Thursday, July 2, 2015

Gladly Paying $1.10 For a Dollar Bill...

....or why rational choices ain't always so rational:

Another lovely puzzle/paradox today from Greg Ross's "Futility Closet" volume. It's known as the "dollar auction" paradox created by economist Martin Schubik. The setup (I've adapted from Wikipedia):

An auctioneer is to auction off a single dollar bill with the following rule: the bill goes to the highest bidder, AND the second-highest bidder LOSES the amount that they bid (to the auctioneer). The winner could gain a dollar for say 20 cents, for example, but only if no one else bids higher. The second-highest bidder is the biggest loser since they pay out their bid and get nothing in return.
The opening, minimum bid is 5 cents (with 5-cent increments thereafter) from one player, who would make a 95-cent profit if no one else bid. But it's sensible for another player to bid, say 10 cents, and still make a 90-cent profit. Then similarly, another bidder may now bid 15 cents, making 85-cents profit.
Whoever is the second-highest bidder at any point in time will wish to convert his potential loss to a gain by bidding higher than the highest-bidder, and so on. Obviously, if this keeps up, at some point, the dollar will COST someone a dollar to purchase -- but at least they will suffer no loss, while the 2nd highest bidder will lose 95 cents, giving them an incentive to bid $1.05 and thus decrease their loss to a nickel... at which point, the other bidder loses a whole dollar... and on and on. Bids beyond $1.00 mean that both top bidders lose money, thus minimizing the amount of loss then becomes the focus. A series of rational bids will reach and ultimately surpass the one dollar point, as the bidders seek to minimize their losses. Thus, "rational" bidding leads inevitably to both the two highest bidders losing money (while the auctioneer makes out well).
No wonder some call economics "the dismal science." ;-)

Wednesday, July 1, 2015

Of Sheep, Literalism, and Language

I got a kick out of Evelyn Lamb's latest posting at her "Roots of Unity" blog:

I enjoyed two things in particular:

First she employs one of my favorite old math jokes... about the astronomer, physicist, and mathematician (in her Wikipedia-rendition) who see a black sheep in a Scottish field... I won't repeat it here (if you're unfamiliar with it just check out her post). What I love about this joke is not so much the humor, which is good, not great, but what it so succinctly says about how mathematicians approach the world, and are set apart from other scientists. Mathematicians want PROOF (or something akin to it)... other scientists deal in, and are satisfied with, evidence, generalization, induction (precarious indicators of truth). But no, no, not we math-types. Show us the proof! So what if a trillion silly values confirm the Riemann Hypothesis; get me some dang proof; enough of this idle speculation!

Secondly, I enjoyed learning that Evelyn is a "literalist," since I've used that term all my adult life, to describe myself, but never met another person employing it. The tendency to take words literally is an annoying way to go through life because of the sloppy, imprecise, ambiguous ways language is routinely (and inherently) used every day, but happily mathematics is a refuge from that.

Language, in business, advertising, politics, religion, culture, is very controlling of our lives (and certainly not always in good ways).  I've long been a proponent of General Semantics, wanting for some time to write a post here about Martin Gardner's dissing of G.S. -- one of the greatest mistakes he ever made in my opinion -- since G.S. teaches people to be skeptical of words and language (and as a sort of professional skeptic, Gardner should've appreciated it). One day I'll get around to it.
In the meantime, if you invite me to your party starting at 8pm., expect me to be there at 8pm (or even 7:58pm); if you want me to arrive "fashionably late" then put on the invite, "please arrive fashionably late." ;-)

Monday, June 29, 2015

A Re-run Monday

I was reading the first of Greg Ross's "Futility Closet" volumes when I came across one of my favorite old puzzles, that was last presented here over 2 years ago. I originally learned of it from Richard Wiseman's blog and again I'll opt for Richard's version of it (most readers here are likely already familiar with it):
"Imagine there is a country with a lot of people. These people do not die, the people consist of monogamous families only, and there is no limit to the maximum amount of children each family can have. With every birth there is a 50% chance it's a boy and a 50% chance it is a girl.  Every family wants to have one son: they get children until they give birth to a son, then they stop having children. This means that every family eventually has one father, one mother, one son and a variable number of daughters.  What percent of the children in that country are male?"
.answer below
(p.s.  also, be sure to check out MathTango today for my interview with Siobhan Roberts, author of the brand-spanking new "Genius At Play" biography of John Conway.)
The answer (surprising to some, not to others) is 50%, which I won't explain, but will direct you to Richard's post (and 270+ comments) if you need an explanation:

Sunday, June 28, 2015

Mathematics and the Real World

Sunday's reflection:

"There are voices in the modern mathematical community that bemoan the state of mathematics today. While relishing the intellectual freedom bequeathed by the non-Euclidean revolution, mathematicians of the twentieth century carried their subject farther and farther from a contact with the real world, until their logical constructs became so abstract and arcane as to be unrecognizable by a physicist or engineer. To many, this trend has transformed mathematics into little more than a pointless exercise in chasing tiny symbols across the page. One of the most vocal critics of this trend is mathematics historian Morris Kline, who wrote:
'Having formulated the abstract theories, mathematicians turned away from the original concrete fields and concentrated on the abstract structures. Through the introduction of hundreds of subordinate concepts, the subject has mushroomed into a welter of smaller developments that have little relation to each other or to the original concrete fields.'
"....In response there can be made an intriguing argument that mathematical theories, no matter how seemingly abstract, often have surprising applications to very solid, real world phenomena.  Even the non-Euclidean revolution, the subject that did so much to sever the bond between mathematics and reality, has found its way into modern physics books, for the relativistic theories of today's cosmologies rely heavily on a non-Euclidean model of the universe. Such a reliance was certainly not foreseen by the nineteenth-century mathematicians who investigated the subject for its own sake, yet it now forms a part of applied mathematics necessary for inclusion in the physicist's tool-kit."

-- William Dunham in "Journey Through Genius" (1990)

Monday, June 22, 2015

Friggin' Trig

via WikimediaCommons

Boy, given the comments generated you might have thought that Cathy O'Neil had suggested getting rid of the Confederate Flag or something, when all she did was recommend dropping trigonometry from the high school curriculum:

If you've missed out on all the brouhaha (or, have fond or not-so-fond memories of trig) check out her post and all the pro-and-con commentary.

Sunday, June 21, 2015

A Tenet From Taleb

A quick Sunday reflection from Nassim Taleb in "Antifragile":
"...let me express my rule as follows: what Mother Nature does is rigorous until proven otherwise; what humans and science do is flawed until proven otherwise."

[...recently finished reading Taleb's 2012 "Antifragile" and enjoyed it even more than his prior "The Black Swan" and "Fooled By Randomness". The last third of the irascible volume is especially good. Recommended.]

Thursday, June 18, 2015

Unlikely Movie Star

The George Csicsery film, "Counting From Infinity," recounting Yitang Zhang's surprising, groundbreaking work on the Twin Prime Conjecture was released earlier this year. If you missed it, the latest trailer for it has been going around recently:

also, an older, similar trailer here:
and an earlier review of the film (calling it a "Math-erpiece") here:

Csicsery, by the way, has previously done film work on Paul Erdös, Julia Robinson, and Paul Halmos.

Wednesday, June 17, 2015

Sweet Surrender?

"Gödel's Lost Letter..." offers a take today, "Security Via Surrender," which employs "judo" for the worsening problem of security in our digital world:

The authors launch their case from this quotation of a martial arts disciple:
"...resisting a more powerful opponent will result in your defeat, whilst adjusting to and evading your opponent’s attack will cause him to lose his balance, his power will be reduced, and you will defeat him. This can apply whatever the relative values of power, thus making it possible for weaker opponents to beat significantly stronger ones."
They forgo the emphasis on "golden key" and secrecy approaches to security in favor of a "knowledge-based authentication" format. But in the end they also admit that "the bad news in all of this is that assuring one’s identity is becoming a battle and there seems to be no simple way to assure victory."

Indeed, can't help but be reminded here of the old Murphy dictum that, "It is impossible to make anything foolproof because fools are so damned ingenious."

Monday, June 15, 2015


Summery stuff is eating up so much of my time these days, not finding as much opportunity for blogging, thus posts dropping off somewhat ("Sunday reflections" are pre-set for the next 6+ months however, so no change in their regularity!).

...So instead a little politics, 'cuz who don't love pontificating on (or in this case, predicting) politics....

No idea who the Democratic presidential nominee will be (too early and sketchy to deduce much there), but the Republican side has never in my lifetime seen such a c-r-r-razy crop of wannabes.

Around 11 Repubs have officially declared for the race, with closer to 20 considered prospective candidates. Given the nature of the Republican base, the workings (and order) of the primary system, the campaign styles of those running, and the fact that the last two Republican moderate nominees went down to defeat, certain conclusions may shake out:

Only 3 candidates strike me as viable for the Party through the primary process (though I won't list the reasons why): Cruz, Rand Paul, and Paul Ryan (who hasn't declared and may not). Walker, Santorum, perhaps Huckabee, are possibles for the VP slot (which could easily go to someone not in the primary run), but I can't fathom a realistic scenario for anyone else making it to the finish line -- some are too moderate, some too baby-faced (not-ready-for-prime-time), some too inexperienced or weak-on-issues, some simply poor campaigners, and others simply too oddball... and several don't have the 'fire-in-the-belly' required to prevail in such a long, knock-down-drag-out competition. But it should certainly be entertaining along the way (...if only in the same manner that race-car crashes are entertaining).

Anyway, in almost 40 years of trying, I've NEVER called a competitive political primary race accurately this early on, so no need to let me know if everything I say above proves false.

....Maybe by Wednesday back to some math!

Sunday, June 14, 2015

Reflecting on Perelman

Sunday reflection... (yesterday was Grigori Perelman's birthday):

"[Grigori] Perelman's aversion to public spectacle and to riches is mystifying to many... I want to say I have complete empathy and admiration for his inner strength and clarity, to be able to know and hold true to himself. Our true needs are deeper -- yet in our modern society most of us reflexively and relentlessly pursue wealth, consumer goods, and admiration. We have learned from Perelman's mathematics. Perhaps we should also pause to reflect on ourselves and learn from Perelman's attitude toward life."

-- Bill Thurston, quoted in Michael Harris's "Mathematics Without Apologies"

Tuesday, June 9, 2015

Constructing Designs...

Writing for Quanta, Erica Klarreich reports today on a 150-year-old combinatoric problem and what Tim Gowers calls “a bit of an earthquake as far as design theory is concerned”:

Sunday, June 7, 2015

"The Exciting Stuff..."

This Sunday's reflection from Ronald L. Graham:
"The trouble with integers is that we have examined only the very small ones.  Maybe all the exciting stuff happens at really big numbers, ones we can't even begin to think about in any very definite way.  Our brains have evolved to get us out of the rain, find where the berries are, and keep us from getting killed.  Our brains did not evolve to help us grasp really large numbers or to look at things in a hundred thousand dimensions."