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Lentil convexity (view source)

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{{A|g|[[file: Lentils.jpg|400px|center|thumb|Out of my cold dead hands.]]}}''Author’s note. I mean no slight on people who buy tinned lentils. Personally, I quite like them.''
{{A|devil|{{image|Lentils|jpg|Out of my cold dead hands.}}}}''Author’s note. I mean no slight on people who buy tinned lentils. Personally, I quite like them.''


So it turns out we haven’t been panic hoarding lentils after all. There is a benign explanation for the sudden disappearance of split peas from the nation’s grocery shelves.  
So it turns out we haven’t been panic hoarding lentils after all. There is a benign explanation for the sudden disappearance of split peas from the nation’s grocery shelves.  
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===Lentils in peacetime===
===Lentils in peacetime===
In ordinary times, our lentil-buying habits are regular: hippies and vegans buy a lot of lentils, and everyone else buys none. Okay, ''almost'' none. The [[Reasonable man|person on the Clapham Omnibus]] might have ''one'' tin, at the back of the cupboard, that someone got in a weak moment years ago, just in case of unexpected apocalypse or visit from long-lost, vegan, cousin from Australia.
In ordinary times, our lentil-buying habits are regular: hippies and vegans buy a lot, and everyone else buys none. Okay, ''almost'' none. The [[Reasonable man|person on the Clapham Omnibus]] might have ''one'' tin, at the back of the cupboard, that someone got in a weak moment years ago, just in case of unexpected apocalypse or visit from long-lost, vegan, cousin from Australia.


But the peacetime lentil-buying motivations ''within'' the community are unrelated. Overall appetite is, within a tight range, predictable. The lentil-acquiring proclivities of each individual are independent: that is to say, ''my'' lentil acquisition does not greatly influence, and is not especially swayed by, ''yours''.  
But the peacetime lentil-buying motivations ''within'' the community are unrelated. Overall appetite is, within a tight range, predictable. The lentil-acquiring proclivities of each individual are independent: that is to say, ''my'' lentil acquisition does not greatly influence, and is not especially swayed by, ''yours''.  


National weekly lentil purchases therefore usually cleave to a [[normal distribution]].  A small part of the population (say 0.5% — the hippies) may buy 8, 9 or 10 tins. A slightly bigger section (say 4.5% — vegans, health-food fanatics etc.) may buy one or two, and the remaining 95% will buy very few (let’s say on average 0.1 tin each: one tin between ten, which is probably generous). Since these lentil buyers’ decisions are ''independent'' events<ref>BUT ARE THEY? I don’t want to spoil the punchline but HOLD THAT THOUGHT.</ref> — it all tends to even out, within a range. Weekly overall tin purchases will fluctuate but overall the individuals will tend to even each other out. In the same week that your hippie stocks up for his bom shankar summer solstice druid’s convention cauldron ''partay'', a vegan might skip a tin, and your 95% who hardly ever buy lentils make little difference to the acquisition rate in any weather.  
National weekly lentil purchases therefore usually cleave to a [[normal distribution]].  A small part of the population (say 0.5% — the hippies) may buy 8, 9 or 10 tins. A slightly bigger section (say 4.5% — vegans, health-food fanatics etc.) may buy one or two, and the remaining 95% will buy very few (let’s say on average 0.1 tin each: one tin between ten, which is probably generous). Since these lentil buyers’ decisions are ''independent'' events<ref>BUT ARE THEY? I don’t want to spoil the punchline but HOLD THAT THOUGHT.</ref> weekly overall tin purchases will fluctuate but overall the random variations within individuals will tend to even out the total. In the same week that your hippie stocks up for his bom shankar summer solstice druid’s convention cauldron ''partay'', a vegan might skip a tin, and your 95% who hardly ever buy lentils make little difference to the acquisition rate in any weather.  


[[File:Lentil distribution.png|300px|left|Lentil buying projections in peacetime yesterday. The average per 100 people is 15, and any more than 28 is genuinely unthinkable.|thumb]]The odds of ''everyone'', including the normals, ''all'' going large on lentils ''in the same week'' is extremely low.<ref>With genuinely unconnected events the probabilities quickly fade into cosmic radiation: the odds of tossing heads just 100 times in a row with a fair coin is one in ''half a googol''. You wouldn’t expect that to happen in ''several lives of the universe''.</ref> The consequence, across the community, is a [[normal distribution]] of weekly lentil acquisition. The random variation in purchases by people in the different demographic groups will cause a small fluctuation in in demand for lentils from week to week, but from a grocer’ perspective, the demand curve is predictable and manageable.
[[File:Lentil distribution.png|300px|left|Lentil buying projections in peacetime yesterday. The average per 100 people is 15, and any more than 28 is genuinely unthinkable.|thumb]]The odds of ''everyone'', including the normals, ''all'' going large on lentils ''in the same week'' is extremely low.<ref>With genuinely unconnected events the probabilities quickly fade into cosmic radiation: the odds of tossing heads just 100 times in a row with a fair coin is one in ''half a googol''. You wouldn’t expect that to happen in ''several lives of the universe''.</ref> The consequence, across the community, is a [[normal distribution]] of weekly lentil acquisition. The random variation in purchases by people in the different demographic groups will cause a small fluctuation in in demand for lentils from week to week, but from a grocers’<ref>DID YOU SEE WHAT I DID THERE?</ref> perspective, the demand curve is predictable and manageable.


===Lentil buying decisions are ''not'' independent===
===Lentil buying decisions are ''not'' independent===
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So, we get ''infected'' with the idea of [[apocalypse]] ''by each other''. Our legume-buying habits ''are not independent after all''. They are all formed out of a collective consensus about the ''non-imminence'' of the second coming.  While each person’s threshold for precautionary lentil purchase in the event of apocalypse will differ, across the group, news of the unchecked spread of [[coronavirus]] will bring each person closer to that threshold and will push some of them over it.  
So, we get ''infected'' with the idea of [[apocalypse]] ''by each other''. Our legume-buying habits ''are not independent after all''. They are all formed out of a collective consensus about the ''non-imminence'' of the second coming.  While each person’s threshold for precautionary lentil purchase in the event of apocalypse will differ, across the group, news of the unchecked spread of [[coronavirus]] will bring each person closer to that threshold and will push some of them over it.  


Back to the 95% normals. As they walk past the tinned goods shelf, it only takes a small proportion of them to decide ''this'' is the week to pick up an extra tin to blow the grocer's expectations out the window. Let’s say, unnerved by all this elbow-touching and obsessive-compulsive hand-washing carry on, just 5 of the 95% decide to buy an extra tin. As the hippies, vegans & health nuts who buy lentils for fun are ''also'' buying their regular quota, the supermarket’s lentil supply quickly diminishes.  
Back to the 95% normals. As they walk past the tinned goods shelf, it only takes a small proportion of them to decide ''this'' is the week to pick up an extra tin to blow the grocer’s expectations out the window. Let’s say, unnerved by all this elbow-touching and obsessive-compulsive hand-washing carry on, just 5 of the 95% decide to buy an extra tin. As the hippies, vegans & health nuts who buy lentils for fun are ''also'' buying their regular quota, the supermarket’s lentil supply quickly diminishes.  


Then an unexpected second order of dependence emerges. For some of the 95% — who have not yet crossed the threshold for precautionary lentil purchase — then ''notice'' that the lentil shelf is nearly empty. They reconsider their apprehension of the apocalypse. They move closer to their threshold. Some cross it. Those who do collect a tin. The shelf is cleaned out.
Then an unexpected second order of dependence emerges. For some of the 95% — who have not yet crossed the threshold for precautionary lentil purchase — then ''notice'' that the lentil shelf is nearly empty. They reconsider their apprehension of the apocalypse. They move closer to their threshold. Some cross it. Those who do collect a tin. The shelf is cleaned out.
 
===Why an item limit does not work, and only makes things worse===
The supermarket management, alarmed at this inexplicable sudden run on lentils, now intervenes. They impose a 3-item limit for each customer. But limiting most customers, who would not normally be seen dead buying ''one'' tin of lentils, to ''three'', won’t help the problem, since the lentils-for-judgement-day-only contingent only wanted one each in the first place. The problem is not these people bulk buying, but that ''all'' of them are ''single''-item buying.  It will also irritate those hippies and rabbit-food munchers who would ordinarily buy more tins than that even when they weren’t panicking.
The supermarket management, alarmed at this inexplicable sudden run on lentils, now intervenes. They impose a 3-item limit for each customer. But limiting most customers, who would not normally be seen dead buying ''one'' tin of lentils, to ''three'', won’t help the problem, since the lentils-for-judgement-day-only contingent only wanted one each in the first place. The problem is not these people bulk buying, but that ''all'' of them are ''single''-item buying.  It will also irritate those hippies and rabbit-food munchers who would ordinarily buy more tins than that even when they weren’t panicking.


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===Implications===
===Implications===
We can see the interconnectedness between human decisions like lentil-buying, at the extremes is not stable. You can’t model it. You can’t predict it. The [[correlation]] ''changes'' on account of the very ''existence'' of each buying decision, and each other people’s reaction to that decision. In ordinary times one person’s buying decision won’t affect another’s. They ''look'' for all the world like truly independent events. What do I care whether you bought lentils? But events which are ''really'' independent ''stay'' independent, however weird things get. The odds of flipping heads on a fair coin stays 0.5 however often you flip it, and whatever the previous results.<ref>Practical point though: the longer your sequence of heads, the greater the probability that the ''coin is not fair''.</ref> This makes the job of modelling independent events much, much easier. Your standard deviation stays put. Modelling dependent events isn’t just a case of more complex maths. It isn’t ''possible''.
Events which are ''really'' independent ''stay'' independent, however weird things get. The odds of flipping heads on a fair coin stays 0.5 however often you flip it, and whatever the previous results.<ref>Practical point though: the longer your sequence of heads, the greater the probability that the ''coin is not fair''.</ref> This makes the job of modelling truly independent events much, much easier. Your [[standard deviation]] stays put.  
 
Interconnected events don’t. They go from stable, most of the time, to flat-out nutso in extreme times. Fair coins don’t go nutso. Dice don’t go nutso. Lentil buying ''can'' go nutso. You can’t model it. You can’t predict it. The [[correlation]] between events then changes further ''because it’s gone nutso''. There’s a feedback loop.
 
The point? Modelling normal distributions of independent events is easy, and safe. Modelling distributions of interconnected events isn’t. It isn’t just a case of more complex maths. It isn’t ''possible''. Now, mis-modelling overall lentil demand is a relatively low-stakes game: liable to annoy peaceniks — who are by nature unlikely to foment insurrection, and annoying them is kind of amusing anyway — plus, realistically (unless it ''is'' Armageddon, in which case lentil shortage is not the problem) actual consumption of lentils won’t change, so the supply-shortage will quickly sort itself out.  


Mis-modelling overall lentil demand is a relatively low-stakes game. It is liable to annoy peaceniks — but they are dispositionally unlikely to foment insurrection — and (unless armageddon does arrive after all, in which case a lack of lentils is the least of our problems) actual bodily consumption of lentils won’t change, so the supply-shortage will quickly sort itself out, as the 95% find themselves unexpectedly long more lentils than they know what to do with and their part of the demand curve hits absolute rock bottom.
So a spot of [[convexity]] might not matter for the world’s lentil purveyors — but how about those in the multi-billion dollar global transport and hospitality industries? What would ''they'' do if everyone, all around the world, without warning — you know, ''billions'' of people — as one, ''indefinitely'', stayed indoors?


But it is a nice illustration of how badly a normal curve can serve you when the chips are down.
Like ''that'' would ever happen.


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{{Technical Tuesday|26/1/21}}
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