The natural sciences study things beyond the ken of ordinary people- electrons, genes, black holes- whereas social sciences study things which some ordinary people know much more about than any Professor. Thus, 'natural language' terms are richer and more useful than the terms of art of the economist or philosopher. Nevertheless, for specific, protocol bound, purposes, those terms of art play a vital role in improving communication and decision making. But this requires practitioners not only to use their terms of art correctly but also to give them restricted, qualified, natural language interpretations or 'models' . Failure to be scrupulous in this matter leads quickly to the spouting of mischievous nonsense.
Consider the English word 'choice'. It is related to having freedom to select from among different options. We wouldn't think we had freely chosen a card forced on us by a magician or that a 'bait and switch' contract we were tricked into signing really represented something we freely chose. This highlights the strategic element in 'choice', or- in the language of Continental philosophy- the 'mise en abyme' of two consciousness facing each other. What I mean is that we very quickly get to infinity in a series of the following sort- you know that I know. I know that you know that I know, I know that you know that I know that you know, etc. The assumption of 'common knowledge' subsumes an infinite regress of this sort. It is useful so as to avoid 'money pumps' or 'intuition pumps' or 'cobwebs' or other obvious pathologies. Yet, shitty academics can make a career out of ignoring utility and just talking bollocks. This may be justified under the rubric of 'virtue signalling' or 'activism' or 'wokeness' or whatever- but it is a waste of time. Maths has elaborated tools so as to be more useful, not to disguise stupidity and laziness. This is precisely why some academics deliberately choose to appear Mathsy while working hard to ignore all contemporary developments and thus remain in their Ivory Tower Eden of ignorance.
To be fair, it is certainly true that Society is riddled with cognitive biases and we too, however enlightened we might wish ourselves to be, routinely succumb to stupid stereotypes. More 'philosophically', I suppose one could also say that there are habitual or customary things we do which aren't really our choices at all. Here our intuition surrounding the word 'choice' conflicts with some ways in which it might be used. More generally, questions about 'free will' raise their ugly head. To an infinite intelligence, could our poor finite reasoning look like anything but the reflex actions of a hairless ape? However, if there is any thought behind that last sentence, surely this finite mind of ours has something of the infinite about it?
Consider what mathematicians call the axiom of choice- For any set X of nonempty sets, there exists a choice function f that is defined on X and maps each set of X to an element of the set. This turns out to be equivalent to 'For any set A there is a function f such that for any non-empty subset B of A, f(B) lies in B.' For finite sets, this is an inductive result. This means for a specific configuration space- e.g the market for onions- we can either say that strategic behaviour (onions are only planted if onion-planters believe most farmers won't plant onions and so its price will be high) 'crashes' the market (there are wild price swings and so prudent farmers abandon onion growing) or else that 'expectations' will be 'Muth rational'- i.e. instead of some infinitistic mise en abyme of strategy, farmers just look at the cost of production and the demand schedule and work out the equilibrium price and make their planting decisions accordingly. This is an example of a smart, mathematically interesting, 'restriction' on 'sets' or 'choice functions' such that you get a useful and sensible result. Thus, since the problem has become 'finite', induction gets us to a 'canonical solution' without our having to lend credence to any metaphysical shite.
We might say- 'this assumption of Muth Rationality diminishes us! We like the notion of an infinite mise en abyme- mirrors endlessly reflecting each other- because that captures something god-like about our notions of 'freedom'! Sartre wasn't a boring pile of shite. We wanna live in his Universe- not that of Adam Smith.'
The problem here is that inductive reason can do useful stuff in a finite, or constructivist, world without our having to place our faith in 'axioms' or 'necessary truths' only justifiable in the Mind of God. Thus, for finite sets, the proof that f(B) lives in B is inductive. That's good, because Muth Rationality or 'common knowledge' can solve coordination problems in a highly utile, cheap, and decentralized manner. But there is no inductive derivation of this result for infinite sets. There the thing is an axiom- i.e. a secular substitute for a Divine commandment.
On the one hand, this suggests that what is finite is qualitatively different- more can be known about it by purely logical means- from what is infinite and that the very act of choosing is in some sense implicated in this. Briefly, if options, or 'sets' containing elements which could be options for a choice function, can be 'well ordered' then many useful results flow by assuming a 'law of continuity'- "whatever succeeds for the finite, also succeeds for the infinite". Here lie the foundations of analysis- i.e. calculus. On the other hand, however, there seems something arbitrary or even counterfactual about the manner by which we get to this type of useful result. Does this represent a scandal? To our current thinking, no, provided we are scrupulous in how we interpret our results. There is some 'Reverse Mathematics' such that 'axioms' are winnowed down and so there is a 'natural' or 'canonical' common knowledge configuration space of a wholly pragmatic type. A Christian may speak of the Katechon, a Hindu might babble of Krishna but Category theory would give both the Theist and the Atheist a playground where they could do stuff delightful to us all.
Such of my readers as are poetic adolescents or prosy idealists might well complain that my deflationary account leaves aside the wider philosophical question- a question about the Human Spirit- which arises out of the consideration that we know uncountable infinities are 'bigger' than countable ones; might there be some further hierarchy of infinities such that there is more to our choice, some greater penalty or opportunity attaching to the exercise of our free will, than our savants can currently envision? These are mystical thoughts but mystics- like Rabindranath Tagore- are perfectly capable of living very useful lives. But Tagore had talent. My poetry- being fractally Socioproctological- is shit- more particularly when I'm genuinely not being an asshole and really am, for a moment, animated by lurve for all- for all love sustains the lurve of even the utterly unlovable cunt writing this stupid shite. That's right; I get a hard on for my own haecceity as asshole coz... youse guys!... youse just so fucking lovely mate! each of you ablaze in precisely that bonfire of your own inextinguishably unique being such that not in you, but the Cosmos, every blemish, every mote in, or jaundice of, the eye, is fulminated or set free to flower as a fresh Eden of restored innocence or painless parturition.
Fuck me. Sometimes I embarrass myself.
Returning to the topic of this post, I must briefly dwell on an intuitionistic type of mathematics created by, a great admirer of Tagore, L.E.J Brouwer which features 'choice sequences' which are not fully determined. Brouwer's motivation was his intuition of the continuum which, it turns out, is connected in a very deep way with measure theory and thus probability and stochastic dynamics.
It should be noted, Brouwer's intuitionistic type of math does not have a 'law of the excluded middle' whereby things must either be x or not x. However, an axiom of 'countable choice'- which is a weak version of the axiom of choice- gets us to something pretty close to 'excluded middle'. But 'countable choice' is problematic because it is 'intensional choice' and has arguments which have 'intensional properties' which can vary in possible worlds (how do we know there are not infinitely many?) without any change in the extension of the set which presents before us. This means that incautious reasoning could lead to our babbling nonsense because the underlying property is not univocal.
Turning to the subject of this post, I wonder whether, if economists really want to talk about 'choice' in a manner which corresponds to our intuition, then perhaps they should use 'choice sequences'. To speak of a 'choice function' is misleading if you are trying to capture a 'natural' intuition regarding what constitutes choice. Moreover, you may end up talking nonsense about 'choice' if you try to derive 'theorems' by relying on the 'law of the excluded middle'. The result would be the same as if you were wholly ignorant and incompetent in both English and Mathematics.
As a case in point, I suggest that Sen's being shit at English comprehension also led to his being shit at using Math. But, this was a choice which paid off for him because there is a market for simple-minded Brown dudes who, while pretending to be good little atheists, actually believe in magical remedies and Maharishi type metaphysics and Mother Theresa type miracles.
Consider Sen's one mathematical innovation- the introduction of a quasi-transitive binary relation. Does it actually exist in a useful manner?
In 1969, Sen published a paper defining a binary quasi-transitive relation which is symmetric (i.e. a equals b means b equals a) in some places and transitive (if a is bigger than b and b is bigger than c then a is bigger than c) in other places. Any transitive binary relation could be used to create multiple quasi-transitive binary relations by relaxing transitivity at some point. However these quasi-transitive relations would not yield a unique preorder or 'non strict' partial order unless they were also antisymmetric- i.e. there is no pair of distinct elements, one of which is related to the other in exactly the same way the other is related to it. In this case, we know that everything in the set is related to everything else almost as well as if transitivity obtained. The problem is that we would no longer be sure that the relation really is binary because the nature of the set or, in certain cases, the class of relevant elements may both satisfy and not satisfy the axiom of extensionality (i.e. sets with the same members are the same) because an element may be 'intensional' in some sense. This means that we can't rely on our definition of the relation to prove any theorem because we are begging the question re whether there really is a 'Choice set' with a given property. We are merely making foolish assertions of an unreasonable and unreasoning kind.
Sen's paper begins thus-
What if a better alternative gets chosen by including in S something not in X? This can happen if an element in X is 'intensional' in one way but extensional in another. In a sense, it is in X and thus satisfies the axiom of extensionality but, in some other sense, it doesn't have the same property or isn't related to other elements at all. In other words, looked at one way it has the same property and belongs to the same 'class' but looked at in another way it has some different property. This may be because the 'reflexivity' relation breaks down at some point or else anti-symmetry breaks down or maybe something weird happens when the Choice function takes it as an input.
I suppose, if we want to talk nonsense with less fear of detection, we might deliberately, or systematically, include elements of this type in our supposedly mathematically rigorous depiction of a binary relation and its associated choice function.
On the other hand, if we are merely concerned with applying mathematical discoveries to some useful end, we would note that Szpilrajn had shown that for 'any asymmetric and transitive relation, there exists an asymmetric, transitive and complete relation that contains the original relation. An analogous result applies if asymmetry is replaced with reflexivity—that is, any transitive relation has an ordering extension'. Suzumura (1976) strengthened this result considerably by establishing that the transitivity assumption can be weakened to consistency without changing the conclusion regarding the existence of an ordering extension. Moreover, he showed that consistency is the weakest possible property that guarantees this existence result.' Essentially, Suzumura eliminates cyclicity or 'money pumps' where there is at least one strong preference. However, costless arbitrage would have the same effect. In a sense, a speculator may be both a seller and a buyer of a commodity and his 'net' preference depends on the preferences of others. But this may also be true of a moralist seeking to set an example. This leads us to the conclusion that no Preference or Choice relation is antisymmetric if there are arbitragers or moralists or others who have an 'intensional' or 'strategic' interest in preferences and choices. Simply put, there is no choice function for a set of people at least some of whom actually make choices in the 'natural' sense of the word.
Specifically, a person can't exist who is 'indifferent between 7 and 8 grams of sugar and indifferent between 8 and 9 grams of sugar, but who prefers 9 grams of sugar to 7.' Why? This person will take the larger amount and sell the excess to an arbitrageur, or else, as a moralist, will always prefer the smallest possible amount. Thus, 'quasi-transitivity' is useless for studying any economic process involving rational beings because either there is no binary relation or it is unknowable. Any stipulation made re. its existence is merely a stipulation we are free to reject. Nothing is entailed by it by reason of its supposed mathematical properties.
Sen, unlike Srinivasan or Sukhamoy Chakraborty, seems to have dropped out of growth theory after his initial foray. Thus he seems unaware or uninterested in the Pontryagin maximum principle whose importance arises from the fact that maximizing the 'Hamiltonian' (instantaneous increment on the Lagrangean) is computationally much easier than the original infinite-dimensional control problem. Instead of maximizing over a function space, the problem is converted to a pointwise optimization. A similar logic leads to Bellman's principle of optimality, a related approach to optimal control problems which states that the optimal trajectory remains optimal at intermediate points in time.In 'Maximization and the act of choice' Sen wrote-
'Behavior characterized by choice may be differentiated from conditioned behavior in which choice is merely imputed, not actual- by awareness of uncertainty sources and knowledge of how such uncertainty may 'cascade'. In particular, the use of stochastic differential equations yields models with a precise interpretation of choice as either 'adapted' or ergodic, or 'non-adapted' and 'insider' (i.e. uncorrelated asymmetries obtain) or otherwise hysteresis based and thus not purely economic in nature.
'Iff an act of choice is an adapted process- i.e. can't see into the future- then maximization is well defined as an Ito stochastic integral. This permits useful results in mathematical finance more particularly by incorporating non-standard analysis. Optimization is different from maximization because it invokes backward induction or, alternatively, involves the construction of notional 'risk-free' measures using Girsanov's theorem- e.g. Black-Sholes Merton option pricing. Only if a theory is sufficiently fine-grained and informationally well-specified could there be a functional relationship between maximization and optimization. Otherwise, multiple realizability will obtain- i.e. supervenience is an empty intuition not a functional relationship. Since Economics is only concerned with choice under scarcity- i.e. where there is an opportunity cost- divergence between maximization and optimization is evidence that either a 'free good' obtains- i.e. the budget set is not unique- or else that existing information has not been fully incorporated and some 'signal' is being mischaracterized as noise. This is a non-excludable 'bad'. Where this happens, no normative issue is involved. It is simply the case that economists need to do more research of an alethic kind so as to correctly specify the components of the underlying Levy process.'
I'm kidding. Sen didn't write that. I just made it up. But, I imagine, it is the sort of thing a mediocre, but mathsy, Economist might usefully have written back in the Nineties.
What Sen wrote was simply stoooopid-
The act of choosing can have particular relevance in maximizing behavior
But 'maximizing behavior' does not involve choice. It is merely behavior with a particular property- viz. it can be predicted by an operation on a mathematical function.
The reverse however is possible. Studying how to maximize a mathematical function can have relevance for acts of choice.
for at least two distinct reasons: (1) process significance (preferences may be sensitive to the choice process, including the identity of the chooser),
Preferences depend entirely on the choice process (e.g- if it involves having your dick bitten off by a rabid dog, don't do it) and the identity of the person or entity concerned (e.g I am happy to choose for you to give away all your wealth but not happy to choose to do so myself).
But it is not the case that 'the act of choosing' has any relevance to the maximization of a mathematical function even though it may itself rely on an axiom of choice or an inductive result to the same effect. Either the function already exists and there is an algorithmic method to perform the operation or else it is wholly irrelevant to any 'act of choosing'.
and (2) decisional inescapability (choices may have to be made whether or not the judgemental process has been completed).
This is foolish. We know the that the solution to a maximization problem may be more or less resource and time intensive depending on the degree of accuracy or likelihood of the desired result. Indeed, some solutions may be in a higher time class than the life of the universe. But the workaround is simple. Depending on the time frame for the decision, you chose the technique which gets a good enough result. However there is no 'decisional inescapability'. You may choose to do nothing.
The general approach of maximizing behavior can- appropriately formulated- accommodate both concerns,
Both 'concerns' are wholly imaginary. It is not the case that we engage with 'choice processes' which involve our getting our dicks bitten off or our suddenly turning into a French-Cambodian ladyboy. Nor is there any great difficulty in using an approximating procedure which is good enough given the relevant time frame.
but the regularities of choice behavior assumed in standard models of rational choice will need significant modification.
Nonsense! All you need is a bit of common-sense to dispose of these two entirely bogus 'concerns' which Sen has mentioned.
These differences have considerable relevance in studies of economic, social, and political behavior
Twenty years later we know they had zero relevance. The thing was a waste of time.
Sen begins his essay with a typical piece of fake erudition
IN 1638, WHEN PIERRE DE FERMAT sent to Rene Descartes a communication on extremal values (pointing in particular to the vanishing first derivative),
which is a necessary not sufficient condition because saddle points exist. Interestingly, Fermat translated a term from Diophantus as 'adequality' or almost equality to get rid of infinitesmals to calculate maxima and minima.
the analytical discipline of maximization was definitively established.
No. Fermat was an amateur mathematician. The 'discipline' was established later as the utility of the method became clear. On the other hand, I suppose you could say this type of analysis was a 'discipline' in the Kerala School of Mathematics at an earlier point in time. Roddam Narasmiha has written about this.
Fermat's "principle of least time" in optics was a fine minimization exercise (and correspondingly, one of maximization). It was not, however, a case of maximizing behavior,
yes it was. We use the word 'behavior' for what an animal, a plant but also an elementary particle, does.
since no volitional choice is involved (we presume)
Behavior may not involve volitional choice and volitional choice may not affect behavior.
in the use of the minimal-time path by light. In physics and the natural sciences, maximization typically occurs without a deliberate "maximizer."
This is irrelevant. Either a mathematical function captures behavior or it is useless.
This applies generally to the early uses of maximization or minimization, including those in geometry, going back all the way to "the shortest arc" studied by Greek mathematicians, and other exercises of maximization and minimization considered by the "great geometers" such as Apollonius of Perga.
But geometry was being used by architects and topographers and so forth. If one guy told another guy the shortest route to the next town- that was 'optimizing' behavior involving consciousness and volition. If some math was involved, then we could speak of a mathematical function being maximized or minimized.
The formulation of maximizing behavior in economics has often paralleled the modelling of maximization in physics and related disciplines.
If a mathematical function can predict what a population will do, then there is 'maximizing' behavior regardless of whether that population consists of people or elementary particles. On the other hand, it is true that the application of Ito's stochastic calculus had opened up new vistas for Econ. But Sen chose to remain ignorant of this development. So 'the formulation' he is speaking had been superseded. But nothing 'normative' can be said about what is obsolete.
But maximizing behavior differs from nonvolitional maximization because of the fundamental relevance of the choice act,
Nope- unless we are speaking of Quantum weirdness- Schrodinger's cat, Wigner's friend etc.
which has to be placed in a central position in analyzing maximizing behavior.
No. It is irrelevant provided the mathematical predictions are accurate and useful.
A person's preferences over comprehensive outcomes (including the choice process) have to be distinguished from the conditional preferences over culmination outcomes given the acts of choice.
Only if that is what the person actually wants and the person has a lot of power over what sort of menu of choice faces him. I'd prefer all the various TV companies to get together and just offer me a subscription to all the shows I might want to view on demand. I hate having to switch from Apple TV to Disney TV to Netflix and so on. On the other hand, many people prefer 'unbundling'. But because I'm poor and unimportant I can't change my menu of choice. No doubt, some entrepreneur is figuring out a way to cater to people like me- i.e. permit us to make just one big choice rather than lots of little ones.
The responsibility associated with choice can sway our ranking of the narrowly-defined outcomes (such as commodity vectors possessed), and choice functions and preference relations may be parametrically influenced by specific features of the act of choice (including the identity of the chooser, the menu over which choice is being made, and the relation of the particular act to behavioral social norms that constrain particular social actions).
Sure. So what? The bigger problem is 'Knightian Uncertainty' and the fact that for most of us it aint worthwhile to develop preferences and do ranking. We just imitate what smart peeps be doing.
All these call for substantial analytical attention in formulating the theory of choice behavior.
No. Such attention is a waste of time unless what you really want to do is air your sense of grievance about being brown rather than White or a stupid economist rather than a smart mathematician.
Also from a practical point of view, differences made by comprehensive analysis of outcomes can have very extensive relevance to problems of economic, political, and social behavior whenever the act of choice has significance.
Yes. From a practical point of view, since mimetic effects predominate once a tipping point is reached, we don't have to bother with a 'comprehensive analysis' which highlights male partiality to things which look like big knobs and female distaste for such things.
Illustrations can be found in problems of labor relations, industrial productivity, business ethics, voting behavior, environment sensitivity, and other fields.
But the solutions to those problems did not involve Sen-tentious shite. Nudge theory- maybe. But Tardean mimetics rules over all. Once enough smart peeps are doing something, the rest of us follow suit for fear of being thought dumb and ignorant. This is always the case provided Tardean mimetics gets reinforced by higher material wellbeing.
The characterization of maximizing behavior as optimization,
holds if a mathematical function can be identified which captures that behavior.
common in much of economic analysis, can run into serious problems in these cases,
only if- as with Sen-tentious shite- no such mathematical function can exist
since no best alternative may have been identified for choice. In fact, however, optimization is quite unnecessary for "maximization," which only requires choosing an alternative that is not judged to be worse than any other.
In which case there may be no unique preorder associated with any given partial order. This means there is no warrant that 'maximization' occurs and what is chosen is not worse than any alternative. This is because cylicity may obtain.
This not only corresponds to the commonsense understanding of maximization (viz. not rejecting an alternative that would be better to have than the one chosen),
No. The commonsense understanding of maximization is 'having the most you could possibly have'. This may not involve any accepting or rejecting. It is purely an objective question of whether there is a mathematical function which captures what is relevant and whether it really is the case that a maximum has been achieved.
it is also how "maximality" is formally defined in the foundational set-theoretic literature (see, for example, Bourbaki (1939, 1968), Debreu (1959, Chapter 1)).
But this begs the question. It assumes the axiom of choice or Zorn's lemma in which case a maximal element must exist. But this is circular. Thus the definition does not matter in the slightest. It is what it is because it is what it is. Nothing further is entailed.
Consider the following-
An example may help to illustrate the role of "comprehensive" description of choice processes and outcomes, in particular the "chooser dependence" of preference. You arrive at a garden party, and can readily identify the most comfortable chair. You would be delighted if an imperious host were to assign you to that chair. However, if the matter is left to your own choice, you may refuse to rush to it. You select a "less preferred" chair. Are you still a maximizer?
You are a cretin. The thing to do is to take the good chair and then offer it up to a pretty lady with a great chivalric flourish. Sen's stupidity prevents him from seeing that his description of the choice situation wasn't 'comprehensive' at all.
Quite possibly you are, since your preference ranking for choice behavior may well be defined over "comprehensive outcomes," including choice processes (in particular, who does the choosing) as well as the outcomes at culmination (the distribution of chairs).
But taking the best chair and then disposing of it to whom you prefer is the best outcome.
To take another example, you may prefer mangoes to apples, but refuse to pick the last mango from a fruit basket, and yet be very pleased if someone else were to "force" that last mango on you.
Take the mango and force it on whoever it best suits your interest to favor. Sen is merely displaying his own blinkered stupidity and narcissism. It never occurs to him that the best chair should go to the most deserving and you can ensure this outcome or that the mango should go to some humble person who will truly relish it.
In these choices, there is no tension at all with the general approach of maximizing behavior, but to accommodate preferences of this kind, the choice act has to be internalized within the system.
No. The choice menu should be expanded till even a cretin like Sen gets to see that having command over scarce resources means you can, if you want, do the right thing even if nobody else is inclined to do so.
This can require reformulation of behavioral axioms for "rational choice" used in economic and political theory.
What is relevant for 'rational choice' is the best and smartest choice. Axioms can't help you locate it.
The influence of the choice act on preferences, and in particular the dependence of preference on the identity of the chooser,
which makes no difference if the guy is smart. Gain command over a scarce resource and then use it as you think fit.
can go with rather different motivations and may have several alternatiue explanations. The comprehensive descriptions may be relevant in quite different ways and for quite distinct reasons. (i) Reputation and indirect effects: The person may expect to profit in the future from having the reputation of being a generally considerate person, and not a vigilant "chair-grabber."
I think Sen's account shows his own deep insecurity. He is a small man- a brown one in a white world. He is probably terrified of being rudely hauled out of a comfy chair by some hulking rugby player. But agents of this description would be better advised not to attend social events where such outcomes are likely. That would be the 'regret minimizing' course.
Sen next muddies the waters of what 'Muth rationality' had clarified. Rationality means acting in consonance with the correct theory.
the act of voting in an election may be very important for a person because of the significance of political participation. This has to be distinguished from whatever may be added by a person's vote to the likelihood of the preferred candidate's chances of winning (the addition could be negligible when the electorate is large).
But the size of the vote for a losing candidate sill has signalling value
It is possible that the voter may enjoy participation, or that she may act under some "deontic" obligation to participate whether or not she enjoys it. So long as she attaches importance to the participatory act of voting, the analysis of the rationality of voting must take note of that concern, whether that concern arises from anticipated enjoyment, or from a sense of duty (or of course, both). In either case, it can be argued that the well-known literature on "why do rational people vote" may have tended to neglect an important concern underlying voting behavior, viz. the choice act of voting. There may, in fact, be no puzzle whatever as to why people vote even when the likelihood of influencing the voting outcome is minuscule
The answer is Muth rationality. We follow the prediction of the correct theory if we are concerned with what that theory promotes- voting promotes Democratic Social Choice. If that's what you want, then you vote. If you don't really care, don't bother.
Sometimes the connection between preference and choice acts may be rather subtle and complex,
not if we are speaking of 'revealed preference' which is identical with what is chosen.
and turn on the exact nature of the actions involved. For example, in the context of work ethics, there may be a substantial difference between (i) actively choosing to "shirk" work obligations, and (ii) passively complying with a general atmosphere of work laxity.
But, in that case, the difference arises only because the two situations are completely different. It is one thing me for me to actively choose to assert a Tamil identity- e.g. by wearing veshti and speaking only Tamil- in London and a completely different thing to do so in Chennai. People might think me eccentric for my behavior in London. But, in Chennai, I'd be just like everybody else. I wouldn't be 'actively' doing anything at all. I'd just be passively complying with prevailing customs. True, I might say 'hey guys! look at me! I'm actively asserting my Tamil identity!' but everybody else would say, 'so what? You aren't a white man. You are a Tamil in a Tamil city. Get over yourself. There's nothing special about you at all.'
The latter may happen much more readily than the former, and the exact nature of the choice act can be very important in this difference.
This is utterly foolish. In the one case you are choosing to be a dick and this may get you fired. In the latter case your company may go bankrupt but you run no higher risk than anyone else of being rendered unemployed.
In fact, "herd behavior" not only has epistemic aspects of learning from others' choices (or being deluded by them, on which see Banerjee (1992)), but can also be linked with the possibility that joining a "herd" makes the choice act less assertive and perspicuous.
Where is the 'herd behavior' here? Either you are going against the prevailing work culture and shirking or else you are acting rationally by cheating your employer in the same way everybody else is doing.
The diminished use of forceful and aggressive volition in (ii) may make it much harder to resist than (i).
This is crazy shit. (i) may get your fired. (ii) won't. It is not the quality of your volition which matters, it is the employer's response.
Such differences may be of great importance in practice,
because Chennai really is different from London. But this has nothing to do with 'volition' or 'choice situations'
even though they may be difficult to formalize completely. Some types of influences of choice acts are more easy to formalize than others, and these include: (i) chooser dependence,
A choice depends on the chooser making that choice. What an amazing discovery! What's next? Will Sen discover that the existence of a fart depends on there being a farter?
and (ii) menu dependence.
Because when you order an aircraft carrier at Olive Garden the waiter doesn't say 'sorry, it's not on the menu'. He brings you an aircraft carrier. Choices don't depend on Menus at all. On the other hand, Sen is such a genius that he has found a 'subtle and complex' manner in which 'menu dependence' might lead to the waiter bringing you a bigger aircraft carrier than your rectum can easily accommodate.
The fact is menu means 'all available options'. Sen however isn't looking at menus at all. He is falsely claiming that some restricted set of his own invention was the actual menu facing his hypothetical beings.
Consider the preference relation Pi of person i as being conditional on the chooser j
We can't do so. If Smith's preferences are conditional on Jones's preferences then no mathematical 'preference relation' can be attributed to Smith. Why? There is no way of constructing an ordering- weak or otherwise- specific to Smith. We may prefer to just impute a preference relation to him. But that is our preference, not his.
Consider chooser dependence first-already introduced in the motivational discussion. To return to one of the earlier examples, in choosing between alternative allocations of fruits from a set S = (ml, al, a2} of one mango and two apples for two persons i and k, person i who prefers mangoes may like the allocation ml that gives the mango to him (and an apple to k), over the allocation a' whereby i gets an apple (say, a'), so long as the choice is made by someone else
Sen has incorrectly specified the set S. It has many more elements including one's where one guy hands the other guy a mango and the other guy hands it back saying 'I lurve u' and then the other guy goes 'I lurve you even more' and then one lovingly inserts an apple in the other's rectum and so on and so forth.
Once the set is fully specified you can have proper Preference Relations and weak orderings and so forth provided no element has an 'intensional' property the others lack. But, at that point, 'chooser dependence' disappears. Why? Every element has exactly the same significance for every agent.
Along with this chooser dependence, there is a related feature of menu dependence, particularly in the case of self-choice. If the set of available options is expanded from S to T containing two mangoes and two apples, person i himself may have no difficulty in choosing a mango, since that still leaves the next person with a choice over the two fruits. On the other hand, menu-dependence of preference is precisely what is ruled out by such assumptions as the weak axiom of revealed preference (WARP) proposed by Paul Samuelson (1938), not to mention Houthakker's (1950) strong axiom of revealed preference (SARP).
Sen says in a footnote 'WARP demands that if an alternative x is picked from some set S, and y (contained in S) is rejected, then y must not be chosen and x rejected, from some other set T to which they both belong. Property a demands that if some x is chosen from a set T and is contained in a subset S of T, then x must be chosen from the subset S. Property T demands that if x is chosen from each of a class of sets, then it must be chosen from their union.
The problem here is that Sen mis-specified the set S which actually featured states of the world where the other guy offers you the mango while tenderly pushing an apple up your bum.
Effectively, if more fruit are available, there is no scarcity. Choice does not matter. There is no opportunity cost. There is just behavior. Guys who like mangos eat mangos.
Indeed, even weaker conditions than WARP, such as Properties a and r (basic contraction and expansion consistency), which are necessary and sufficient for binariness of choice functions over finite sets (see Sen (1971)), much used in general choice theory as well as social choice theory, are violated by such choices.
Only if the set S is wrongly specified. Sen is comparing a situation where there is scarcity to one where there is no scarcity and thus no opportunity cost to choice and thus nothing of interest to economic theory. It is certainly the case that we give up our seat to a pregnant lady in a crowded train. But we wouldn't do so in an empty train. It would look decidedly odd.
How are these basic conditions of intermenu consistency violated by the concerns we are examining?
They aren't. In one case the menu featured scarcity and opportunity cost. In the other it didn't.
Consider the same example again. While an apple a' is what person i may pick if he is choosing from S (as given by (3.2)), he may sensibly go for one of the mangoes (say, min) from the enlarged set T = {ml, nm2, al, a2}: (3.3) mlpij al . The combination of (3.2) and (3.3) violates Property a as well as WARP and SARP,
no it does not because the initial set S had an element such that one guy misses out on mango. The second set S did not have this element by Sen's own explicit stipulation. The fellow is a fucking cretin.
and it can be easily shown with further examples that this type of menu dependence can lead to the violation of the other standard consistency conditions.
It can show that a guy on an empty train who does not offer his seat to a pregnant lady who gets on is a bounder and a cad. It can also show that if I speak Tamil and wear veshti when I am in Chennai then I'm probably some sort of extreme Tamil chauvinist rather than simply a Tamil dude who, quite naturally, dresses and speaks like other Tamils in the capital city of Tamil Nadu.
Menu dependence-when true-may be quite a momentous characteristic of choice functions.
No. It is stupid shit Sen pulled out of his arse. In a footnote he writes
My experience in presenting this paper in seminars has alerted me to the possibility that some readers will seek explanation of the alleged "inconsistency" in the influence of "framing" (in line with Kahneman and Tversky's (1984) important findings). But these two problems are quite distinct. The influence of "framing" arises when essentially the same decision is presented in different ways,
which is why that is an interesting result. Sen's result is not interesting because it compares a situation where there is scarcity with one where there is no opportunity cost and thus nothing of interest for economic theory.
whereas what we are considering here is a real variation of the decision problem, when a change of the menu from which a choice is to be made makes a material difference.
because scarcity disappeared.
There is, in fact, no inconsistency here, only menu dependence of preference rankings
No. There is no 'menu dependence'. There is scarcity which in turn means opportunity cost which in turn means that there is some point to pressing the other guy to have the mango and then him pressing it on you and then you both get smoochy and insert apples in each other's bums.
The above discussion concentrates only on one kind of reason for menu dependent preferences (related to the direct relevance of the choice act), but there can be other reasons for such dependence (on this see Sen (1993)). One connection may come from the value we place on our autonomy and freedom of decisions. We may value not merely the alternative we eventually choose, but also the set over which we can exercise choice. In valuing the "autonomy" of a person, it is not adequate to be concerned only with whether she receives what she would choose if she had the opportunity to choose; it is also important that she actually gets to choose herself.
It may also be important that we don't get to gas on about the importance of her getting to choose whether to take it up the bum or to buy contraception. She may resent this greatly and exercise her autonomy by kicking us in the nuts.
The fact is we may be prepared to pay money for an 'option' to choose though we may not in fact exercise that option. That is a fit matter for economic theory. Indeed, there's a lot of money in options and so forth.
Also, when our knowledge is limited, the menu may have epistemic importance, and we may "learn" what is going on from the menu we face.
Morishima made that point. He noticed that his colleagues, when dining in a Japanese restaurant took 'price as an indicator of quality'. This is about information asymmetry. We may also mention the Monty Hall problem. Additional information should be incorporated in our decisions. This does not mean our preferences have changed or are inconsistent.
For example, if invited to tea (t) by an acquaintance you might accept the invitation rather than going home (0), that is, pick t from the choice over {t, O}, and yet turn the invitation down if the acquaintance, whom you do not know very well, offers you a wider menu of either having tea with him, or some heroin and cocaine (h), that is, you may pick 0, rejecting t, from the larger set {t, h, 0}.
Most people would do the reverse. However, this is not a case of 'menu dependency'. It is a case of your having received information that a particular person is engaged in illegal behavior.
The expansion of the menu offered by this acquaintance may tell you something about the kind of person he is, and this could affect your decision even to have tea with him (see Sen (1993)).
Why? The answer is obvious. If the guy has heroin and cocaine, he might also have guns or be visited by violent criminals who have guns. Furthermore, he might shoot your pecker off while high on crack.
Sen is falsely claiming that an invitation to tea is a menu. It isn't. Sen may have an expectation of what would be involved but if it turned out sodomy was on his host's mind, he can't say that the menu offered to him had made no mention of this possible outcome. Why? There was no fucking menu.
A different type of example of epistemic use of menus can be found in using one's own menu to judge the opportunities that others would have to undertake similar behavior.
We shall soon see that there is neither a menu, or anything 'epistemic' as opposed to bigoted here.
In explaining "corrupt" behavior in business and politics in Italy, a frequent excuse given has been: "I was not alone in doing it."
And thus did not expect to be punished or to suffer reputational damage.
A person may resist seizing a unique opportunity of breaking an implicit moral code, and yet be willing enough to break that code if there are many such opportunities, on the indirect reasoning that the departures may be expected to become more "usual."
This is not the case. A unique opportunity is one where detection is less likely. Furthermore, a 'first time offender' is likely to get off lightly more particularly if there is little chance of reoffending. If there are many such opportunities and you transgress, risk of detection and punishment is much higher.
Similarly, a unique opportunity of "crossing the picket line" may be rejected
because you'll get your fucking head kicked in
by someone, who may nevertheless not hesitate to do that crossing if he expects others to do the same.
in which case there's a bunch of you who can beat the fuckers who come to kick your heads in.
If there is only one opportunity xl of crossing a picket line,
why bother? You'd only get one day's pay. Only if the opportunity is not unique- it is ongoing- would it be worth risking getting your head kicked in more particularly because there may be a lot of people in the same boat as yourself and you can stand together against the Teamsters.
a person may refrain from grabbing that (knowing that she would be alone in this), and yet she may choose that very opportunity x1 if there are other opportunities x2, etc. (expecting others to take them).
How is this 'epistemic'? What does it have to do with menus? Sen won't explain. The fact is a one-off offer of day labor is not a menu. You have to work out whether there is a risk your head will be kicked in. Research is involved. No extra knowledge is communicated by the offer.
Another type of epistemic relevance of the menu is illustrated by a "moderate" voter who tends to choose a middle-of-the-way candidate among the ones offered for choice, for example, some "median" alternative according to some politically perspicuous ranking (such as "relative conservatism").
There is no 'epistemic relevance' here whatsoever. The moderate guy does a bit of research and hits on a moderate guy to vote for. He doesn't care whether there are lots of crazy people on the ballot paper. The menu is irrelevant. It aint the case that the British voter decided not to vote for John Major just because they saw that Screaming Lord Sutch was a rival candidate. Maybe Sen is thinking of tactical voting- e.g guys in a Tory shire voting for the Monster Raving Loony party to protest John Major's Europhile views.
The range of options offered in the menu may give the person a "reading" of the real policy options in the country at that time,
Sen was living in England at that time. Was his vote (Indian citizens can vote in British elections) really influenced by the fact that the Maharishi's 'Natural Law Party' (which promoted world peace through yogic levitation) was represented on the ballot paper?
and the menu-dependent choice of a "moderate" candidate may, thus, reflect that epistemic reading.
Sheer nonsense!
It may, again, be tempting to think that the violation of the standard "consistency conditions" (such as WARP) can be eliminated by some suitable redefinition, for example by defining an alternative in terms of choosing a fruit from a set. The alternative mn/S, taking ml from set S, can be seen as a different alternative from m'/T, taking ml from set T. But that would make all inter-menu conditions, such as WARP, SARP, a, r, etc., vacuous, since these have cutting power only when "the same" alternative can be picked from two different sets-precisely what is ruled out by this recharacterization.
But WARP is defined in terms of a budget set. If this varies- as it does when mangoes stop being scarce- then something previously infeasible ceases to be so. But WARP only applies where there is scarcity. Thus it has no application here. We can't say preferences are inconsistent or menu-dependent because where there is no opportunity cost, the budget set constrains nothing. The theory has no purchase.
Similarly, if we try to apply conditions like WARP, SARP, a, etc., to alternatives defined as complete allocations of commodities for everyone in the community, these conditions have severely reduced discriminating power, because of the tendency of each option to become a unique alternative.
Provided there is scarcity. If price falls to zero or there is no opportunity cost- as happens with free goods- then no alternative presented by the theory is unique. Any feasible allocation can be associated with an infinite number of allocations of free goods. Thus though everyone's income is limited, they are welcome to breathe in and out more frequently- if that is what they want to do. They can also spend more time looking at the moon rather than at the neighbor's cat.
Much would depend on the exact circumstances.
Economic theory can only be usefully applied where there is scarcity- i.e. opportunity cost obtains. Sen has made a career of writing vacuous, virtue signalling shite, by ignoring the exact circumstances where economic analysis has purchase.
Not surprisingly, Samuelson (1938) and others employed their choice consistency conditions, in general, by defining an "alternative" for the choice of a person to be his or her own commodity basket (independently of the overall menu and of the allocations to everyone else in the community).
if there are no externalities, no free goods, and open markets prevail, then endowments generate a budget set and what is chosen is 'revealed preference' provided it meets a consistency requirement. Otherwise it might be error or 'discovery' or drunken behavior or whatever.
It is in this form that these conditions have been used, with much force and profit, both in consumer theory and also to obtain results in general equilibrium theory (see, for example, Samuelson (1947), Debreu (1959), Arrow and Hahn (1971)).
Shite taught to undergrads which they quickly forget because Knightian uncertainty obtains in the real world.
The kinds of influences considered here suggest the need for limiting the domain of applicability of such conditions.
But those influences are artefacts of Sen's own stupidity and failure to properly specify set S.
But we should also consider a different type of argument which says that while menu dependence may occur and may be important for some problems (such as "social choice" judgments), an individual chooser need not really worry about it, since it is not relevant to her decisions.
Individual choosers needn't worry about Economic theory. It is stupid shit.
It could be argued that menu dependence cannot affect the form of maximizing behavior for an individual, since the individual does not get to choose the menu from which she can select an alternative.
There is no fucking menu. It is not possible to list out all possible states of the world- indeed they are unknowable. That's what Knightian Uncertainty means.
Menu dependence, in this view, may be true, but irrelevant for the individual's choice problem, since the person always faces a choice over a given menu, rather than having to choose between menus.
Quite false. People choose between such alternatives as they have knowledge of or regarding which they have some information.
This line of argument is faulty for two distinct reasons. First, we do have occasion to make choices that affect our own future choices (or future menus),
but we don't know when those occasions arise or what the outcome will necessarily be. This is even true of choice of academic specialization. Sen may have thought studying Econ would help India overcome poverty. But, because he was as stupid as shit, he helped India go the other way.
and indeed the literature on "preference for flexibility" (see Koopmans (1964) and Kreps (1979)) has extensively considered just such choices.
but that literature is shit because it neglects Knightian uncertainty.
We do not live in a world of a "one-shot choice."
Though we may die as a result of such choices.
Kreps (1979, 1988) has presented illuminating analyses of preference for flexibility in choosing between future menus.
But Kreps had the good grace to admit he had been talking krap about TOTREP. Still, his text-books for kids who want to go into Fintech are useful because his exposition is clear. However, a regret minimizing approach would be better for general purposes.
Such concerns may be important in strategic choice in many games as well, and an example of this will be considered presently (Section 4).
It is regret minimizing to pay a little ahead of time to have the option to shifting to a different game.
Second, the issue is not just whether the chooser herself has to "do something" about menu dependence,
by which Sen means whatever stupid shit he thinks is the 'menu' as opposed to the true set of feasible options.
but whether in the study of choice behavior the possibility of menu dependence has to be included.
If does have to be included if Sen is doing the studying because he is too stupid to correctly identify the actual menu.
It is the behavioral scientist who has to consider how a person's choices vary with alterations of the menu,
No. It is obvious that choice changes as options change. This is as true of cats as cabbages or Kings.
and in particular whether a canonical binary relation of preference can be used to predict choices of that person over different menus.
Provided the menus are sufficiently comprehensive and well specified such a relationship must exist so long as preferences don't change- i.e. are consistent.
The point is that even when the option set (or the menu) S is given, the nature of the menu can influence the ranking of the alternatives x in S,
No. The menu has no nature. It is a pure abstraction. On the other hand, a shitty menu pressed on you by a stupid Sen-tentious cunt might cause you to rapidly exit the restaurant.
and this relationship is of immediate relevance in understanding and predicting choice behavior.
Except it isn't at all. Mimetic effects predominate. Preferences aint consistent coz Tardean effects fluctuate as some rise while others fall.
Menu-independence as a formal characteristic of preference can be defined in terms of Rs in the following way.
Sen means, if we assume axiom of choice or Zorn's lemma, then we can define a mathematical function and pretend it represents 'preference' though it doesn't really.
Menu-independent preference: There exists a binary relation Rx defined over the universal set X
which is unknown and unknowable save perhaps at the end of time
such that for all S cX, Rs is exactly the "restriction" of Rx over that S: (3.4) Rs = RxIs.
But that 'restriction' from something unknowable only exists by express stipulation. We can't be sure it really is a function. The fact that some such function exists doesn't mean anything we stipulate is necessarily a restriction on that particular function.
The condition of menu-independence is a standard assumption-typically made implicitly-in mainstream utility theory and choice theory. In Bourbaki's language, Rs is simply "induced by" an overall ordering RX, and Rx is an "extension" of Rs on X (Bourbaki (1968, p. 136)).
So, some guys say 'assume we aint talking crap' and Sen is saying 'aha, but what if we assume the crap I'm talking raises serious ethical questions about the crap you are talking?'
This relationship is implicitly presupposed when a utility function U(x) is defined just over the culmination outcome x, as is the standard practice (see, for example, Hicks (1939), Samuelson (1947), Debreu (1959), Arrow and Hahn (1971), Becker (1976)).
In other words, you get pleasure from the cake you ate, not the rasmalai you did not eat.
In what follows, I shall consider choice functions based on optimization, that is, choosing an element from the optimal set B(S, R) (that is, choosing a "best" element) from each menu set S, according to a weak preference relation R (interpreted as "preferred or indifferent to"), which ranks the set of available alternatives X of which each "menu" S is a nonempty subset.
There is a configuration space corresponding to choice from a menu specifying all states of the world under the contingency of it itself being chosen. An example is the menu at the Chinese restaurant you decide to enter which tells you all the things its kitchen can cook for you. There is no configuration space for the menu of the best menus of all the restaurants where you could have decided to dine. This is because you can't order pizza in a Chinese restaurant. True, you could say, I'd have preferred to have a slice of pizza with my peking duck rather than the pancakes they have provided me. But that isn't a preference recognized by Revealed Preference or any other Economic theory. It is just a yearning of a vague sort.
(3.5) B(S, R)=[xlx E S & for all y E S: xRy]. While (3.4) defines menu-independent preference, taking preference to be the primitive, there is an analytically different problem of characterizing a menu-independent choice function.
No. Preferences for economic theory were made identical to choices. What you chose was your revealed preference provided you choose consistently.
For this it is convenient to define the "revealed preference" relation Rs of a choice function C(S) over a given menu S. Although the revealed preference relation Rc is standardly defined without restricting the observation of choice to one particular set S only (see, for example, Samuelson (1938), Arrow (1959)),
In which case it is entirely meaningless. If the domain is unknowable, there is no provable mathematical relation.
it is of course possible to consider the revealed preference Rs for a specific menu S.
Either this menu S is properly specified in which case the maths works out fine, or it was shittily specified in which case Sen can talk stupid shit while posing as some sort of philosopher.
Menu-specific revealed preference: For any x, y in X, and any S cX, (3.6) xRsy *[x E C(S) &y E S]. Obviously, there would tend to be much incompleteness in the relation Rs for any given S, since any two unchosen alternatives in S would not be ranked vis-ai-vis each other; we must take note of this elementary fact in using Rs.
The Szpilrajn extension theorem or Suzumura consistency or some such thing gets rid of the problem.
Menu independence of choice can now be defined in terms of there being a canonical, menu-independent R0, not varying over option sets, in terms of which we can explain the choices over every menu.
Nope. We don't know if the thing is computable. If it isn't we can't explain shit. There may be a mathematical function which describes everything that has and will happen in this universe. But it doesn't explain why shitty things keep happening to us because we don't know it and have no means of finding out very much about it.
Menu-independent choice function: There exists a binary relation Ro over X such that for all S cX: (3.7.1) for all x,y in S: xRsy entails xRoy; (3.7.2) C(S) = B(S, R0).
And there exists a binary relation S, such that for all Sen-tentious shitheads there is a dog whose existence entails their eating dog turds. True, I am assuming there is a mathematical function which describes how, for complex Quantum Mechanical reasons involving tachyonic particles , the correct chanting of a Latin prayer will cause everything to turn into a dog shitting into the mouth of hungry Nobel Prize winners named Amartya Sen.
How does menu-independence of choice relate to menu-independence of preference?
Through all the dog turds Sen is swallowing.
If preference is defined simply as "revealed preference," there is obviously no gap between the two, given the constructive form of (3.7.1) and (3.7.2). But this is trivial, since "revealed preference" is only a reflection of choice itself, and gives no real role to conscious use of preference.
But pretending this isn't so aint trivial, it is stupid.
To consider a nontrivial problem, consider a person who makes conscious optimizing decisions on the basis of a potentially menu-dependent preference RS, and the choice function that results from it is given by C(S) = B(S, RS).
In that case those decisions aren't 'optimizing' anything. It is easy to have a rule like 'always chose the second item on the menu'. But that is arbitrary. There is no 'canonical' binary relationship here because there is no adjointness, not 'naturality', nothing truly mathematical of any sort.
The menu-independence of Rs and that of C(S) would not then necessarily coincide. However, the following relation will hold, denoting Rx as in (3.4) and Ro as in (3.7.1) and (3.7.2), when those respective conditions are satisfied. We take both Rx and Ro to be complete, acyclic, and reflexive rankings (CARR for short).
So, by happenstance, it is possible that always choosing the second item might be 'optimal' or else the thing may be prearranged by an Occassionalist God.
THEOREM 3.1: Menu-independence of preference entails menu-independence of the generated choice function, but menu-independence of a choice function need not entail menu-independence of the preference that generated this choice function.
PROOF: Suppose preference is menu-independent with an RX which "induces" (in Bourbaki's sense) Rs for each S.
In which case each S has a metric and topological properties of a certain type. For this to happen it must make sense to say 'I'd prefer a slice of pizza with my Peking Duck'. Sen is relying on the fact that in ordinary English we do sometimes say things like that though, for Economics, 'preference' is a term of art and thus this formulation is incorrect or meaningless. However, obviously, since we couldn't actually choose pizza at the Chinese restaurant, the choice function is independent of the menu in some other restaurant. This is what Sen is getting at next
It follows immediately that for Ro = RX, (3.7.1) and (3.7.2) will be satisfied, so that the choice function is menu-independent.
This is not the case. We may leave the Chinese restaurant and go to the Italian place. This is the problem with having an unknown and unknowable binary relation. We may think it entails something we find useful but what is unknown entails nothing we can know. Of course, after the fact, we may say 'well that unknown relation did hold up after all.' But that too is pure speculation.
To check why the converse does not work, suppose with a menu-independent choice function, we get a binary relation Ro
which is unknown
that would be menu-independent had it been a preference relation.
we can't know that. We may make such a stipulation but it has no epistemic content.
But it is possible for a menu-dependent reflective preference relation Rs to generate exactly the same choice function as a menu-independent binary relation Ro.
We can't know that unless we can actually study the properties of that binary relation.
A simple example establishes this. Consider a definitely menu-dependent reflective preference relation Rs defined over T = {x, y, z} and its subsets, such that: xI(X Yly; yP(Y z)z; zp(x,zzx; ypTx; ypTz.
Defined by whom? Sen himself. But why should we believe he has correctly specified the actual 'menu' or possible 'states of the world'? He may be comparing a situation where there is scarcity and opportunity cost to one where good are 'free'.
Maximization according to this reflective preference relation will yield the following choices: C({x, y}) = {x, y}; C({y, z}) = {y}; C({x, z}) = {z}; C(T) = {y}. This is a menu-independent choice function, and will correspond to the complete, acyclic and reflexive relation R,, given by: xI,y; yP0z; zPOx.
But only if we take Sen's word for it. But why should we? He was wrong about mangoes.
Indeed, this Ro is Samuelson's "revealed preference" relation for this choice function (even though Samuelson's "weak axiom of revealed preference" is violated).
No it isn't. Samuelson was only talking about situations were scarcity exists and preferences are consistent.
But menu-independent R0 is not congruent with menu-dependent Rs (even though the two generate the same choice function).
Because the menu is mis-specified by a fucking cretin.
So a menu-independent choice function C(S, Rs) may be generated by a menu-dependent preference relation RS.57 THEOREM 3.2: A choice function is binary if and only if it is menu-independent.
In which case there is some Preference Relation, on a properly specified set, such that Sen's misology disappears.
It can also be seen that menu-independence of the choice function is not really different from the binariness of the choice function.
But if the choice function has an unknown, unknowable domain, then there is no binary relation because there is no set with knowable properties. The fact is, suppose this magical choice function exists might not knowledge of it endow us with the property to levitate? We'd know exactly how to choose our footing so that like, kung fu action heroes, our own chi can provide us with sufficiently dense air-pockets to enable us to run through the air.
Binariness of a choice function is the condition that guarantees
nothing at all if the relevant sets are not knowable.
that for every set S that is chosen C(S) is exactly what would be chosen if the best elements of S were picked using the ranking given by the revealed preference relation RC of the choice function as a whole
Binariness sure is magical! If only we could know exactly what to do at every moment we could become billionaire Presidents of the World Federation who sleep only with Super Models. We'd also discover the secret to immortality and intergalactic faster-than-light travel. Wouldn't that be cool?! Sen was a fool to just use his wonderful discovery to score off his equally cretinous colleagues.
Sen thought the difference between 'maximization' and 'optimization' is that the former is achieved if there is no better alternative. This is not the case. It is foolish to speak of maximization unless there really is a unique maximum. Optimization is more interesting because there would always be a mathematical dual or adjoint functor and thus notions of naturality or canonicity etc gain purchase.
Sen writes-
The basic contrast between maximization and optimization arises from the
fact that the latter has a dual which is a minimization problem. Not the
possibility that the preference ranking R, may be incomplete, that is, there may be a pair of alternatives x and y such that x is not seen (at least, not yet seen) as being at least as good as y, and further, y is not seen (at least, not yet seen) as at least as good as x.
This is irrelevant because information may be available from the dual. I don't need a complete ranking of my preferences because when I get to the shop I see that the 'dual' of my problem (faced by sellers seeking to get my money in return for something useful to me) has restricted my choice to something quite easily managed.
However, there are subjects outside Econ which are interested in 'maximization'. Thus a sports scientist may want to know the best possible diet for a particular type of athlete whereas what the food industry is concerned with is something tasty, reasonably nutritious and attractively priced and packaged.
What possible 'epistemic' value does Sen think attaches to his stupid lucubration?
What are the lessons from all this? Theorem 5.1 tells us that while a best alternative must also be maximal, a maximal alternative need not be best.
only if you have mis-specified the Set of options.
Case 1 covers the case in which there are no best alternatives whatever, but a maximal choice can still be made. This is easily seen by considering the situation in which neither xRy nor yRx, so that B({x, y}, R) = 0, whereas M({x, y}, R) = {x, y}. A classic example of Case 1 is given by one interpretation of the story of Buridan's ass: the tale of the donkey that dithered so long in deciding which of the two haystacks x or y was better, that it died of starvation . There are two interpretations of the dilemma of Buridan's ass. The less interesting, but more common, interpretation is that the ass was indifferent between the two haystacks, and could not find any reason to choose one haystack over the other. But since there is no possibility of a loss from choosing either haystack in the case of indifference, there is no deep dilemma here either from the point of view of maximization or of that of optimization. The second-more interesting-interpretation is that the ass could not rank the two haystacks and had an incomplete preference over this pair. It did not, therefore, have any optimal alternative, but both x and y were maximal-neither known to be worse than any of the other alternatives. In fact, since each was also decidedly better for the donkey than its dying of starvation z, the case for a maximal choice is strong. Optimization being impossible here, I suppose we could "sell" the choice act of maximization with two slogans: (i) maximization can save your life, and (ii) only an ass will wait for optimization.
Sen is being foolish. Buridan's ass is stupid and has a short memory. It sees a pile of hay but glances around, out of caution, before advancing towards it. Then it sees the other pile of hay and the same thing happens. It dies of starvation because of its very limited memory and self-conscious reasoning power. Thankfully, evolution has solved the underlying problem. As an animal's hunger increases, it becomes reckless and stops glancing around and just rushes towards the food source.
Sen mis-specifies the menu facing the ass. It isn't 'this bale of hay, that bay of hay, or starve to death'. It is 'hay! but have to check for a predator' followed by 'omg, hay! but have to check for a predator' 'omg, hay! etc'.
Case 2 is more subtle. Consider a preference ranking that consists exactly of xIy and yIz, with no other pair in the set S = {x, y, z} being ranked (where I is the symmetric factor-indifference-of the weak preference relation R). Clearly, B(S, R) = {y), and M(S, R) = {x, y, z}. But the real force of Theorem 5.1 lies in showing that maximization may work even when optimization does not (Case 1), with the added lesson that sometimes maximization may permit a wider set of possible choices than optimization would (Case 2).
Sure. We know that sports scientists or guys figuring out what to feed astronauts, may come up with stuff we would find it beneficial to buy. But that depends on people doing actual science not bogus econ or bogus moral philosophy which au fond is just an exercise in magical thinking similar to that of the Maharishi. But that dude died a billionaire.
Perhaps I'm unfair to Sen. He was a brown dude doing the sort of stupid shite white dudes were doing to earn a little money. He was just conforming to the conventions of the shitty little milieu to which he'd managed to escape from India (along with his best friend's wife). This gives a certain pathos to his incessant invocation of Adam Smith
The practice of enjoining rules of conduct that go beyond the pursuit of specified goals has a long tradition. As Adam Smith (1790)
a moral philosopher who successfully moved into Econ- while Sen did the reverse
had noted, our behavioral choices often reflect "general rules" that "actions" of a particular sort "are to be avoided" (p. 159). To represent this formally, we can consider a different structure from choosing a maximal element, according to a comprehensive preference ranking (incorporating inter alia the importance of choice acts), from the given feasible set S (allowed by externally given constraints).
No we can't. Not if we wish to be faithful for Smith's empiricism and associationalism. His 'general rules' are just heuristics which assign higher expected utility to some options than to others.
Instead, the person may first restrict the choice options further by taking a "permissible" subset K(S), reflecting self-imposed constraints, and then seek the maximal elements M(K(S), R) in K(S). The "permissibility function" K identifies the permissible subset K(S) of each option set (or menu) S. How different an approach is the use of such a permissibility function in comparison with incorporating our concerns fully in the preference ranking itself?
It could be as different or as similar as you liked- depending on your degree of stupidity. You can replace Preferences with 'as if' Preferences or you could simply say that there is no such thing as an individual. That's just bourgeois false consciousness. Also dicks don't really exist. They were invented by evil Neo-Liberals. Sadly, Sen never made this great discovery. No doubt some of his woker acolytes are slowly coming to this realization.
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