I’m in the middle of reading Andrew Crumey’s rather intruiging novel Mr Mee at the moment. One minor point of interest is that this may be the first work of fiction to contain a description of the Monty Hall problem (see Brian’s post below ) in the form of a letter, supposedly written in 1759 from a Jean-Bernard Rosier to the Encyclopedist d’Alembert:
Sir, you may know that many years ago one of our countrymen was taken prisoner in a remote and barren region of Asia noted only for the savagery of its inhabitants. The man’s captors, uncertain what to do with him, chose to settle the issue by means of a ring hidden beneath one of three wooden cups. If the prisoner could correctly guess which cup hid the gold band, he would be thrown out to face the dubious tenderness of the wolves; otherwise he was to be killed on the spot. By placing bets on the outcome, his cruel hosts could enjoy some brief diversion from the harsh austerity of their nomadic and brutal existence.
The leader of the tribe, having hidden his own ring, commanded that the unfortunate prisoner be brought forward to make his awful choice. After considerable hesitation, and perhaps a silent prayer, the wretch placed his trembling hand upon the middle cup. Bets were placed; then the leader, still wishing to prolong the painful moment of uncertainty which so delighted his audience, lifted the rightmost cup, beneath which no ring was found. The captive gave a gasp of hope, and amidst rising laughter from the crowd, the leader now reached for the left, saying that before turning it over he would allow his prisoner a final opportunity to change his choice. Imagine yourself to be in that poor man’s position, Monsieur D’Alembert, and tell me, what would you now do?
Anyone considering the Monty Hall problem might like to look at this photograph
http://www.fotolog.net/av_producer/?pid=8134635
So what is the reasonable answer for Monseiur D’Alembert? Let’s take it as given that using the mathematical and philosophical resources we now have at our disposal it is provable that swapping is the uniquely justified and justifiable play. The theory of probability was nowhere near as well developed back then as it was now, and from memory conditional probability was a particularly bad mess. Bayes’s paper which started sorting things out wasn’t published until 1764, so D’Alembert wouldn’t have had access to that. Now one could argue that the reasonable thing for D’Alember to have done would have been to figure out on the spot (and with the murderous tribe looking on!) the theory of conditional probability and then correctly apply it. But here’s a more interesting (if slightly vague) question.
Given just the resources available in 1759, is there a mathematical argument that swapping is the right play? I don’t know the answer, but I think such an argument would be quite tricky to come up with. (This being CT I bet someone will now produce such an argument in seconds…)
Not sure when the book you mention was published. Mark Haddons “The Curious Incident of the Dog in the Night-time” describes the Monty Hall Problem in some detail. Its about a child with aspergers syndrome. Its also wonderful
dermot
I think Crumey has priority, as his book came out in 2000. But yes, I have heard that Curious Incident is v. good and intend to get round to it.
Many times gambling is less about mathematics and probability than about correctly assessing the character of your opponent. The leader of the tribe presumably knows where he hid his own ring. Clearly, he wants to prolong the game. But why give you the option to change your mind? What would bring his men more pleasure – the thrill of victory or the agony of your defeat? Humans seem to take more pleasure in seeing others fail than they do in seeing others succeed. Which would be the greater agony – steadfastly making the wrong choice or knowing that you abandoned the correct choice because you panicked? For the brief moment before your death the tribe would revel in your agony. That’s what the leader wants. Stand your ground. And since you really don’t win either way…..
It seems like the Monty Hall problem can be explained without deploying any real mathematical framework. Perhaps I’m missing something? Here’s what I would argue:
In the case of the game show, one has a 1/3 chance of initially selecting the right door and a 2/3 chance of a wrong selection. Monty will never open the “right” door. Thus if you select a “wrong” door the other alternative he leaves is certain to have the car behind it — which is 2/3 of the time.
(If the tribal chief hadn’t initally known that the right was empty, but simply seized upon an opportunity for a twist on the game, nothing is changed. The situation is exactly as if he had known. The captive not knowing of the twist in advance, as the contestants presumably do, doesn’t change anything, either, as the initial choice is arbitrary and unaffected by such knowledge.)
…and I apologize for being redundant in saying “the other alternative.”
More difficult to explain, I think, is why this result seems counterintuitive. Initially each door has equal probability of hiding the car, and I suppose the assumption is that opening one door doesn’t change that the two other doors each have a 1/3 chance of hiding the car. Call it “probabilistic independence” or something.
But, in fact, the initial choice is a choice of which door to exclude from the information revelation round where Monty shows that one of two doors doesn’t have the car…
There is a further problem, which is that the story gives no clear indication that the chieftain was, like Monty Hall, sure to pick a cup that had no ring underneath it.
If you ignore this problem, you could do the whole thing without an explicit theory of conditional probabilities, just by counting, as follows.
Initially, there are three possible locations for the ring. Now consider the chief’s choices. If the ring was under the right or left cup, he has only one choice of cup to lift. If the ring is under the middle cup, the chief has two choices. So there are four cases.
By lifting the right cup, the chief eliminates one case, leaving only three, of which two are favorable to switching. Now a valid application of the principle of insufficient reason will do the job.
Of course, if you did this kind of exercise more than a few times, you’d be bound to discover conditional probability and Bayes theorem. Maybe we’ll discover new evidence that the good reverend was an active follower of 18th century game shows.
I’m not sure that’s right John - I mean the “no clear indication” bit. We’re told (a) that the leader had hidden the ring and (b) that he wished to prolong the painful moment of uncertainty.
Chris, at this point we may need to call in a professional epistemologist. Crumey has the fictional Rosier tell the fictional d’Alembert that the doubly-fictional chief wants to prolong the uncertainty, but Rosier gives no clear indication that the doubly-fictional French captive knows this.
Meanwhile, we, your readers, have been asked to place ourselves in the position of a pre-Bayesian d’Alembert and respond to Rosier. How should we treat the aside about the chief’s intentions?
Hre’s my response:
A Bayesian captive would presumably have a prior which placed weight on both a Monty-Hall chief and a random choice. Since there’s zero effect from switching if the chief had made a random choice, a dominance argument suggests acting as if the chief was Monty Hall.
The fact that no ring was under the cup lifted by the chief increases the posterior weight on the Monty hypothesis, but this is just a bonus.
But I wouldn’t like to try and formulate this part of the argument in anything other than Bayesian terms.
À Gauche
Jeremy Alder
Amaravati
Anggarrgoon
Audhumlan Conspiracy
H.E. Baber
Philip Blosser
Paul Broderick
Matt Brown
Diana Buccafurni
Brandon Butler
Keith Burgess-Jackson
Certain Doubts
David Chalmers
Noam Chomsky
The Conservative Philosopher
Desert Landscapes
Denis Dutton
David Efird
Karl Elliott
David Estlund
Experimental Philosophy
Fake Barn County
Kai von Fintel
Russell Arben Fox
Garden of Forking Paths
Roger Gathman
Michael Green
Scott Hagaman
Helen Habermann
David Hildebrand
John Holbo
Christopher Grau
Jonathan Ichikawa
Tom Irish
Michelle Jenkins
Adam Kotsko
Barry Lam
Language Hat
Language Log
Christian Lee
Brian Leiter
Stephen Lenhart
Clayton Littlejohn
Roderick T. Long
Joshua Macy
Mad Grad
Jonathan Martin
Matthew McGrattan
Marc Moffett
Geoffrey Nunberg
Orange Philosophy
Philosophy Carnival
Philosophy, et cetera
Philosophy of Art
Douglas Portmore
Philosophy from the 617 (moribund)
Jeremy Pierce
Punishment Theory
Geoff Pynn
Timothy Quigley (moribund?)
Conor Roddy
Sappho's Breathing
Anders Schoubye
Wolfgang Schwartz
Scribo
Michael Sevel
Tom Stoneham (moribund)
Adam Swenson
Peter Suber
Eddie Thomas
Joe Ulatowski
Bruce Umbaugh
What is the name ...
Matt Weiner
Will Wilkinson
Jessica Wilson
Young Hegelian
Richard Zach
Psychology
Donyell Coleman
Deborah Frisch
Milt Rosenberg
Tom Stafford
Law
Ann Althouse
Stephen Bainbridge
Jack Balkin
Douglass A. Berman
Francesca Bignami
BlunkettWatch
Jack Bogdanski
Paul L. Caron
Conglomerate
Jeff Cooper
Disability Law
Displacement of Concepts
Wayne Eastman
Eric Fink
Victor Fleischer (on hiatus)
Peter Friedman
Michael Froomkin
Bernard Hibbitts
Walter Hutchens
InstaPundit
Andis Kaulins
Lawmeme
Edward Lee
Karl-Friedrich Lenz
Larry Lessig
Mirror of Justice
Eric Muller
Nathan Oman
Opinio Juris
John Palfrey
Ken Parish
Punishment Theory
Larry Ribstein
The Right Coast
D. Gordon Smith
Lawrence Solum
Peter Tillers
Transatlantic Assembly
Lawrence Velvel
David Wagner
Kim Weatherall
Yale Constitution Society
Tun Yin
History
Blogenspiel
Timothy Burke
Rebunk
Naomi Chana
Chapati Mystery
Cliopatria
Juan Cole
Cranky Professor
Greg Daly
James Davila
Sherman Dorn
Michael Drout
Frog in a Well
Frogs and Ravens
Early Modern Notes
Evan Garcia
George Mason History bloggers
Ghost in the Machine
Rebecca Goetz
Invisible Adjunct (inactive)
Jason Kuznicki
Konrad Mitchell Lawson
Danny Loss
Liberty and Power
Danny Loss
Ether MacAllum Stewart
Pam Mack
Heather Mathews
James Meadway
Medieval Studies
H.D. Miller
Caleb McDaniel
Marc Mulholland
Received Ideas
Renaissance Weblog
Nathaniel Robinson
Jacob Remes (moribund?)
Christopher Sheil
Red Ted
Time Travelling Is Easy
Brian Ulrich
Shana Worthen
Computers/media/communication
Lauren Andreacchi (moribund)
Eric Behrens
Joseph Bosco
Danah Boyd
David Brake
Collin Brooke
Maximilian Dornseif (moribund)
Jeff Erickson
Ed Felten
Lance Fortnow
Louise Ferguson
Anne Galloway
Jason Gallo
Josh Greenberg
Alex Halavais
Sariel Har-Peled
Tracy Kennedy
Tim Lambert
Liz Lawley
Michael O'Foghlu
Jose Luis Orihuela (moribund)
Alex Pang
Sebastian Paquet
Fernando Pereira
Pink Bunny of Battle
Ranting Professors
Jay Rosen
Ken Rufo
Douglas Rushkoff
Vika Safrin
Rob Schaap (Blogorrhoea)
Frank Schaap
Robert A. Stewart
Suresh Venkatasubramanian
Ray Trygstad
Jill Walker
Phil Windley
Siva Vaidahyanathan
Anthropology
Kerim Friedman
Alex Golub
Martijn de Koning
Nicholas Packwood
Geography
Stentor Danielson
Benjamin Heumann
Scott Whitlock
Education
Edward Bilodeau
Jenny D.
Richard Kahn
Progressive Teachers
Kelvin Thompson (defunct?)
Mark Byron
Business administration
Michael Watkins (moribund)
Literature, language, culture
Mike Arnzen
Brandon Barr
Michael Berube
The Blogora
Colin Brayton
John Bruce
Miriam Burstein
Chris Cagle
Jean Chu
Hans Coppens
Tyler Curtain
Cultural Revolution
Terry Dean
Joseph Duemer
Flaschenpost
Kathleen Fitzpatrick
Jonathan Goodwin
Rachael Groner
Alison Hale
Household Opera
Dennis Jerz
Jason Jones
Miriam Jones
Matthew Kirschenbaum
Steven Krause
Lilliputian Lilith
Catherine Liu
John Lovas
Gerald Lucas
Making Contact
Barry Mauer
Erin O'Connor
Print Culture
Clancy Ratcliff
Matthias Rip
A.G. Rud
Amardeep Singh
Steve Shaviro
Thanks ... Zombie
Vera Tobin
Chuck Tryon
University Diaries
Classics
Michael Hendry
David Meadows
Religion
AKM Adam
Ryan Overbey
Telford Work (moribund)
Library Science
Norma Bruce
Music
Kyle Gann
ionarts
Tim Rutherford-Johnson
Greg Sandow
Scott Spiegelberg
Biology/Medicine
Pradeep Atluri
Bloviator
Anthony Cox
Susan Ferrari (moribund)
Amy Greenwood
La Di Da
John M. Lynch
Charles Murtaugh (moribund)
Paul Z. Myers
Respectful of Otters
Josh Rosenau
Universal Acid
Amity Wilczek (moribund)
Theodore Wong (moribund)
Physics/Applied Physics
Trish Amuntrud
Sean Carroll
Jacques Distler
Stephen Hsu
Irascible Professor
Andrew Jaffe
Michael Nielsen
Chad Orzel
String Coffee Table
Math/Statistics
Dead Parrots
Andrew Gelman
Christopher Genovese
Moment, Linger on
Jason Rosenhouse
Vlorbik
Peter Woit
Complex Systems
Petter Holme
Luis Rocha
Cosma Shalizi
Bill Tozier
Chemistry
"Keneth Miles"
Engineering
Zack Amjal
Chris Hall
University Administration
Frank Admissions (moribund?)
Architecture/Urban development
City Comforts (urban planning)
Unfolio
Panchromatica
Earth Sciences
Our Take
Who Knows?
Bitch Ph.D.
Just Tenured
Playing School
Professor Goose
This Academic Life
Other sources of information
Arts and Letters Daily
Boston Review
Imprints
Political Theory Daily Review
Science and Technology Daily Review