Invariance
16th April 2009, 13:17
The Irrationality of Rationality.
I apologize for this being as long and as disorganized as it is – I began with several paragraphs, but the more I wrote, the more I needed to write! I have divided it up so that it is more readable.* Some of the maths symbols might have been altered by copying it from word. - Vinnie.
Introduction:
At the centre of modern micro-economic theory are its claims about rationality, and its depiction of the consumer as a utility optimizing, selfish creature (some economists such as Veblen have criticized this). I won’t bore you with the mathematical ‘proofs’ of these rationality and psychological assumptions (they are known as the ‘completeness’, ‘transitivity’, ‘non-satiation’ and ‘convexity’ assumptions). Essentially, the principle of rationality can be summed up in the statement: if aRc and bRc then aRc, which translated means ‘If our agent thinks that bundle a is at least as good as bundle b and that bundle b is at least as good as bundle c, then they also think that bundle a is at least as good as bundle c.’ This is all lovely, but isn’t really in dispute at this particular time (but will be when trying to aggregate demand from an individual and then it becomes very, very funny. But that’s a joke for another day). However, the rationality principle, as I will explain below, essentially assumes an undynamic, static and non-evolutionary view of consumer’s tastes which is up for criticism.
Optimization:
What I want to look at is the concept of optimizing – that consumers will choose, from a number of commodities, the bundle that best maximizes their utility (‘enjoyment’ in lay terms. Economists say that consumers can ‘ordinally measure’ the utility of a commodity in ‘utils’). The possible bundles are restricted by income – hence, consumers maximize their utility in what is known as the economically feasible consumption set (economists assume that consumers spend all their income, savings are a form of ‘delayed consumption.’)
Okay, here’s an example: let’s say there are two commodities – Marx’s Capital and a Che T-shirt (which any typical lefty is deemed to have). The possible combinations of these commodities (the bundles) are: 0 Capital, 0 Che-shirt; 1 Capital, 0 Che-shirt; 0 Capital, 1 Che-shirt and; 1 Capital, 1 Che-shirt. In other words, there are four possible bundles, of which one will maximize the consumer’s utility.
Okay what’s my point? My point is that let’s forget the guy on the corner who sells copies of Green Left and let’s go to a leftist supermarket. Suddenly, instead of being faced with two commodities we are faced with… a thousand – Lenin’s collected works, the writings of Trotsky, Mao’s little red books, and even the rarities of Enver Hoxha. Well, how many bundles do we have? Instead of writing each possible I’ll do the short-hand maths: the number of choices is equal to the number of units being considered + 1 to the power of how many goods. So, in our above model it was 2^2 = 4. If there was 3 goods, it would be 2^3 = 8 possible bundles.
1000 goods? Well, that would equal 2^1000. That seems like a small number – it isn’t. I’ll ‘spell’ it out for you:
10, 715, 086 , 071, 862, 673, 209, 484, 250, 490, 600, 018, 105, 614, 048, 117, 055, 336, 074, 437, 503, 883, 703, 510, 511, 249, 361, 224, 931, 983, 788, 156, 958, 581, 275, 946, 729, 175, 531, 468, 251, 871, 452, 856, 923, 140, 435 ,984 ,577 ,574 ,698 ,574, 803, 934, 567, 774, 824, 230, 985,421, 074, 605, 062, 371, 141, 877, 954, 182, 153, 046, 474, 983, 581, 941, 267, 398, 767, 559, 165, 543, 946, 077, 062, 914, 571, 196, 477, 686, 542, 167, 660, 429, 831, 652, 624, 386, 837, 205, 668, 069, 376
Yeah, it’s a big one.
How Big is Your Brain?
How big would your brain need to be to remember that many combinations? Well, let’s pretend that each neuron can remember the utility of 100,000,000,000 different bundles (I’m no brain scientist, but this sounds like a very optimistic assumption/myth!). The grey matter comprises about a kilo of your brain and contains 100,000,000,000 neurons, each weighing 1/100,000,000 grams. Following me?
So…how heavy would your brain be if it could remember that many combinations?
A) The weight of a normal human brain?
B) The weight of a small truck?
C) The weight of Earth?
D) The weight of the Universe?
E) Heavier?!
Well… it would weigh 10^244 times as much as the universe. Now people can justifiably call you big-headed, and vice-versa.
How Long Does It Take You to Shop?
Another question – how long would it take to optimize – i.e. recall the utility of each combination – supposing that we could recall the utility of a single bundle in 1/10,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000,000,000,000,0 00,000,000,000,000,000,000,000,000th of a second (this is rather fast!)?
It would take 10^200 seconds. In other terms, 10^180 times the age of the universe. Raise this next time someone hassles you about your slow shopping habits.
Let’s be Rational:
Okay, enough fun with numbers. My point is, that the idea that consumers consider the utility of each particular bundle is patently false – and absurd – I dare say its irrational! Consumers do consider their budgets/income when deciding tastes (no thanks homothetic preferences!).
Neoclassical economics (in order to avoid the problems with aggregating individual demand curves) declared the golden principle of homothetic tastes. To quote my mircroeconomics book, Microeconomics – a Modern Approach, Andrew Schotter, 3rd ed., 2001, page 68:
‘When a consumer has homothetic preferences, all goods are superior and purchased in the same proportion no matter what the consumer’s income. In a world where all consumers have homothetic preferences, we might think of rich people as simply expanded versions of poor people. The tastes of such rich people do not change as their incomes change. They allocate their incomes exactly the way they did when they were poor. They just buy proportionally more of each good as their income grows.’
I hope you found that as humorous as I did when I first read it. In effect, Schotter is saying that if Bill Gates spent 10% of his income on pizza when he just another university computer science nerd, then he now spends 10% of his total income on pizza. In other words, he spends hundreds of millions on buying pizza for his own consumption.
Alternatively, the poor man who happens to get rich (we all know how often that happens!) did not change his spending habits at all – he still lives in a cardboard box (just an extremely large one). Likewise, the rich man spends the same amount he did on healthcare when he was poor – none. Going from the opposite direction, how many pieces of art work would a poor person have if they spent the same proportion that a wealthy person spends on art? A thousandth of a Mona Lisa?
It means that your spending habits will not change from when you are 16 to when your 60.
Clearly such statements are absurd and contrary to all empiricism. Yet for Neoclassical economics to work it essentially assumes that (1) there is only one consumer in society – or a endless number of drones and (2) there is only one commodity in society (I’ll cover that in another thread). This comes from the paradigm which asserts the uniqueness of the individual! The great theory that the market maximizes social welfare depends on these assumptions. It falls with them also.
To get back on point, we rely on habit, convention, culture, even flat-out ignoring / being ignorant of potential commodities – and a variety of things which may very well inhibit our ability to make utility optimizing decisions! When faced with such vast options we may use segmented optimizing – e.g. 100 possible combinations of food with a buy/not buy approach. Sure, there would still be 10^31 possible combinations, and your brain would still weigh about a million tonnes, but we’re getting there!
Optimisation: a good example of neo-classical economics using mathematics which are aesthetically pleasing but empirically laughable – time and what is physically/mental possible do matter.
Further Examination of Rationality Assumptions:
The Weak Axiom of Revealed Preference (WARP) stated that if a consumer chooses bundle A once when bundle B is also affordable, then the consumer will always choose A instead of B, regardless of relative prices. As Varian puts it: ‘If a bundle (x1, x2) is purchased at prices (p1, p2) and a different bundle (y1, y2) is purchased at prices (q1, q2) then if p1 x1 + p2 x2 ≥ p1 y1 + p2 y2, then it must NOT be the case that q1 y1 + q2 y2 ≥ q1 x1 + q2 x2. If the y-bundle is affordable when the x-bundle is purchased, the x-bundle must not be affordable.’
The Strong Axiom of Revealed Preference (SARP) was what I stated in the introduction– If (x1, x2 ) is revealed preferred to (y1, y2) and (y1, y2) is different from (x1,x2), then (y1, y2) cannot be directly or indirectly revealed preferred to (x1, x2). There is also a third axiom, known as the Generalised Axiom of Revealed Preference, which I won’t get into here, but is the weakest of the three.
How have these rationality assumptions held up to testing?
A test undertaken by Sippel attempted to replicate neoclassical standards. Ten sets of budgets and relative prices were presented. Consumers were to chose from 8 goods at each budget/price ratio, where the computer calculated the budget costs. The consumers were given a period of one hour to ‘optimize.’
The results weren’t good (for neoclassical economics).
The first experient contained 12 subjects, of which 11 of 12 subjects violated the SARP and WARP assumptions. 5 violated the weaker test GARP. The second experiment contained 30 subjects, of which 22 of the 30 subjects violated SARP and WARP, 19 of 30 violated the weaker GARP. The experiments conclusion: “We conclude that the evidence for the utility maximisation hypothesis is at best mixed. While there are subjects who do appear to be optimising, the majority of them do not… we … call the universality of the maximising principle into question.”
Other experiments have shown that consumers rely more on habit to reduce the vast array of potential choices – and hence do not optimise – or do not act ‘rational’ as the economist puts it. But it is not the consumer whom is acting irrationally, but the economists.
Undynamic Nature of Rationality:
When we take time into account, two of the four economic concepts of rationality (these four being, completeness, transitivity, non-satiation and convexity) fall into question – non-satiation and transitivity.
The principle ‘more is always preferred to less’ disregards time – two bannanas may be preffered to one, and three to two, but twenty bannanas to be consumed is not an ideal situation to be in . When we consider consumption as a function of time, anyone who behaves in what economists call ‘rational’ – preferring more to less – is a nutjob.
Likewise, a consumer may prefer bundle A to bundle B at one time, and yet at another time she may prefer bundle B to bundle A; consumer’s tastes cahnge. As Steve Keen says:
“Economists might protest that these comparisons are only valid when they are all made at the same instant, but that is precisely the problem: economic theory requires that everything happens at once, when in the real world an individual’s consumption is sequential, not simultaneous.”
Completness requires an impossible level of computation by the individual consumer. And lastly, as Keen points out:
“Convexity is also a by-product of the timeless analysis of consumption, which permits economists to ignore the manner in which the consumption of one commodity interacts with the consumption of another. Convexity implies that, to keep a consumer’s utility constant, it is necessary to subtract units of one commodity while adding units of another. But if the commodities are “sugar” and “tea”, and we are considering the act of drinking tea, then it is not true that a reduction in the amount of sugar can be compensated for by an increase in the amount of tea.”
The Scientific Status of Neoclassical Economics:
Schotter makes the claim:
‘If economic theories are correct, they should be verifiable or falsifiable in the lab. In other words, if the behaviour predicted by economic theory cannot be replicated in appropriately designed experimental setting, where human subjects make decisions under a set of monetary incentives that approximate those described by the theory, then the theory must be considered severely lacking.’
What a fantastic indictment of Popper’s falsifiability and unintended endorsement for Kuhn’s view of normal science as fundamentally dogmatic. Neoclassical economics has been replicated ‘in the lab.’ People were overwhelmed trying to optimize with just 8 commodities, taking over half an hour to reach a decision.
Unsuprisingly Schotter claims that
“Normative or welfare economics deals with what ought to be rather than what is and involves prescriptive statements that may be based on value judgement. Positive economics deals with what is rather than what ought to be and involves descriptive statements that are objective and verifiable.”
Marx once said that “no school of thought has thrown around the word ‘science’ more haphazardly then that of Proudhon, for ‘where thoughts are absent, words are brought in as convenient replacements.” I think Neoclassical economics is certainly a contender for that position – claiming to be a positive, non-biased, objective and verifiable discipline, all the while being naught but the most dogmatic religion.
(I would address alternative models for analyzing economic behaviour, but it is more appropriate elsewhere. Needless to say, neo-classical economics was a distinct departure/dissent from the classical (e.g Smith, Ricardo, Malthus) view of analyzing society from the viewpoint of classes – which Marx expropriated and turned it against its creators. Any model which takes into account classes – capitalists, workers, peasants, landlords - or even uses appropriate income groups not altogether removed from classes – is far likely to do a better job at analyzing aggregate market behaviour, rather than starting from the isolated individual. Marxists, post-Keynesians/neo-Ricardians, feminists, evolutionary economists take this approach.)
* I make no claim at being credited for pointing out the absurdity of this – the mathematics/humour is entirely borrowed from economist Steve Keen. Other experiments, which replicated neo-classical paradigm in all its splendour , showed that a consumer was overwhelmed in trying to optimize with just 8 commodities. I’m sure many of us can sympathize with that. Google the ‘curse of dimensionality’ for why.
I apologize for this being as long and as disorganized as it is – I began with several paragraphs, but the more I wrote, the more I needed to write! I have divided it up so that it is more readable.* Some of the maths symbols might have been altered by copying it from word. - Vinnie.
Introduction:
At the centre of modern micro-economic theory are its claims about rationality, and its depiction of the consumer as a utility optimizing, selfish creature (some economists such as Veblen have criticized this). I won’t bore you with the mathematical ‘proofs’ of these rationality and psychological assumptions (they are known as the ‘completeness’, ‘transitivity’, ‘non-satiation’ and ‘convexity’ assumptions). Essentially, the principle of rationality can be summed up in the statement: if aRc and bRc then aRc, which translated means ‘If our agent thinks that bundle a is at least as good as bundle b and that bundle b is at least as good as bundle c, then they also think that bundle a is at least as good as bundle c.’ This is all lovely, but isn’t really in dispute at this particular time (but will be when trying to aggregate demand from an individual and then it becomes very, very funny. But that’s a joke for another day). However, the rationality principle, as I will explain below, essentially assumes an undynamic, static and non-evolutionary view of consumer’s tastes which is up for criticism.
Optimization:
What I want to look at is the concept of optimizing – that consumers will choose, from a number of commodities, the bundle that best maximizes their utility (‘enjoyment’ in lay terms. Economists say that consumers can ‘ordinally measure’ the utility of a commodity in ‘utils’). The possible bundles are restricted by income – hence, consumers maximize their utility in what is known as the economically feasible consumption set (economists assume that consumers spend all their income, savings are a form of ‘delayed consumption.’)
Okay, here’s an example: let’s say there are two commodities – Marx’s Capital and a Che T-shirt (which any typical lefty is deemed to have). The possible combinations of these commodities (the bundles) are: 0 Capital, 0 Che-shirt; 1 Capital, 0 Che-shirt; 0 Capital, 1 Che-shirt and; 1 Capital, 1 Che-shirt. In other words, there are four possible bundles, of which one will maximize the consumer’s utility.
Okay what’s my point? My point is that let’s forget the guy on the corner who sells copies of Green Left and let’s go to a leftist supermarket. Suddenly, instead of being faced with two commodities we are faced with… a thousand – Lenin’s collected works, the writings of Trotsky, Mao’s little red books, and even the rarities of Enver Hoxha. Well, how many bundles do we have? Instead of writing each possible I’ll do the short-hand maths: the number of choices is equal to the number of units being considered + 1 to the power of how many goods. So, in our above model it was 2^2 = 4. If there was 3 goods, it would be 2^3 = 8 possible bundles.
1000 goods? Well, that would equal 2^1000. That seems like a small number – it isn’t. I’ll ‘spell’ it out for you:
10, 715, 086 , 071, 862, 673, 209, 484, 250, 490, 600, 018, 105, 614, 048, 117, 055, 336, 074, 437, 503, 883, 703, 510, 511, 249, 361, 224, 931, 983, 788, 156, 958, 581, 275, 946, 729, 175, 531, 468, 251, 871, 452, 856, 923, 140, 435 ,984 ,577 ,574 ,698 ,574, 803, 934, 567, 774, 824, 230, 985,421, 074, 605, 062, 371, 141, 877, 954, 182, 153, 046, 474, 983, 581, 941, 267, 398, 767, 559, 165, 543, 946, 077, 062, 914, 571, 196, 477, 686, 542, 167, 660, 429, 831, 652, 624, 386, 837, 205, 668, 069, 376
Yeah, it’s a big one.
How Big is Your Brain?
How big would your brain need to be to remember that many combinations? Well, let’s pretend that each neuron can remember the utility of 100,000,000,000 different bundles (I’m no brain scientist, but this sounds like a very optimistic assumption/myth!). The grey matter comprises about a kilo of your brain and contains 100,000,000,000 neurons, each weighing 1/100,000,000 grams. Following me?
So…how heavy would your brain be if it could remember that many combinations?
A) The weight of a normal human brain?
B) The weight of a small truck?
C) The weight of Earth?
D) The weight of the Universe?
E) Heavier?!
Well… it would weigh 10^244 times as much as the universe. Now people can justifiably call you big-headed, and vice-versa.
How Long Does It Take You to Shop?
Another question – how long would it take to optimize – i.e. recall the utility of each combination – supposing that we could recall the utility of a single bundle in 1/10,000,000,000,000,000,000,000,000,000,000,000,000 ,000,000,000,000,000,000,000,000,000,000,000,000,0 00,000,000,000,000,000,000,000,000th of a second (this is rather fast!)?
It would take 10^200 seconds. In other terms, 10^180 times the age of the universe. Raise this next time someone hassles you about your slow shopping habits.
Let’s be Rational:
Okay, enough fun with numbers. My point is, that the idea that consumers consider the utility of each particular bundle is patently false – and absurd – I dare say its irrational! Consumers do consider their budgets/income when deciding tastes (no thanks homothetic preferences!).
Neoclassical economics (in order to avoid the problems with aggregating individual demand curves) declared the golden principle of homothetic tastes. To quote my mircroeconomics book, Microeconomics – a Modern Approach, Andrew Schotter, 3rd ed., 2001, page 68:
‘When a consumer has homothetic preferences, all goods are superior and purchased in the same proportion no matter what the consumer’s income. In a world where all consumers have homothetic preferences, we might think of rich people as simply expanded versions of poor people. The tastes of such rich people do not change as their incomes change. They allocate their incomes exactly the way they did when they were poor. They just buy proportionally more of each good as their income grows.’
I hope you found that as humorous as I did when I first read it. In effect, Schotter is saying that if Bill Gates spent 10% of his income on pizza when he just another university computer science nerd, then he now spends 10% of his total income on pizza. In other words, he spends hundreds of millions on buying pizza for his own consumption.
Alternatively, the poor man who happens to get rich (we all know how often that happens!) did not change his spending habits at all – he still lives in a cardboard box (just an extremely large one). Likewise, the rich man spends the same amount he did on healthcare when he was poor – none. Going from the opposite direction, how many pieces of art work would a poor person have if they spent the same proportion that a wealthy person spends on art? A thousandth of a Mona Lisa?
It means that your spending habits will not change from when you are 16 to when your 60.
Clearly such statements are absurd and contrary to all empiricism. Yet for Neoclassical economics to work it essentially assumes that (1) there is only one consumer in society – or a endless number of drones and (2) there is only one commodity in society (I’ll cover that in another thread). This comes from the paradigm which asserts the uniqueness of the individual! The great theory that the market maximizes social welfare depends on these assumptions. It falls with them also.
To get back on point, we rely on habit, convention, culture, even flat-out ignoring / being ignorant of potential commodities – and a variety of things which may very well inhibit our ability to make utility optimizing decisions! When faced with such vast options we may use segmented optimizing – e.g. 100 possible combinations of food with a buy/not buy approach. Sure, there would still be 10^31 possible combinations, and your brain would still weigh about a million tonnes, but we’re getting there!
Optimisation: a good example of neo-classical economics using mathematics which are aesthetically pleasing but empirically laughable – time and what is physically/mental possible do matter.
Further Examination of Rationality Assumptions:
The Weak Axiom of Revealed Preference (WARP) stated that if a consumer chooses bundle A once when bundle B is also affordable, then the consumer will always choose A instead of B, regardless of relative prices. As Varian puts it: ‘If a bundle (x1, x2) is purchased at prices (p1, p2) and a different bundle (y1, y2) is purchased at prices (q1, q2) then if p1 x1 + p2 x2 ≥ p1 y1 + p2 y2, then it must NOT be the case that q1 y1 + q2 y2 ≥ q1 x1 + q2 x2. If the y-bundle is affordable when the x-bundle is purchased, the x-bundle must not be affordable.’
The Strong Axiom of Revealed Preference (SARP) was what I stated in the introduction– If (x1, x2 ) is revealed preferred to (y1, y2) and (y1, y2) is different from (x1,x2), then (y1, y2) cannot be directly or indirectly revealed preferred to (x1, x2). There is also a third axiom, known as the Generalised Axiom of Revealed Preference, which I won’t get into here, but is the weakest of the three.
How have these rationality assumptions held up to testing?
A test undertaken by Sippel attempted to replicate neoclassical standards. Ten sets of budgets and relative prices were presented. Consumers were to chose from 8 goods at each budget/price ratio, where the computer calculated the budget costs. The consumers were given a period of one hour to ‘optimize.’
The results weren’t good (for neoclassical economics).
The first experient contained 12 subjects, of which 11 of 12 subjects violated the SARP and WARP assumptions. 5 violated the weaker test GARP. The second experiment contained 30 subjects, of which 22 of the 30 subjects violated SARP and WARP, 19 of 30 violated the weaker GARP. The experiments conclusion: “We conclude that the evidence for the utility maximisation hypothesis is at best mixed. While there are subjects who do appear to be optimising, the majority of them do not… we … call the universality of the maximising principle into question.”
Other experiments have shown that consumers rely more on habit to reduce the vast array of potential choices – and hence do not optimise – or do not act ‘rational’ as the economist puts it. But it is not the consumer whom is acting irrationally, but the economists.
Undynamic Nature of Rationality:
When we take time into account, two of the four economic concepts of rationality (these four being, completeness, transitivity, non-satiation and convexity) fall into question – non-satiation and transitivity.
The principle ‘more is always preferred to less’ disregards time – two bannanas may be preffered to one, and three to two, but twenty bannanas to be consumed is not an ideal situation to be in . When we consider consumption as a function of time, anyone who behaves in what economists call ‘rational’ – preferring more to less – is a nutjob.
Likewise, a consumer may prefer bundle A to bundle B at one time, and yet at another time she may prefer bundle B to bundle A; consumer’s tastes cahnge. As Steve Keen says:
“Economists might protest that these comparisons are only valid when they are all made at the same instant, but that is precisely the problem: economic theory requires that everything happens at once, when in the real world an individual’s consumption is sequential, not simultaneous.”
Completness requires an impossible level of computation by the individual consumer. And lastly, as Keen points out:
“Convexity is also a by-product of the timeless analysis of consumption, which permits economists to ignore the manner in which the consumption of one commodity interacts with the consumption of another. Convexity implies that, to keep a consumer’s utility constant, it is necessary to subtract units of one commodity while adding units of another. But if the commodities are “sugar” and “tea”, and we are considering the act of drinking tea, then it is not true that a reduction in the amount of sugar can be compensated for by an increase in the amount of tea.”
The Scientific Status of Neoclassical Economics:
Schotter makes the claim:
‘If economic theories are correct, they should be verifiable or falsifiable in the lab. In other words, if the behaviour predicted by economic theory cannot be replicated in appropriately designed experimental setting, where human subjects make decisions under a set of monetary incentives that approximate those described by the theory, then the theory must be considered severely lacking.’
What a fantastic indictment of Popper’s falsifiability and unintended endorsement for Kuhn’s view of normal science as fundamentally dogmatic. Neoclassical economics has been replicated ‘in the lab.’ People were overwhelmed trying to optimize with just 8 commodities, taking over half an hour to reach a decision.
Unsuprisingly Schotter claims that
“Normative or welfare economics deals with what ought to be rather than what is and involves prescriptive statements that may be based on value judgement. Positive economics deals with what is rather than what ought to be and involves descriptive statements that are objective and verifiable.”
Marx once said that “no school of thought has thrown around the word ‘science’ more haphazardly then that of Proudhon, for ‘where thoughts are absent, words are brought in as convenient replacements.” I think Neoclassical economics is certainly a contender for that position – claiming to be a positive, non-biased, objective and verifiable discipline, all the while being naught but the most dogmatic religion.
(I would address alternative models for analyzing economic behaviour, but it is more appropriate elsewhere. Needless to say, neo-classical economics was a distinct departure/dissent from the classical (e.g Smith, Ricardo, Malthus) view of analyzing society from the viewpoint of classes – which Marx expropriated and turned it against its creators. Any model which takes into account classes – capitalists, workers, peasants, landlords - or even uses appropriate income groups not altogether removed from classes – is far likely to do a better job at analyzing aggregate market behaviour, rather than starting from the isolated individual. Marxists, post-Keynesians/neo-Ricardians, feminists, evolutionary economists take this approach.)
* I make no claim at being credited for pointing out the absurdity of this – the mathematics/humour is entirely borrowed from economist Steve Keen. Other experiments, which replicated neo-classical paradigm in all its splendour , showed that a consumer was overwhelmed in trying to optimize with just 8 commodities. I’m sure many of us can sympathize with that. Google the ‘curse of dimensionality’ for why.