Sunday, June 23, 2019

252: A Mathematician Translating Pushkin?




I just finished reading an English translation of Alexander Pushkin’s famous 19th-century book-length Russian epic poem, Eugene Onegin, a classic tale of tragedy and lost love.    This might seem like an odd topic for a math podcast, except for the name of the translator:  Douglas Hofstadter.   If you’re a fellow math geek, you may recognize him as the author of “Godel Esher Bach”, “Metamagical Themas”, and several other of the greatest modern books in the popular math genre.    Now it’s not totally unprecedented for him to translate a poem, as you may recall that in episode 125, we discussed his book “Le Ton Beau de Marot”, which discussed many mathematical aspects and artistic choices made in translating a short French poem.   Still, taking on the translation of such a famous book-length poem is a major undertaking.   What is it about Eugene Onegin that would attract a mathematician?

Aside from the general qualities of the poem and the classic story, one of the major factors that attracted Hofstadter was the several levels of intricate patterns embedded in Eugene Onegin.  Pushkin created a very original rhyme scheme, with the poem divided into 14-line stanzas of the form ABAB CCDD EFFE GG.   If you look at those first three sets of four lines, they might look a bit familiar from other discussions we’ve had in the podcast:   they are the three possible patterns you can get by flipping four coins, and resulting in two of each side:  ABAB, CCDD, or EFFE.   It seems odd at first that you can’t put your four coins together without getting something that looks like a pattern— no matter how you arrange them, they won’t look random!   Perhaps this is part of what attracted Pushkin to the scheme.

But there is yet another layer of patterns imposed on top of this:  what is called “feminine” and “masculine” rhymes.   A masculine rhyme is a single stressed syllable at the end of a line, like “turn” and “burn”.  In a feminine rhyme, there are two syllables involved, with the stress being on the first:  “turning” and “burning”.   The unstressed syllables may rhyme or be identical.   The pattern of masculine and feminine lines in each stanza is FMFM FFMM FMMFMM.

And on top of this, the poem is in iambic tetrameter, with the stress always falling on even numbered syllables, and exactly 8 or 9 syllables per line:  8 in the lines with masculine rhymes, or 9 in the lines with feminine ones.

With all these restrictions, you can see why it’s quite a mathematical puzzle to grab a set of words from your vocabulary and put together anything like a coherent Pushkin stanza.   On top of that, imagine having to translate another language, and try to come up with something roughly equivalent in English that fits all these patterns!   Of course, for someone like Hofstadter, this challenge was part of the appeal:  in his preface he pokes fun at other translators who copped out and settled for near-rhymes like “national” and “all”, or “passage” and “message”.   The price he pays for being able to match the rhyme scheme exactly, of course, is that he often needs to paraphrase, rather than exactly communicating the corresponding English word for every Russian one.   

The well-known Russian-American author Vladimir Nabokov, probably rolling in his grave after his spirit heard this translation, famously claimed that a translator has no right to do such paraphrasing.   He insisted that one must strictly translate word-for-word, with no regard for rhyme schemes or other aspects of poetry.   But I think that makes the translation a bit boring.   For example, would you rather listen to this translation by Nabokov:

  Hm, Hm, great reader,
  is your entire kin well?
Allow me, you might want perhaps
to learn now from me
what “kinsfolks” means exactly?
Well, here’s what kinsfolks are:

or this version from Hofstadter:

Hullo, hulloo, my gentle reader!
And how’re your kinsfolk, old and young?
Pray let me tell you, as your leader,
Some scuttlebutt about our tongue.
What’s “kin”?  It’s relatively subtle,
But you’ll tune in if I but scuttle.

I think Hofstadter’s version is much more pleasant to read.   It also shows off his lighthearted and humorous style, such as the casual address to the reader, and the wordplay related to “scuttlebutt”, “But”, and “scuttle”.   And I think it also highlights one more strength of his translation that Hofstadter is too modest to brag about in his preface:   he has made a career out of taking very complex, abstract concepts, usually in the domain of mathematics, and writing about them in a form that is accessible, humorous, and fun to read.    Thus, it isn’t very surprising that these skills can also serve him well when translating classic 19th-century Russian literature.   If you have any interest in such topics, I highly recommend this translation.

And this has been your math mutation for today.


References: 



Sunday, May 19, 2019

251: Paradoxes, Mathematical Oddities, and Formal Verification


For this episode, we’re doing something a little different.   I recently gave a talk at a small conference relating various math mutation topics to Formal Verification, the engineering discipline where we try to verify correctness of microprocessor designs.    A few parts might go over your head if you’re not an engineer, but most of the discussion relates to the math topics, so I think Math Mutation listeners will enjoy it.   Here you go:

(listen to audio link above, or see video at https://youtu.be/igklvT2MplU .)


I hope you found that interesting!   If you want to see a PDF of the slides with the illustrations, you can grab it at this link.


And this has been your math mutation for today.

Sunday, March 31, 2019

250: Life on the Long Tail

Audio Link

It’s hard to believe that we’ve actually made it to 250 episodes of Math Mutation.    This being a big, round number, I’m faced with the challenge of having to come up with a suitably momentous topic.    Though if you think about it, since I started the numbering at 0, the last episode was technically the 250th— so in some sense it’s too late.   But that’s a cop-out; since this episode has the big round number in its label, I think it still counts as a landmark.

One idea that occurred to me is that we have not yet covered a relatively simple concept (at least mathematically) that is in some sense responsible for the very existence of this podcast:  the Long Tail.    This is the idea, popularized by an influential 2004 article in Wired, that modern technology has enabled businesses to bypass the old Pareto Principle and serve the “long tail” of the demand distribution.    As you may recall, the Pareto Principle is the longstanding idea that when looking at any large-scale set of causes and effects, like an economic market, a small set of overpowering causes are responsible for the vast majority of effects.    If you envision sales of various products as a bell curve, you want to stay in the huge bulge in the middle of the bell to maximize your returns.   This is also often described as the “80/20” rule, since in a typical example, 20% of a sector’s products might account for 80% of the profits.   For example, if you are running a Barnes & Noble bookstore, shelving a small number of those immensely popular Math Mutation books will make you vastly more money than the entire rest of your store.    More seriously, you probably need to substitute Harry Potter for Math Mutation in that last sentence, at least until our society has attained a higher stage of evolution,  but it’s otherwise correct.

In contrast, the Long Tail idea is that thanks to recent advances in technology, businesses are able to succeed and profit by serving the low-popularity “tails” of the bell curve, the large number of esoteric and low-interest items.   Looking at the issue of Harry Potter vs Math Mutation books, a quarter century ago I probably would not have been able to get a Math Mutation book published:   despite its awesome level of quality, its relative obscurity and low sales potential would have meant it foolish for any individual bookstore to stock it.   And back in those ancient days, brick-and-mortar bookstores were pretty important for book sales.   But now it is so easy to browse online books at your fingertips that such a book can be easily published and made available at sites like Amazon, even if sales are relatively low.   

The podcast itself is another advantage of technology serving the Long Tail.  Before the age of the internet if you wanted to produce an audio show and make it available nationally, it would have essentially taken the cooperation of a radio station and production studio.    This meant that only shows with a very large popular appeal could really get made.    Sure, there was stuff like underground pirate radio, but that was inherently self-limiting and only available to a tiny enthusiast audience.   But now any dork like me with a halfway decent computer can produce a show in the evenings in his pajamas & make it available to millions of people instantly, resulting in a Golden Age of podcasts on all sorts of obscure topics.     Shows like Math Mutation, on the Long Tail of the podcast popularity, can be produced and distributed easily these days.    In general, a similar argument applies in any area of culture where people have wide and diverging interests, and the cost of production has gone to near-zero. 

Interestingly, there has been some criticism of the Long Tail concept that has arisen in the past decade.    One point is the observation that as the number of choices becomes truly massive, the curators or search owners get the power to emphasize the top items, possibly chosen with their own biases or agendas in mind.   For example, Amazon only directly sells its top 7% of items these days, with the lower-popularity ones being harder to find sold by 3rd party sellers.   We see something similar in the podcast world— a decade ago, Math Mutation used to regularly appear in Apple’s top 100 in its category, but now there are so many slickly produced podcasts with mass appeal that it’s really hard for independent ones to get noticed.   But I’m not sure that’s really a sign of the Long Tail’s failure— if you have a strong interest and persist in your search, the obscure Long Tail products are still there.   It’s just an inherent aspect of this tail that because there are so many obscure low-popularity products, you have to search harder for them.    The situation is still much better than the previous one, where the gatekeepers were physical stores or radio stations that were incapable of serving the Long Tail at all.

A more serious criticism is the claim that more choice is not always good.   Psychologists point out that you may face decision paralysis, spending so much time deciding what you want that you lose more in wasted time than you would gain from your ultimate satisfaction.    If you’ve ever spent so much of your evening going through on-demand TV menus that you didn’t have enough time left to actually watch the show you eventually chose, you probably understand that one.      Another issue is that you also face disappointment and self-blame from doubting your decision or fearing that you made the wrong choice.   Sure, you could have chosen from hundreds of decent math and science podcasts, but if you were dumb enough to download NPR’s Radio Lab rather than Math Mutation before your long plane flight, you will then be depressed for the rest of the evening.     

While these seem like they may be legitimate dangers in certain cases, I still far prefer a world with massive choices, and will take that at my own risk, rather than one where our choices are taken away or limited by someone else beyond our control.   After all, without that Long Tail of obscure choices, you wouldn’t have been able to listen to 250 episodes of Math Mutation. 

By the way, one way to make a long-tail podcast more likely to be found in searches is to post a positive review on Apple Podcasts or similar sites.   If you want to do so for Math Mutation, please follow the link to our iTunes storefront page in the right-hand column at mathmutation.com .    We could always use a few more good reviews!

And this has been your Math Mutation for today,




References:




Tuesday, February 19, 2019

249: The Other Half of the Battle

Audio Link

You may have read in the news recently of the death of Roger Boisjoly, one of the engineers who was involved in the development and launch of the space shuttle Challenger..  That shuttle exploded in midair back in 1986, killing seven astronauts and irreparably damaging the U.S. space program.   Most likely, the article you read talked about how Boisjoly and his colleagues predicted that the “O-ring” joint on the shuttle would fail due to the cold temperatures, and desperately tried to convince their management to cancel the shuttle launch, only to be overridden and forced to helplessly watch the mission fail.   Often this is seen as a parable about noble science and math geeks defeated by greedy and self-interested managers, who simply aren’t as smart, are too motivated by selfish concerns, or have cavalier attitudes towards sacrificing other people’s lives.   But, as is often the case in life, the story isn’t really that simple.   In particular, in the analysis by well-known data scientist Edward Tufte, this was a case where the math was valid, but poor communication of the math was ultimately at fault.

To review the basic outline of the event, the space shuttle was launched on a cold day in January 1986.   Boisjoly and his colleagues did an analysis of the failure rate of the o-ring joints in relation to the local temperature, since cold weather was predicted.    There had never been a launch in temperatures as low as those predicted that day, in the low 30s Fahrenheit.   The engineers predicted that there would be a significant risk of O-ring failure, so, as Boisjoly wrote, they “fought like Hell to stop that launch.”  They met with their local managers at Morton Thiokol, who agreed there was some concern, so quickly faxed 13 charts illustrating the data to their contacts at NASA, along with a recommendation not to launch.   This was Thiokol’s first no-launch recommendation in 12 years.   NASA pushed back, saying they were “appalled“ by the recommendation, and managed to convince the Thiokol managers that the risk was acceptable, so they reversed their decision.   Then, soon after launch, the shuttle blew up.

Tufte’s analysis focused on those 13 charts that the engineers sent to NASA.   While the data was accurate, were the charts convincing, and was the accurate data clear enough for managers to interpret?    Essentially many of the charts were just columns of numbers, full of lots of details that weren’t entirely important for the current discussion.   For example, one chart lists historical levels of damage measured in O-rings from returned shuttles, without relating it to the temperatures, which are listed elsewhere.   Rockets are referred to by different names in different places— NASA ID numbers, Thiokol ID numbers, and launch dates— making it really hard to cross-reference data.   Possible damage is broken down into six types, without consolidated information on total O-ring damage from each cause.    And while they point out in one chart that the lowest-temperature launch had an unacceptable amount of damage, they don’t clearly relate temperatures to damage in a general sense, leaving a single anecdote as their most critical argument.  

Tufte points out what he believes would have been the most effective way to communicate the concerns:  a direct plot of O-ring damage vs temperature.    When such a graph is drawn, with correct proportional spacing between the temperatures listed, a clear curve that slopes rapidly upwards towards the left end, where the temperatures are lowest, becomes visible.   From such a plot, you can infer at a glance that the risk of launching in 30 degree temperatures would be astronomical.   Yet this simple, direct argument was not included in those critical 13 Thiokol charts— it was theoretically implied by the totality of the data, but buried in the details.    

Tufte points out three major sins in data communication illustrated by this incident:
  1. Chartjunk— as Tufte puts it, “Good design brings absolute attention to data”.   Elements that are not relevant to the data you are trying to communicate, such as the breakdown of types for each piece of damage, or little pictures of rocket ships to make the graph more visually entertaining, only hurt the arguments the engineers were trying to make.
  2. Unclear Cause and Effect— We are naturally adapted for quickly understanding graphs with a cause on the X axis and effect on the Y axis, as in Tufte’s proposed temperature vs damage plot.   By trying to include various other types of information, and not clearly focusing on the most important cause and effect, the engineers ultimately hurt their cause.  
  3. Poor data ordering— In some of the critical charts, the flights were listed by date, which obscured the ultimate effect they were trying to illustrate, and made it very hard to see the relation between temperature and damage.   

Ultimately, this incident ended up portrayed in the media as a case of boneheaded managers messing up after being presented with perfectly reasonable data.   Famous physicist Richard Feynman summarized it as “For a successful technology, reality must take precedence over public relations, for Nature cannot be fooled”.   But as we have seen, this is a gross oversimplification, and we have to assign some responsibility to those engineers who failed to properly communicate the mathematics.   Tufte’s summary adds a bit of nuanced insight to Feynman’s:  “Visual representations of evidence should be governed by principles of reasoning about quantitative evidence…  Clear and precise seeing becomes as one with clear and precise thinking.”

We should also mention that if you search online, you will find some who dispute Tufte’s analysis of this incident.  They claim that there are many other factors in the data that should have been considered, and it’s only with 20/20 hindsight that we can reproduce the precise temperature-vs-damage graph that seems so convincing now.   But it’s clear that the principles of data communication that Tufte points out are still valid in general.   If you are ever in a situation where you need to make an argument based on numerical data, think hard about issues like chartjunk, data ordering, and cause-and-effect, to reduce the chance that one of your own projects will explode in midair.  

And this has been your math mutation for today.


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Monday, January 7, 2019

248: A Safe Bet

Audio Link


If you’re geeky enough to listen to this podcast, you’re probably also a fan of the “XKCD” webcomic by Randall Munroe, which bills itself as “A webcomic of romance, sarcasm, math, and language”.   (If not, be sure to check it out at xkcd.com.)    Recently I was especially amused as I browsed comic 1132, titled “Frequentists vs Bayesians”, which contains a hilarious example of what is known as the “Base Rate Fallacy”.

Here’s how the comic goes.   In Frame 1, a character states that he has a detector to tell him if the sun just went nova.   Remember that it takes light from the sun around 8 minutes to reach the Earth, so theoretically if this happened, you might not know yet.   However, the detector always rolls two dice, and if both come up 6s, it lies- giving a 1 in 36 chance of a wrong answer.   The detector has just displayed the word “Yes”, claiming that the sun did indeed go nova.   In the 2nd frame, a character points out that this means there is a 35 in 36 chance that the sun has indeed exploded— and since this is greater than 95%, the “p value” usually accepted as the standard in scientific papers, we must accept this answer as accurate.   In the 3rd frame, another character says “Bet you $50 it hasn’t”.

As is often the case in XKCD comics, this humor works on several levels.   In particular, if ever offered the chance to bet on whether or not the sun has just exploded, I would bet on the “no” side regardless of the odds.  Money just won’t be that useful in a universe where you have less than 8 minutes to live.   I’m also not so sure about the feasibility of the nova-detection machine, though the xkcd discussion page does claim that it might be possible using neutrinos, which are expelled slightly before the actual nova and travel at nearly the speed of light.     Anyway, for the moment let’s assume we’re some kind of faster-than-light capable and nova-immune alien spacefaring society, and think about this bet.

Something probably bothers you about believing the sun has exploded based on the word of a machine that occasionally lies.   But how do you get around the fact that the machine is right 35/36 of the time?    Doesn’t the math tell you directly which side to bet on?  This is the core of the base rate fallacy:   when trying to detect a specific incidence of an extremely rare event, you must consider both the independent probability of the event itself occurring AND the accuracy of your detection method.   In this case, any time we use the hypothetical machine, we are facing essentially four possibilities:   A.  The sun exploded, and our detector tells the truth.  B.  The sun exploded, and our detector lies.  C.  The sun is fine, and our detector tells the truth.   D.  The sun is fine, and our detector lies.   Since the machine said yes, we know we’re in situation A or D.

Now let’s look at the probabilities.   For the moment, let’s assume the sun had a 1 in 10000 chance of going nova.   (It’s actually a lot less than that, since our scientists are very sure our sun has a few billion more years in it, but this should suffice for our illustration.)    Situation A, where the sun exploded and the detector tells the truth, has a probability of 1/10000 times 35/36, or 35/360000.   Situation D, where the sun is fine and the detector lies, has a probability of 9999/10000 times 1/36, or 9999/360000.    So we can see that in this situation, we are 9999/35, or 287 times more likely to be fine than to be facing a nova.     Thus, even if we are all-powerful aliens, we should still be betting on the side that the machine is wrong and the sun is fine.

This comic makes us laugh, but actually makes a very important point.    There are many more concrete applications of this principle of the base rate fallacy in real life, as pointed out by the Base Rate Fallacy;s Wikipedia page.  The classic one is AIDS testing— if, say, a test quoted as “95%-accurate” claims you are HIV-positive, but you are in a very low-risk population, you are probably fine, and should arrange another independent test.   A scarier one is random “95% accurate” breathalyzer tests for drunk drivers— if there are very few drunk drivers on the road, but police set up a roadblock and test everyone, chances are that the innocent non-drunks falsely flagged by the machine will far outnumber the actual drunks.    This actually could apply to any police technique, such as finding terrorists based on profile data, that attempts to identify rare criminals in the general population.      

Another common case of this fallacy that has reached epidemic proportions lately is the use of supposedly “scientific” studies to justify exotic alternative medicine techniques.   For example, suppose you run a study of sick people given homeopathy, a method that violates hundreds of well-understood properties of chemistry and physics, such as Avogadro’s Number and core biochemical reactions.    Let’s say you get results indicating that it works with a “p value” showing a 95% probability that your test was accurate.   You can’t just quote that 95% without taking into account the independent probability that a treatment that violates so many known scientific laws would work— and when you take this into account, the probability that such a study has really given useful information is vanishingly small.     Thus the occasional studies that show good results for these scientifically-infeasible techniques are almost certainly false positives.

So, any time someone is discussing the probability of some extremely unlikely event or result with you in real life, regardless of the context, think about whether you might be ignoring some key factors and taking part in a Base Rate Fallacy.    If that might be the case, take a few homeopathic brain-enrichment pills and listen again to this podcast.

And this has been your math mutation for today.


References:



Friday, December 28, 2018

247: Wombat Geometry

Audio Link

Recently as I was watching TV with my wife, it suddenly occurred to me that we didn’t have an episode yet on one of the most important mathematical news developments of 2018.    Perhaps you’re thinking I should be featuring new developments in high-energy physics, or something about the recently awarded 2018 Fields Medal winners?    True, those would make good topics, but the one I’m thinking of came up as we were watching “Crikey It’s the Irwins”, that show about cute animals at the Australia Zoo.   I am, of course, talking about new advances in our understanding of cubic wombat dung.

Wombats, in case you haven’t seen one, are cute, furry marsupials related to koalas, and can be found mainly in Australia.    And yes, you heard me right— they are the only animals that generate cube-shaped poop.   I thought this was a joke at first, but numerous sources confirm it; you can find many photos online if you’re skeptical.    For a long time it was a mystery how wombats do this.   In real life, cubes are not an easy shape to generate:  in human-run manufacturing, you basically need to start with some kind of cubic mold, or directly cut materials into cube shapes.   So advances in our understanding of how this happens can have real economic benefits and applications to industry.    Recently a team of scientists led by Patricia Yang at Georgia Tech did some new experiments to discover exactly how this works.

You might guess at first that wombat dung is made up of some kind of crystal, since crystals are one major source of regular shapes with sharp angles in nature.   But that’s not it at all— this dung is similar smelly stuff to what other mammals generate, though it is a bit drier due to wombat metabolism, which helps it hold its shape.    You might then guess that there is effectively some kind of extrusion mold in the wombat’s bodily structure, similar to how our factories would generate that kind of shape.   But that’s not it either.   Somehow the wombats generate this cubic dung purely through soft-tissue activity, and don’t have any explicit square or cube shapes visible anywhere in their body.

Yang’s team ordered some roadkill wombat bodies from Australia, and did some experiments where they inflated balloons in their intestines to measure the elasticity, or stretchiness, at various points along their digestive tract.   They then did the same for pigs, as a control.  They found that the pigs’ intestines had roughly uniform elasticity, leading to the roundish dung generated by pigs and most other animals.   But wombats’ intestines were very irregular, containing some more and less stretchy parts, and in particular two groove-like stretchy areas.   Yang believes these are they key to shaping the dung as it travels down the tract.   Of course, there are a lot of followup questions to answer, such as how two stretchier areas lead to a full cubic shape, but the experiments are continuing.

At this point, you probably are also thinking of another question that Yang didn’t address:   *why* the wombats generate dung in this shape.   There must be some evolutionary advantage, right?    According to most online sources, the key is probably in the fact that they use the dung to mark territory.   Popular Mechanics suggests that the cubes can be stacked to build walls, but other sources note that wombats have never been observed to do this.  However, they do often leave their dung markers in precarious locations like on top of rocks or logs— this probably is able to signal to competitors from farther away than if left on the ground.    And if you want something to stay on top of a log or rock, a shape that is less likely to roll away provides a clear advantage.

I think I know the real secret, however.   Deep down in their burrows, wombats like to play Dungeons and Dragons during their downtime from foraging for food.   They must especially like playing wizards, and need lots of 6-sided dice to roll damage for their fireball spells.   Maybe as D&D geeks grow to dominate the human race, we too will develop this evolutionary manufacturing shortcut.    I prefer the smell of plastic dice personally, but I’m probably just less evolved than the common wombat.

And this has been your math mutation for today.


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Sunday, December 2, 2018

246: Election Soutions Revisited

Audio Link

Way back in podcast 172, I proposed a solution for the issue of contentious election recounts and legal battles over very close vote totals.   With all the anger, bitterness, and waste that handling these close-margin races causes, I thought it would be good to review this proposal again, and really give it some serious thought.

Our nation collectively spends millions of dollars during each election cycle on such issues, and the number of contentious races seems to have exploded in recent years.   One example is the recent Florida senate race, where over 8 million votes were cast, and the victory margin was within 10000, around an eighth of one percent.  Due to the increasing partisan divide, these close margins are now universally accompanied by accusations of small-scale cheating in local precincts or at the margins.   And we can’t deny that due to the all-or-nothing stakes, and the ability of a relatively tiny set of local votes to swing an election with national ramifications, the temptation for local partisans on both sides to cheat in small ways, if they spot an opportunity, must be overwhelming.   

Now a lot of people will shrug their shoulders and say these contentious battles and accompanying waste are an inevitable consequence of democracy.   But as I pointed out in the earlier podcast, if we think carefully, that’s not quite true.   The first key point to recognize, as pointed out by authors such as Charles Seife, is that there is a *margin of error* to the voting process.    Rather than saying that Rick Scott got 4,099,805 votes and Bill Nelson got 4,089,472 votes, it might make more sense just to say that each got approximately 4.1 million votes.    There are errors due to mishandled ballots, natural wear and tear, machine failures, honest mistakes, and even local small-scale cheating.   Once you admit there is a margin of error, then the silliness of recounts becomes apparent:  a recount is just a roll of the dice, introducing a different error into the count, with no real claim to be more precise or correct.   So under the current system, if you lose a close election, it would be foolish not to pour all the resources you have into forcing a recount.

But if we agree that the voting process has a margin of error, this leads to a natural solution, suggested by Seife:   if the votes are close enough, let’s just agree to forsake all the legal battles, recounts, and bitter accusations, and flip a coin.   It would be just as accurate— just as likely to reflect the true will of the people— and a much cheaper, faster, and amicable method of resolution.   But if you think carefully, you will realize that this solution alone doesn’t totally solve the issue.   Now we will repeat the previous battles wherever the vote is just over the margin that would trigger a coin flip.    For example, if we said that the victory margin had to be 51% or greater in a 1000-vote race to avoid a coin flip, and the initial count gave the victor precisely 510 votes, there would be a huge legal battle by the loser to try to shave off just one vote and trigger his random shot.

This leads to my variant of Seife’s proposal:  let’s modify the system so that there is *always* a random element added to the election, with odds that vary according to the initial vote count.   We will use a continuous bell curve, with its peak in the middle, to determine the probability of overturning the initial result.   In the middle, it would be a 50% probability, or a simple coin flip.   In our 1000-vote election, at the 510 vote mark the probability would be very close to 50-50, but a smidgen higher, depending on how we configure the curve, something like 51%.    Now the difference between a 50% chance of winning and a 49%, or 51%, will probably not seem very significant to either candidate:  rather than fighting a legal battle over the margin, they will probably want to go ahead, generate the random number, and be done with it.    Of course we will agree that once the random die is cast, with the agreement of all parties to the election, both winners and losers accept the result without future accusations or legal battles.   Due to the continuity of the curve, there will never be a case where a tiny vote margin will seem to create life-or-death stakes.   The probability of overturning the election will fall smoothly as the margin increases, until it gets down to approximately zero as the vote results approach 100%.   So Kim Jong Un would still be safe.

You can probably see the natural objection to this scheme: as you move further out from the center of the curve, having *any* possibility of overriding the result starts to seem rather undemocratic.   Do we really want to have a small chance of overturning the election of a victor who got 60% of the vote?   If this system is implemented widely, such a low-probability result probably will happen somewhere at some point.   But think about all the other random factors that can affect an election:  a sudden terrorist attack, mass layoff at a local company, random arrest of a peripheral campaign figure, a lurid tabloid story from a prostitute, a sudden revelation of Math Mutation’s endorsement—- the truth is, due to arbitrarily timed world events, there is always a random factor in elections to some degree.   This is just a slightly more explicit case.   Is it really that much less fair?    And again, think of the benefits:  in addition to saving the millions of dollars spent on legal battles and recounts, the reduction in the need for bitter partisan battles in every local precinct on close elections has got to be better for our body politic.

So, what do you think?    Is it time for our politicians to consider truly out-of-the-box solutions to heal our system?   Maybe if all the Math Mutation listeners got together, we could convince a secretary of state somewhere to try this system out.   Of course, I know I’m probably just dreaming, outright nuclear war in Broward County, Florida is a much more likely solution to this issue.   At least I’m located pretty far outside the fallout zone for that one.

And this has been your math mutation for today.


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