Here's The Winner In Our Pick-A-Number Game

Last week, we asked readers to participate in our experiment:

This is a guessing game. To play, pick a number between 0 and 100. The goal is to pick the number that's closest to half the average of all guesses.

So, for example, if the average of all guesses were 80, the winning number would be 40.

Thanks very much to the 15,322 people who entered.

We'll get to the winning number (and the winners) in a minute. But first, here's why we were interested in this game in the first place.

What's fun about this game is it forces you to think about how everybody else thinks. And then it forces you to think about how everybody else thinks about how everybody else thinks. Etc.

If you assume everybody else is just picking a number at random, you would expect the average guess to be 50. In that case, the winning number would be 25 — half of 50.

But if everybody is thinking that way, and everybody guesses 25, then then answer should be half of that — somewhere around 12.

Why stop there? You can use this kind of logic to keep cutting the number in half. So eventually you get down near zero.

You can see each of these strategies play out in the entries in our contest. We graphed all of the responses, and you see big spikes around 25, 12 and near zero.

The 15322 numbers listeners guessed i i
Jess Jiang/NPR
The 15322 numbers listeners guessed
Jess Jiang/NPR

(Here's a spreadsheet with every single guess.)

I talked with Ben Polak, a Yale economist who plays this game each year with his intro to game theory class. He said he likes the game because it shows the importance of strategic thinking — to think about not only what others are thinking, but what others think that you're thinking. (Like in Keynes's beauty contest, or this Friends episode.)

Polak guessed that the answer to our contest would be somewhere around 10. In other words, he's saying Planet Money readers think other Planet Money readers are rational — but not that rational.

Over at Cheap Talk, the economist Jeff Ely has a very nice discussion of this sort of experiment. He does a little math, a little handwaving, and guesses 16.

Ok. Here's the moment you've been waiting for:

The winning number in our experiment is 11.53

Five people guessed 11.5. They are our winners.

BJ
Amanda
Joel Togstad
Jami L. Bennett
Greg Cunningham

A few people gave decimal answers that were also pretty close — Bolivia Red, Spencer J Padilla, Steven Burt, Aaron Chinn, and Jasen.

Finally, an honorable mention goes to all 772 people who guessed 12 — the closest integer to the correct answer. Here's the full list:

61north
Aaron
Aaron
Aaron Couch
Aaron Rouby
ababcoc
Abbie
Abby
Abby Brody
Abby Thomas
Adam
Adam Sholom
Adam Snyder
Adam Webb
Adeline Basile
Aditya Dwivedula
Adriel
Aimee
Al Good
Al Lopez
Alan Goos
Alanna Turner
Alex
Alex Barrett
Alex Chaice
Alex Delgado
Alex Hopmann
Alex MacDonald
Alexa Squire
Alexander Vennebush
Aline Fader
Aliotsy Andrianarivo
Alison
Alison Klejna
Alissa Browning-Couch
Alix Grabowski
Allison
Alyssa
Amanda
Amanda Hannan
Amanda Holmes Duffy
Amanda Yates
Amber Huff
Amos
Amy Bitner
Amy Brandenburg
Amy Rossello
Amy S.
Amy Stroth
Amy Taylor
Andrea
Andrea Palmer
Andrew
Andrew
Andrew Conner
Andrew Hodges
Andrew J
Andrew McDaniel
Andrew Payzant
Andrew Peterson
Andrew Ramirez
Andy Cohen
Angie Boyter
Anissa Core
Anne
Annie Adams
Annie Clement
Anurag Saxena
Apple Kabob
April Neigut
April Stevenson
Armando Duran
Arnold Fox
Art Ryel-Lindsey
Ash Haeger
Ashley Trimble
Audrey
Avi
Avi
Avi Adler
Barb C,
Barb Kilgust
Barbara Berkley
Barbara Gooby
Barbara Langstaff
Barbara Targum
Becky LaFlamme
becky nelson
Beki S.
Ben A
Ben B
Ben Gladwell
Ben Luippold
Ben Thompson
Benjamin Morrill
Benjamin Reichstein
Beth
Beth Forbes
beth h.
Beth Meredith
Beth Ruedi
BHC
bill
Bill
Bill Palkowsky
Bill Siderski
Billie
Bobby!
Boris Mindzak
Brad Abernethy
Braden Mogler
Brandon
Brandon Leung
Bredan McGovern
Brenda Cooper
Brendan Simons
Brian
Brian Archibald
Brian Exelbierd
Brian Francis
Brian Hilburn
Brian Jacoby (Tallahassee, if there's more than one)
Brian Landberg
Brian Roberg
Brian Schermerhorn
brian weindling
britt
Britt Smith
Brittany Burkett
Brittany-Renee
Bronislava Lee
Bryan Newman
Buck
Byron Combs
Caitlin Rohrer
Calvin Gosla III
Cameron Gilroy
Cao Jingshu
Cara McClintock-Walsh
carina
Carl
Carla Kilgore
Carly Baxendale
Carole
Carolyn K.
CarterB
Casey Primel
cassandra lenhard
Cassia
Cassie
Catherine Merrbach
Catie
CB Jarvis
Charles Borgerding
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Chase P S
Chelan
Chelsea Paice
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chris
Chris
Chris Benham
Chris Burnett
chris hardee
chris moore
Chris Morin
Chris S.
Chrissy
Christie Jarrio
Christie Jett
Christina Hevko
christina steen
Christine Gunsaullus
Christopher Coleman
Christopher Wheat
Cindie Carroll-Pankhurst
Cindy
Cindy
Clark
Clayton Kinsler
Clifford Beshers
Clint
Collin
Corey Tadlock
Corina Jaisle
Cory Erlandson
Courtney Cudlip
Craig R.
Cris Whetstone
cucumber17
Cynthia
Cynthia Beard
Dale Borgeson
Dale Howard
Dali
Dan
Dan Cieplinski
Dan S
Daniel Fletcher
Daniel Michelson-Horowitz
Danny A
Danny Swan
Darryn
Dave Cooper
Dave Grace
Dave Hagen
Dave Kramberg
david
David Bauer
David G.
David Hilson
David Lakin
David Myer
David Onder
David Palchak
David Roberts
David robertson
David Schiffrin
David Tuman
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Dawn Leavy
denise
Derek Martinez
Destin perrin
Diana Hellman
Diane
Diane Carlson
diane h. james
Diane Johnson
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Dorothy
Doug Berry
Doug Cornelius
drew woods
Duane Wilkey
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Duncan Ellis
dustin skarda
Dusty Reed
Dylan J.J.
Ed Briscoe
Ed Gulczynski
Elizabeth
Elizabeth Fahs
elizabeth impossible
Ella
Emily Bull
Emily Shelton
Emily Watts
emma
Eric Kitterman
Eric McCormick
Eric The Awesome Nord
Eric Werner
Erik Johnsgard
Erin Richey
Erin Stutelberg
Evan N
Farid
Fernando Miranda-Mendoza
Francisco Melli-Huber
Frank McD
frank newell
fuzzless
Gabriel Martin
Gary Oshust
Genesis Ponce
Geoff Harrison
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George Hamzik
Gianna Borkhuis
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Gorg Huff
gough.bob
greg
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Greg Sand
Greg Schaeffer
Greg T
Gregory Lobo Jost
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Gwynne Ash
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Harlyn Pacheco
Harris Kalat
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Hoang
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Holly Burke
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Jacob Heidenreich
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James Pace
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Janice Bossart
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Jason
Jason
Jason Baek
Jason Blanchard
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Jason Mayer
Jason Robles
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Jay Feldman
Jayanth Rangaraju
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jean coggan
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Jeff Myers
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Jennifer Jula
Jennifer Little
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Jennifer Marzullo
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jenny hall
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Jesse
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Ji12
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John
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Joy Washington
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Justin Ropella
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katie
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Kevin Hunt
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Kristen
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Linda Nielsen
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Lisa
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Lynn
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M. Papadopoulos
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Martin Fisher
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MaryBeth
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Max Corchin
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Mike_L
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