheus' objections are not addressed to any views prominent in mainstream economic teaching for at least half a century. And to assert that diminishing marginal utility is consistent with utility reaching only an ordinal-level scale requires an argument. (I'm actually intrigued by J. Huston McCulloch's 1977 attempt to make such an argument, the one example of which I know in the last quarter of the last century.)

Mises incorrectly asserted that much of his theory could be deduced from a single postulate.

"The only axiom is 'man acts' and we draw the entire body of economic science spanning a thousand pages." -- Mattheus von Guttenberg

"...I have always been interested in rewriting [Human Action] 'as a set of numbered axioms, postulates, and syllogistic inferences using, say, Russell's Principia.' I believe it can be done." -- Mattheus von Guttenberg

I think such a rewriting, as it starts from the above informally stated premise, would be unconvincing.

Furthermore, the current state of decision theory suggests that analyses other than Mises' approach, are consistent with this axiom. The Austrian school approach is roughly akin to Samuelson's revealed preference theory. (One important difference is that Austrian advocates have some silly things to say about the impossibility of indifference.) Anyways, the idea is that an acting human, when presented with two lists of goods, decides between them. But social choice theory, as developed by, say, Amartya Sen in the late 1960s and early 1970s, has shown how to dispense with the formalization of choice as a binary relation as a primitive notion. Instead, one can start with a choice function, that is, a mapping from each menu that an agent might be presented with to a set of best choices for that menu. The derivation of a complete and transitive binary preference relation from a choice function requires additional structure on how menu choices relate across menus. And why the imposition of those additional requirements follows from human action needs to be argued. For example, why are not increasingly prevalent models, at least in research literature, of divided selves consistent with human action?

Update (3 July 2014): The blog free radical has a blog post pointing out Austrian confusions about mainstream teaching on ordinal utility.

Robert Nozick, The Refutation Of Rational Choice, Etc.

"Robert Nozick has a unique place in the annals of rational choice theory: he refuted it." -- Ian Hacking (1994)

My reaction, when reading this, was, "What?" Hacking is referring to a paper by Robert Nozick¹ on Newcomb's Paradox. I'm fairly sure I've read something about this paradox, but I had to look it up.

Suppose there exists a psychic that has shown themselves to be extremely reliable in their predictions. And the psychic has presented you with a choice, based on one of their predictions. You are presented two boxes, one transparent and one wrapped such that you cannot see the contents. The rules are that you can take either:

• Just the opaque box, or
• Both boxes.

The transparent box contains $1,000, as you can plainly see. If the psychic has predicted you will pick just the opaque box, they have placed $1,000,000 in it. If they have predicted you will pick both boxes, they have ensured that the opaque box contains nothing. The prediction has been made, and the boxes have been sealed. You know all these conditions but not what the prediction was. What should you do?

Apparently many initially are very decided on what they would do. But people split half-and-half on what that is. Anyways, Hacking states that this example shows that two principles of rational decision-making are not necessarily consistent². I guess he is correct, and I'm in no position to challenge that this is of philosophical interest³. But, since no such psychic can exist, I find other examinations of rational choice theory of more practical import.

By the way, I want to give a qualified defense of Stephen Metcalfe's comments in Slate on Nozick's Wilt Chamberlin example⁴. Strictly speaking, Metcalf's confusion about which Keynes comment was on which Hayek book is irrelevant to these comments later in the article⁵. And I accept that he doesn't describe the logic of Nozick's argument⁶. Neither did I. It is perfectly legitimate to argue that the rhetorical force of the argument comes from elements of the argument extraneous to its strict logic. And that is what Metcalf does⁷.

Footnotes

1. Nozick's "Reflections On Newcomb's Paradox" (in Knotted Doughnuts and Other Mathematical Entertainments (ed. by M. Gardner), W. H. Freeman, 1986).
2. Choose dominant strategies. Maximize mathematical expected utility.
3. I find Wittgenstein perennially fascinating.
4. Metcalf's Slate followup is here.
5. So is the fact that Nozick was smoking dope during the period in which he wrote Anarchy, State, and Utopia; I was startled to find he mentions in his book his experiences while under the influence. More by Brad DeLong on Nozick is here. Even more can be found in the Delong's blog archives.
6. By the way, Yglesias is mistaken in concluding, "Since as best I can tell nobody does hold such a [patterned] theory [of distribution]". Nozick explicitly states that marginal productivity gives such a patterned theory. Nozick is confused, since marginal productivity, correctly understood, is a theory of the choice of technique, not a theory of distribution.
7. Although I am not convinced appealing to guilty regret over the history of race relations in the United States has anything to do with Nozick's rhetoric.

Three Routes To Choice

A theme of this blog is the incorrectness of the neoclassical textbook description of how agents choose. The assumptions of this view can be stated as:

1. An agent knows the complete list of choices from which they must select.
2. Given any two elements from this space of choices, the agent knows whether one of these elements is not preferred to the other.
3. Any element from this space is not preferred to itself.
4. The ranking obtained from the preference relation is transitive.
5. If the space of choices is a continuum, a certain continuity assumption must hold for the preference relation so as to rule out lexicographic preferences.

These assumptions supposedly imply the claim that utility attains at most an ordinal measurement scale level¹. And they allow one to derive the demand for consumer goods and the supply of factors of production.

Economists have transcended this framework. I have previously pointed out models of agents as consisting of multiple selves. I think this approach exhibits a consilience with theories in, for example, cognitive psychology. I have recently stumbled upon two other ways of modeling choice, generalizing the textbook view to an approach more consistent with empirical evidence from behavioral economics and that cannot be justifiably characterized as "irrational".

Nadeem Naqvi has developed an approach of incorporating tertiary information into choice. In the outdated neoclassical theory, one might represent the relationship y is not preferred to x for agent i by:

x R_i y

Naqvi and his colleaques introduce the relation R_i(V_i^j), where V_i^j is the background set for agent i. Parametric variation in the agent’s background set can alter the agent’s preferences. That is, one can have, for l ≠ m:

x R_i(V_i^l) y

and

y R_i(V_i^m) x

One interesting consequence of this modeling strategy is that racial discrimination is formally consistent with Pareto optimality. This "is a surprising, though serious, indictment of relying exclusively on the Pareto principle in social evaluation."

Gul and Pesendorfer consider choice among menus. They consider an agent who is a vegetarian for health reasons, but who is tempted to choose hamburgers, if available. In choosing a restaurant at noon, they would prefer a restaurant with hamburgers on the menu. But in choosing in the morning a restaurant to visit at noon, they will select one with an all-vegtable menu. I hope you can see how this approach allows one to analyze time-consistency of choices.

How long do you think before such approaches are presented in mainstream textbooks in widespread use?

Footnotes:
¹ Nominal, ordinal, interval, and ratio are well-known measurement scale level, where a level is defined up to a set of transformations. I find curious the claim that the expression of the marginal rate of transformation as a ratio of marginal utilities is consistent with an ordinal scale. Mirowski, in More Heat Than Light has also raised questions about the claim that utility only attains an ordinal scale level. I recently stumbled upon Mandler (2006), where he suggests, not necessarily for related reasons, utility be considered to attain a measurement scale level between ordinal and interval.

References

• George A. Akerlof and Rachel E. Kranton (2010) Identity Economics: How Our Identities Shape Our Work, Wages, and Well-Being, Princeton University Press (TO READ).
• Arian Berdellima and Nadeem Naqvi (2011) "Existence of a Pareto optimal social interaction with non-binary preferences".
• Frank Gul and Wolgang Pesendorfer (November 2001) "Temptation and Self-Control", Econometrica, V. 69, N. 6: 1403-1435.
• Michael Mandler (September 2006) "Cardinality versus Ordinality: A Suggested Compromise", American Economic Review, V. 96, N. 4: 1114-1136.
• Nadeem Naqvi (2010) "On Non-binary Personal Preferences, Economic Theory and Racial Discrimination".

Picoeconomics: A New Vocabulary Word For Me

I have previously described models of agents divided in mind. And I have noted that akrasia is defined as the phenomenon of acting against one's own best judgement.

I find that George Ainslie uses the term picoeconomics to describe the study of the interaction of components of a mind in individual behavior and decision-making. Microeconomics is, in some sense, the study of the interactions of individuals in determining economic behavior. Picoeconomics is an analysis on an even smaller scale. I also found a website for this subject¹.

By the way, picoeconomics is not necessarily a non-mainstream field of economics. For example, Glen Weyl (2009), a very young mainstream economist trained at some of the most prestigious economics departments in the United States, adopts a model of an agent as a community. He uses this model to examine political individualism. If a community cannot have group rights and cannot have an unique ordering of choices², how can an individual have such rights when he may be just as divided in mind as a community?

One criticism of mainstream economists relates to their treatment of the literature. A mainstream economist can ignore long-established analytical tools to treat their subject, introduce some related analysis into orthodox models in an ad-hoc way, and never reference the previously-existing heterodox literature. I do not feel I have enough understanding of picoeconomics to say whether this criticism applies to mainstream and non-mainstream contributions to the field³.

Footnotes

• ¹ Is this Ainslie's website? I could not quickly find a name associated with the site?
• ² See the Arrow impossibility theorem.
• ³ I'm not even sure I know the field boundaries. My blogs posts on divided minds build on some literature by Amartya Sen. Some recent papers from Nadeem Naqvi and others build on later literature from Sen. They analyze agent decision-making, but, as I understand it, do not model the mind as composed of subagents. Does this literature fall within picoeconomics?

References

• George Ainslie (1992) Picoeconomics: The Strategic Interaction of Successive Motivation States within the Person, Cambridge University Press. (I haven't read this.)
• George Ainslie (2001) "Breakdown of Will" , Cambridge University Press. (I haven't read this.)
• G. Ainslie (October 2005) "Précis of Breakdown of Will" Behav. Brain Sci., V. 28, N. 5: 650-673.
• Arian Berdellima and Nadeem Naqvi (2011) "Existence of a Pareto optimal social interaction with with non-binary preferences".
• E. Glen Weyl (May 2009) "Whose rights? A critique of individual agency as the basis of rights" Politics, Philosophy & Economics, V. 8, N. 2: 139-171.

James Joyce On Identity Economics

I think that if one looked, one would be able to find in lots of depictions in literature of multiple selves. Here's an example:

"...he had heard about him the constant voices of his father and of his masters, urging him to be a gentleman above all things and urging him to be a good catholic above all things. These voices had now come to be hollow-sounding in his ears. When the gymnasium had been opened he had heard another voice urging him to be strong and manly and healthy and when the movement towards national revival had begun to be felt in the college yet another voice had bidden him to be true to his country and help to raise up her language and tradition. In the profane world, as he foresaw, a worldly voice would bid him raise up his father's fallen state by his labours and, meanwhile, the voice of his school comrades urged him to be a decent fellow, to shield others from blame or to beg them off and to do his best to get free days for the school. And it was the din of all these hollow-sounding voices that made him halt irresolutely in the pursuit of phantoms." -- James Joyce, A Portrait of the Artist as a Young Man

Does how artists depict human beings carry any weight for how economists choose to portray agent's choices? Should it?

Faustian Agents

"Two souls, alas, do dwell within this breast. The one is ever parting from the other" -– Goethe

"He [i.e., Dickens] told me that all the good simple people in his novels, Little Nell, even the holy simpletons like Barnaby Rudge [Slater comments parenthetically that this must have been Dostoevsky's description, not Dickens' -- indeed] are what he wanted to have been, and his villains were what he was (or rather, what he found in himself), his cruelty, his attacks of causeless enmity towards those who were helpless and looked to him for comfort, his shrinking from those whom he ought to love, being used up in what he wrote. There were two people in him, he told me: one who feels as he ought to feel and one who feels the opposite. From the one who feels the opposite I make my evil characters, from the one who feels as a man ought to feel I try to live my life. Only two people? I asked." -- Fyodor Dostoevsky

I have previously described agents that assess an action by ranking outcomes among a number of incommensurable dimensions. By Arrow's impossibility theorem, such an agent in general cannot have a single aggregate ranking of the outcome of actions. I was able to list all best choices for my simple example. That is, for each menu, I listed best choices, with ties being possible. (By the way, a budget constraint is a menu.) If one wants to generalize this approach, one would need to specify methods for specifying best choices when listing all possible menus by hand becomes impractical. Pairwise voting is not a good idea, since the results depend on the voting order in which pairs are compared. Furthermore, one would not want to specify one such method, but allow for many different possibilities. Ulrich Krause has done this. He calls the method for choosing out of these rankings of different aspects an agent's "character". As I understand it, he allows for these rankings to change, based on the agents experience. And so he ends up with a formal model of opinion dynamics. I don't know if or how this relates to Akerlof's identity dynamics, but, I think, that would be an interesting question to explore. References

• K. J. Arrow (1963) Social Choice and Individual Values (2nd. Edition), John Wiley & Sons.
• Ulrich Krause (2010) "Collective Dynamics of Faustian Agents", in Economic Theory and Economic Thought: Essays in Honour of Ian Steedman (ed. by J. Vint, J. S. Metcalfe, H. D. Kurz, N. Salvadori, and P. Samuelson), Routledge.
• S. Abu Turab Rizvi (2001) "Preference Formation and the Axioms of Choice", Review of Political Economy, V. 13, N. 2: pp. 141-159.
• A. K. Sen (1969) "Quasi-Transitivity, Rational Choice and Collective Decisions", Review of Economic Studies, V. 36, N. 3 (July): pp. 381-393.
• A. K. Sen (1970) "The Impossibility of a Paretian Liberal", Journal of Political Economy, V. 78, N. 1 (Jan.-Feb.): pp. 152-157.

To read:

• G. A. Akerlof and R. E. Kranton (2010) Identity Economics: How Our Identities Shape Our Work, Wages, and Well-Being, Princeton University Press.
• J. B. Davis (2003) The Theory of the Individual in Economics: Identity and Value, Routledge.
• A. Kirman and M. Teschl (2004) "On the Emergence of Economic Identity" Revue de Philosphie Économique, V. 9, N. 1: pp. 59-86
• U. Krause (2009) "Compromise, Consensus and the Iteration of Means", Elemente der Mathematik, V. 64: pp. 1-8
• I. Steedman and U. Krause (1986) "Goethe's Faust, Arrow's Possibility Theorem and the Individual Decision-Taker" in The Multiple Self: Studies in Rationality and Social Change (ed. by J. Elster), Cambridge University Press.

Survey of Utility Theory?

1.0 Introduction
I think utility theory has a canonical textbook presentation. Many variations seem to exist. In some, the additional structure is imposed on the (commodity?) space over which agents choose. In others, more basic assumptions are made from which preferences can be derived under certain special cases.

I'd like to know if there are any surveys to read over these variations. I'm not insisting on something critical. And, given the dryness of the subject matter, I might not put such a survey on top of my queue. As can be seen below, I'm not sure of the field that would be demarcated by such a surveys. But literature surveys, in some sense, construct their object.

2.0 Textbook Treatment
Consider a space of n commodities. Each element of the space is a vector x = (x₁, x₂, ..., x_n). Under the usual interpretation, x_i is the quantity of the ith commodity.

An agent is modeled as having a preference relation, ≤, over the space of commodities. A typical question is what assumptions must hold for a utility function to exist. A utility function u(x) exists if, for all x and y in the space of commodities:

x ≤ y if and only if u(x) ≤ u(y)

Typically, the preference relation is taken to be a total order, that is, complete, reflexive, and transitive. A preference relation is complete if, for all x and y in the space of commodities,

x ≤ y or y ≤ x

A preference relation is reflexive if, for all x in the space of commodities

x ≤ x

A preference relation is transitive if, for all x, y, and z in the space of commodities,

if x ≤ y and y ≤ z then x ≤ z

If the quantities of commodities fall along a continuum, a preference relation being a total order is not sufficient for a utility function to exist. Lexicographic preferences are an example of a preference relation for which a utility function does not exist. A continuity assumption rules out this case. This assumption is that for all x in the space of commodities, the sets {y | y ≤ x} and {z | x ≤ z} of commodities not preferred to x and commodities x is not preferred to, respectively, are closed.

Theorem: If a preference relation is a total order and is continuous in the above sense, then a utility function exists.

The utility function is only defined up to a monotonically increasing transformation. In other words, utility is ordinal. Typical exercises are to show certain properties of utility functions, such as ratios of marginal utilities (du/dx_i)/(du/dx_j), are invariant over the set of such transformations.

3.0 Probability
Von Neumann and Morgenstern generalized the commodity space to include vectors of the form: (p₁, x⁽¹⁾; p₂, x⁽²⁾; ..., p_m, x^(m)), where:

p₁ + p₂ + ... + p_m = 1

A commodity, in this sense, is a lottery. Each superscripted commodity vector x⁽ⁱ⁾ is associated with a probability p_i that it will be chosen.

Von Neumann and Morgenstern defined a new set of axioms to go along with their redefined commodity space. One implication is that for any two elements x and y in the commodity space, the linear combination (p, x; (1 - p), y) is also in the space. They obtain that a utility function exists, and it acts like mathematical expectation:

u(p₁, x⁽¹⁾; p₂, x⁽²⁾; ..., p_m, x^(m)) = p₁ u(x⁽¹⁾) + p₂ u(x⁽²⁾) + ... + p_m u(x^(m))

Under Von Neumann and Morgenstern's approach, utility functions are only defined up to affine transformations. That is, they are cardinal. In other words, they attain an interval measurement scale level. The utility for a lottery depends only on the probabilities and the resulting outcomes. It does not depend on how many spins of the wheel or roll of the dice are needed to decide between otherwise equivalent lotteries. Gambling is assumed to have no utility or disutility.

Leonard Savage develops axioms of probability concurrently with axioms of utility theory in his personalistic approach to probability and statistics. I'm not sure how much the survey I would like would go into approaches to probability, even if probability is important to decision theory. The same comment applies to game theory.

4.0 Attributes and Needs
Some see commodities as being chosen as an indirect means to choose something more abstract. As I understand it, Kevin Lancaster depicts a commodity as a bundle of attributes. Different commodities can have some attributes in common. A choice of an element in the space of commodities can then be related to an element in a space of commodity attributes.

The early Austrian school economists thought of goods as being desired for the satisfactions of wants. Water, for example, can be used to water your lawn, to satisfy a pet's thirst, or to drink yourself. One can imagine ranking wants in disparate categories. I am thinking of the triangular tables in Chapter III of Carl Menger's Principles of Economics, in Book III, Part A, Chapter III of Eugen von Böhm-Bawerk's Positive Theory of Capital, and in Chapter IV of William Smart's An Introduction to the Theory of Value. The tables are triangular because the most pressing want in one category typically is less pressing than the most pressing want in another category. An element in the space of commodities corresponds to the set of wants that the agent would choose to satisfy with the quantities of commodities specified by that element.

This mapping from quantities of commodities to sets of wants leads to a redefinition of marginal utility, which one might as well designate by a new name - marginal use. The marginal use of a quantity of commodity is, roughly, the different wants that would be added, with a set union, to the set of wants satisfied by the the given quantities of commodities with that additional quantity of the given commodity. McCulloch shows that a ranking of wants in different categories can arise such that a measure does not exist for the space of sets of wants. (A measure in this sense is a technical term in mathematics, typically taught in courses in analysis or advanced courses in the theory of probability.) He argues that the Austrian theory of the marginal use is thus ordinal. Surprisingly, his argument implies that the law of diminishing marginal utility does not require utility to be measured on a cardinal scale.

I haven't read Ian Steedman's work on consumption, but I think I'll mention it here.

5.0 Choices from Menus
Another generalization of the textbook treatment is to examine how a preference relation can be built out of a more fundamental structure. Imagine the agent is presented with a menu, where a menu is a nonempty set of elements of the commodity space. The agent is assumed to have a choice function, which maps each menu to the set of best choices, in some sense, in that menu. The agent is not postulated to rank either the elements not chosen for a given menu or the elements in the choice set.

A question: what constraints need to be put on choices out of menus such that preferences exist? Since a choice function can be constructed for which no preference function exists, some such constraints exist. I previously noted literature drawing on the logical structure of social choice theory in this context. Alan Isaac emphasizes temporal and menu independence in his overview of abstract choice theory.

6.0 Experimental Economics
I am emphasizing theory. A literature exists on experiments, many of which have falsified the textbook treatment of economics.

7.0 Computatibility, Conservation Laws, Etc.
Some of the above extensions of the textbook treatment seem to postulate some sort of structure within the agent's mind. Computers provide an arguable metaphor of mental processes, and some literature applies the theory of computability to economics. Gerald Kramer, for example, shows that no finite automaton can maximize utility in the simplest setting. I gather others have shown that the textbook treatment postulates that each agent's computation powers exceed those of a Turing machine, that agents compute functions that are, in fact, noncomputable. I turn to Kumaraswamy Velupillai's work for insights into computability, constructive mathematics, and economics. Philip Mirowski is always entertaining. One might also mention the literature on Herbert Simon's notion of satisficing

8.0 Conclusion
This post is a brief overview of some of what would be treated in a survey of variations and approaches to utility theory. Apparently, the notion of economic man can be complicated.

An Incomplete List of References

• Colin F. Camerer (2007) "Neuroeconomics: Using Neuroscience to Make Economic Predictions", Economic Journal, V. 117 (March): C26-C42.
• Alan G. Isaac (1998) "The Structure of Neoclassical Consumer Theory"
• Daniel Kahneman and Amos Tversky (1979) "Prospect Theory: An Analysis of Decision under Risk" Econometrica, V. 47, N. 2 (March): pp. 263-292
• Gerald H. Kramer () "An Impossibility Result Concerning the Theory of Decision-Making", Cowles Foundation Paper 274
• Kevin J. Lancaster (1966) "A New Approach to Consumer Theory", Journal of Political Economy, V. 75: pp. 132-157.
• J. Huston McCulloch (1977) "The Austrian Theory of the Marginal Use and of Ordinal Marginal Utility", Journal of Economics, V. 37, N. 3-4: pp. 249-280.
• Judea Pearl (1988) Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference, Morgan Kaufmann
• Leonard J. Savage (1954, 1972) The Foundations of Statistics, Dover Publications
• Chris Starmer (1999) "Experimental Economics: Hard Science or Wasteful Tinkering?" Economic Journal, V. 109 (February): pp. F5-F15
• Ian Steedman (2001) Consumption Takes Time: Implications for Economic Theory, Routledge
• S. Abu Turab Rizvi (2001) "Preference Formation and the Axioms of Choice", Review of Political Economy, V. 13, N. 12 (Nov.): pp. 141-159
• John Von Neumann and Oskar Morgenstern (1953) Theory of Games and Economic Behavior, Third Edition, Princeton University Press

Nietzsche On The Individual As A Society

I have previously noted the problems for utility theory created by the application of Arrow's impossibility theorem to a single individual. And I had quoted a number of classic authors who wrote of themselves as being composed of more than one mind. Here's another:

"'Freedom of the will' - that is the expression for the complex state of delight of the person exercising volition, who commands and at the same time identifies himself with the executor of the order - who, as such, enjoys also the triumph over obstacles, but thinks within himself that it was really his will itself that overcame them. In this way the person exercising volition adds the feelings of delight of his successful executive instruments, the useful 'underwills' or undersouls - indeed our body is but a social structure composed of many souls - to his feelings of delight as commander. L'effet c'est moi. What happens here is what happens in every well-constructed and happy commonwealth; namely, the governing class identifies itself with the successes of the commonwealth. In all willing it is absolutely a question of commanding and obeying, on the basis, as already said, of a social structure composed of many 'souls'." -- Friedrich Nietzsche, Beyond Good and Evil: Prelude to a Philosophy of the Future (Kaufmann translation), paragraph 19

By the way, the idea of modeling an individual choice with a structure underlying the textbook treatment of preferences over the elements of a linear space of commodities is not necessarily non-mainstream. I cannot say I know much about the relevant literature. However, I stumbled over an example - a paper, "Multiple Temptations", from John E. Stovall, a graduate student at the University of Rochester.

Samuelson’s Revealed Preference: A Failed Research Program

Wong (1978, 2006) grew out of what may have been the last doctorate thesis Joan Robinson supervised. As it is, Wong did not complete his thesis under Robinson's supervision. Luigi Pasinetti and then Geoffrey Harcourt later became Wong's supervisor.

Wong’s study is centered around three publications by Paul Samuelson, in 1938, 1948, and 1950. Samuelson, in 1938, according to Wong, attempted to construct a new theory without any reliance on utility theory or any concept that relies on non-observational phenomena. This theory was intended to be a replacement, not a complement for utility theory.

Samuelson, in 1940, according to Wong, attempted to construct indifference maps from observed consumer choices in a space of price and quantity observations. "The whole theory of consumer's behavior can thus be based upon operationally meaningful foundations in terms of revealed preference." -- Samuelson (1948)

Samuelson, in 1950, according to Wong, was responding to work primarily by Hendrik Houthakker, who showed that ordinal utility theory and revealed preference theory were logically equivalent. Thus, utility theory has the same empirical implications and operational foundations.

Wong interprets the observational equivalence of utility and revealed preference as a defeat for Samuelson's 1938 program. Samuelson, however, asserted this finding was the completion of a victorious research program. And mainstream economists have let him get away with this claim, without ever subjecting it to a critical inquiry.

"I soon realized that [the weak axiom of revealed preference] could carry us almost all the way along the path of providing new foundations for utility theory. But not quite all the way. The problem of integrability, it soon became obvious, could not yield to this weak axiom alone." -- P. A. Samuelson (1950)

References

• Paul A. Samuelson (1938) “A Note on the Pure Theory of Consumer’s Behaviour”, Economica, v. 5: pp. 61-71.
• Paul A. Samuelson (1948) “Consumption Theory in Terms of Revealed Preference”, Economica, v. 15: pp. 243-253.
• Paul A. Samuelson (1950) “The Problem of Integrability in Utility Theory”, Economica, v. 17: pp. 355-385.
• Stanley Wong (1978, 2006) Foundations of Paul Samuelson’s Revealed Preference Theory: A Study by the Method of Rational Reconstruction, Revised Edition, Routledge.

An Experiment Protocol

1.0 Introduction
The point of the experiment described here is to offer empirical evidence for the importance of the distinction between uncertainty and risk, as put forth by Frank Knight and by John Maynard Keynes. People are not "rational", as "rationality" is defined by neoclassical economists.

As usual, I don't claim much originality except, maybe, in details. Daniel Ellsberg described the experiment below, as well as another. He references Chipman as having conducted experiments much like these. (Although Ellsberg's paper is oft cited and has been republished, Daniel Ellsberg is probably best known for having leaked The Pentagon Papers to the New York Times and others. Nixon's "plumbers" illegally broke into and searched Ellsberg's psychiatrist's office.)

2.0 The Protocol
The experimenter shows the test subject two urns, urn I and urn II. The test subject is shown that urn 1 is empty. The experimenter truthfully assures the test subject that urn II contains 8 balls, with some or none of them red and the remainder black. The test subject sees the experimented put one red and one black ball in urn II. The experimenter also puts in five red and five black balls in urn I in the test subject's presence. The urns are shaken.

So the test subject knows that urn number I contains 5 red and 5 black balls. Urn number II contains 10 balls. All are either red or black. At least one is black, and at least one is red.

The experimenter flips two coins so as to offer a gamble to the test subject. The coin flipping ensures the probability of offering each gamble is one in four. The gambles are described to the test subject:

• Gamble A: You pay $5 for a draw from urn number I. You choose before the draw whether to play red or black. If a ball is drawn of your color, you receive a payout of $10.
• Gamble B: You pay $5 for a draw from urn number II. You choose before the draw whether to play red or black. If a ball is drawn of your color, you receive a payout of $10.
• Gamble C: You pay $5. You choose urn number I or urn number II. A ball is drawn from the urn you selected. If the ball is red, you receive $10.
• Gamble D: You pay $5. You choose urn number I or urn number II. A ball is drawn from the urn you selected. If the ball is black, you receive $10.

Each test subject goes exactly once, and no test subject is able to observe previous plays by other test subjects (so urn number II cannot be sampled by a test subject).

The hypothesis is that in gambles A and B, statistically equal numbers of people will choose each color, while in gambles C and D, people will prefer to choose urn nmber I.

3.0 To Do

• Demonstrate mathematically that no assignments of probability in urn number II are compatible with the hypothetical behavior.
• Decide on a sample size. Perhaps a sequential test can be defined in which the sample size is not known beforehand.
• Read Craig and Tversky (1995) and Chipman (1960). Where else is Ellsberg referenced?

References

• J. S. Chipman, "Stochastic Choice and Subjective Probability", in Decisions, Values and Groups (edited by D. Willner), Pergamon Press (1960)
• Daniel Ellsberg, "Risk, Ambiguity, and the Savage Axioms", Quarterly Journal of Economics, V. 75, N. 4 (Nov. 1961): 643-669
• Craig R. Fox and Amos Tversky, "Ambiguity Aversion and Comparative Ignorance", Quarterly Journal of Economics, V. 110, N. 3 (1995): 585-603

Greek To Me

1.0 Introduction
I previously described, in an abstract way, a model in which individuals choose rationally even though they may not have a complete transitive preference relation. In that post, I relied heavily on a paper by S. Abu Turab Rizvi. Searching on some of Turab Rizvi's references, I stumbled upon Jeanne Peijnenburg's doctoral thesis, Acting Against One's Best Judgement: An Enquiry into Practical Reasoning, Dispositions and Weakness of Will. Reading some of this thesis inspired me to revisit my model by presenting a somewhat more concrete example.

2.0 Background
I learned a new word from Peijnenburg's thesis. Acting against one's own best judgement is called "akrasia". Peijnenburg shows that discussion of being divided in mind goes back, at least, to debates among Socrates, Plato, and Aristotle. She provides some amusing quotes about akrasia:

"I do not do what I want to do but what I hate... What happens is that I do, not the good I will to do, but the evil I do not intend." -- Romans 7:15 and 7:19

"The mind orders the body and is obeyed. But the mind orders itself and meets resistance." - Augustine

"Two souls, alas, do dwell within this breast" - Goethe

"Faust complained that he had two souls in his breast. I have a whole squabbling crowd. It goes on as in a republic." -- Otto von Bismarck

3.0 The Example
Consider an individual choosing among three actions. This person foresee an outcome for each action. For my purposes, it is not necessary to distinguish between an action and the outcome the individual believes will result from the action. Accordingly, let A, B, and C denote either the three actions or the three outcomes, depending on context.

3.1 Tastes
Suppose that the individual cares about only three aspects of the outcome. For example, if the action is obtaining an automobile of one of three brands, one aspect of the outcome might be the fuel efficiency obtainable from the car. Another might be the roominess of the car interior. And so on.

In the example, the individual has preferences among these three aspects of the outcomes, but not over the outcomes as a whole. "Preferences" are here defined as in neoclassical economics, that is, as a total order. Let the individual order the actions under each aspect as shown in Table 1. For example, under the first aspect, this person prefers A to B and B to C. Since a total order is transitive, one can conclude that this individual prefers A to C under the first aspect. The individual prefers C to A, however, under either of the other two aspects. (This example has the structure of a Condorcet voting paradox, but as applied to an individual.)

Table 1: Preferences Over Aspects of Outcomes

Aspect	Preference Over Aspect
1st	A > B > C
2nd	B > C > A
3rd	C > A > B

3.2 The Choice Function
The individual is not necessarily confronted with a choice over all three actions. Mayhaps only two of the three needed automobile dealers have franchaises in this person's area. The specification of the example is completed by displaying possible choices for each menu of choice with which the individual may be confronted. That is, I want to specify a choice function for the example:

Definition: A choice function is a map from a nonempty subset of the set of all actions to a (not necessarily proper) subset of that nonempty subset.

The domain of a choice function is then the set of all nonempty subsets of the set of all actions. Informally, the value of a choice function is the set of best choices on a menu of choices with which an agent is confronted. (The above definition is a variation on the one I gave in my previous post.)

Table 2 gives the choice function for this example. The first three rows show that in a menu consisting of exactly one action, the individual chooses that action. In a menu consisting of exactly two actions, the individual is willing to choose only one of those actions. And in a menu with three actions, the individual is willing to choose any of the three.

Table 2: The Choice Function

Choices on the Menu	Best Choice(s)
{A}	{A}
{B}	{B}
{C}	{C}
{A, B}	{A}
{A, C}	{C}
{B, C}	{B}
{A, B, C}	{A, B, C}

3.3 The Conditions of Arrow's Impossibility Theorem
I intend the above example as an illustration of application of Arrow's impossibility theorem to a single individual. The choice function given above is compatible with the conditions of Arrow's impossibility theorem:

• No Dictator Principle: For each aspect, some menu exists in which the choice function specifies a choice in conflict with preferences under that aspect. For example, the choice from the menu {A, C} conflicts with the individual's preferences under the first aspect of the outcomes.
• Pareto Principle: This principle is trivially true in the example. No menu with more than one choice exists in which preferences under all aspects specify the same choices. So the choice function cannot be incompatible with the Pareto principle when it applies, since it never does apply.
• Independence of Irrelevant Alternatives: I think this principle is also trivially true.

In compatibility with Arrow's impossibility theorem, the existence of a single preference relation is not possible for the above choice function. A preference relation applies to all possible pairs of actions, and it must be transitive. But a transitive relation cannot be constructed for the three menus consisting of exactly two actions. So I have defined a choice function, but preferences (one total order) does not exist.

4.0 Conclusions
Neoclassical economists tend to equate rationality with the existence of a unique preference relation for an individual. In other words, rationality for an individual is identified with the existence of one total order (that is, a complete and transitive binary relation) over a space of choosable actions. The example suggests this point of view is mistaken. An orthodox economist can either assert that the individual in the example is not rational or accept that he has been learning and teaching error.

A choice function is a generalization of preferences, as neoclassical economists understand preferences. If such preferences exist for an individual, then a choice function exists for that individual. But individuals can have choice functions without having such preferences, as is demonstrated by the above example. It is up to those asserting the existence of preferences to state their special-case assumptions, to show that models with those assumptions can provide falsifiable predictions about society, and to provide empirical evidence. The evidence from experimental economics, though, is systematically hostile to neoclassical economics. The phenomenon of menu-dependence is particularly apposite here.

So much for prattle about competitive markets yielding efficient outcomes.

How Individuals Can Choose, Even Though They Do Not Maximize Utility

1.0 Introduction
I think of this post as posing a research question. S. Abu Turab Rizvi re-interprets the primitives of social choice theory to refer to mental modules or subroutines in an individual. He then shows that the logical consequence is that individuals are not utility-maximizers. That is, in general, no preference relation exists for an individual that satisfies the conditions equivalent to the existence of an utility function. I have been reading Donald Saari on the mathematics of voting. What are the consequences for individual choice from interpreting this mathematics in Rizvi's terms?

I probably will not pursue this question, although I may draw on these literatures to present some more interesting counter-intuitive numerical examples.

2.0 Arrow's Impossibility Theorem and Work-Arounds
Consider a society of individuals. These individuals are "rational" in that each individual can rank all alternatives, and each individual ranking is transitive. Given the rankings of individuals, we seek a rule, defined for all individual rankings, to construct a complete and transitive ranking of alternatives for society. This rule should satisfy certain minimal properties:

• Non-Dictatorship: No individual exists such that the rule merely assigns his or her ranking to society.
• Independence of Irrelevant Alternatives (IIA): Consider two countries composed of the same number of individuals. Suppose the same number in each country prefer one alternative to another in a certain pair of alternatives, and the same number are likewise indifferent between these alternatives. Then the rule cannot result in societal rankings for the two countries that differ in the order in which these two alternatives are ranked.
• Pareto Principle: If one alternative is ranked higher than another for all individuals, then the ranking for society must rank the former alternative higher than the latter as well.

Arrow's impossibility theorem states that, if there are at least three alternatives, no such rule exists.

Arrow's work has generated lots of critical and interesting research. For example, Sen considers choice functions for society, instead of rankings. A choice function selects the best alternative for every subset of alternatives. That is, for any menu of alternatives, a choice function specifies a best alternative. Consider a rule mapping every set of individual preferences to a choice function. All of Arrow's conditions are consistent for such a map from individual preferences to a choice function.

Saari criticizes the IIA property as requiring a collective choice rule not to use all available information. In particular, the rule makes no use of the number of alternatives, if any, that each individual ranks between each pair. The rule does not make use of enough information to check that each individual has transitive preferences. (Apparently, the IIA condition has generated other criticisms, including by Gibbard.) Saari proposes relaxing the IIA condition to use information sufficient for checking the transitivity of each individual's preference.

Saari also describes a collective choice rule that includes each individual numbering their choices in order, with the first choice being assigned 1, the second 2, and so on. With these numerical assignments, the choices are summed over individuals, and the ranking for society is the ranking resulting from these sums. This aggregation procedure is known as the Borda count. Saari shows that Borda count satisfies the relaxed IIA condition and Arrow's remaining conditions.

3.0 Philosophy of Mathematics
Above, I have summarized aspects of the theory of social choice in fairly concrete terms, such as "individuals" and "society". The mathematics behind these theorems is formulated in set-theoretic terms. The referent for mathematical terms is not fixed by the mathematics:

"One must be able to say at all times - instead of points, straight lines, and planes - tables, chairs, and beer mugs." - David Hilbert (as quoted by Constance Reid, Hilbert, Springer-Verlag, 1970: p. 57)

"Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." -- Bertrand Russell

4.0 An Interpretation
Rizvi re-interprets the social choice formalism as applying to another set of referents. A society’s ranking, in the traditional interpretation, is now an individual’s ranking. An individual’s ranking, in the traditional interpretation, is now an influence on an individual’s ranking. Rizvi’s approach reminds me of Marvin Minsky's society of mind, in which minds are understood to be modular. Rizvi examines the implication’s of Sen’s impossibility of a Paretian liberal for individual preferences under this interpretation of the mathematics of social choice theory.

Constructing natural numbers in terms of set theory allows one to derive the Peano axioms as theorems. Similarly, interpreting social choice theory as applying to decision-making components for an individual allows one to analyze whether the conditions often imposed on individual preferences by mainstream economists can be derived from this deeper structure. And, it follows from Arrow's impossibility theorem, these conditions cannot be so derived in general. Individuals do not and need not maximize utility. On the other hand, Sen's result explains how individuals can choose a best choice from menus with which they may be presented.

References

• Kenneth J. Arrow (1963) Social Choice and Individual Values, Second edition, Cowles Foundation
• Alan G. Isaac (1998) "The Structure of Neoclassical Consumer Theory", working paper (9 July)
• Marvin Minsky (1987) The Society of Mind, Simon and Schuster
• Donald G. Saari (2001) Chaotic Elections! A Mathematician Looks at Voting, American Mathematical Society
• S. Abu Turab Rizvi (2001) "Preference Formation and the Axioms of Choice", Review of Political Economy, V. 13, N. 2 (Nov.): 141-159
• Amartya K. Sen (1969) "Quasi-Transitivity, Rational Choice and Collective Decisions", Review of Economic Studies, V. 36, N. 3 (July): 381-393 (I haven't read this.)
• Amartya K. Sen (1970) "The Impossibility of a Paretian Liberal", Journal of Political Economy, V. 78, N. 1 (Jan.-Feb.): 152-157