From beloved math popularizer Ben Orlin comes a masterfully compiled collection of dozens of playable mathematical games.
This ultimate game chest draws on mathematical curios, childhood classics, and soon-to-be classics, each hand-chosen to be (1) fun, (2) thought-provoking, and (3) easy to play. With just paper, pens, and the occasional handful of coins, you and a partner can enjoy hours of fun—and hours of challenge.
Orlin’s sly humor, expansive knowledge, and so-bad-they’re-good drawings show us how simple rules summon our best thinking.
About these notes I pulled these excerpts for future reference and my own memory. If they interest you, please buy the book to support this lovely author.
All blue emphasis is mine.
Let’s begin with a riddle. What, exactly, distinguishes you from a chimpanzee?
Answer: The chimpanzee began as a baby chimp, then grew up, whereas you began as a baby chimp, then stayed that way.
Seriously, look at yourself: furless skin, tiny jaw, enormous rounded cranium-these are traits that our ape cousins lose as they age, yet you have stubbornly kept. No judgment; I've done it too. We humans retain childlike traits into adulthood, clinging to what biologist Stephen Jay Gould called "eternal youth." The technical term is neoteny, and among primates, it's kind of our calling card. The best part is that we don't just look like baby chimps. We also act like them; mimicking, exploring, puzzling - in a word, playing.
That, my baby-faced friends, is how we became the geniuses of the primate order. That's how we built our pyramids, left our footprints on the moon, and recorded our multiplatinum album Abbey Road. Not by outgrowing foolishness, but by refusing to. The secret to our brilliance is that we never stop learning, and the secret to our learning is that we never stop playing.
“Many animals display flexibility and play in childhood but follow rigidly programmed patterns as adults” wrote Stephen Jay Gould. As a math teacher, I admit that our math lessons often seem designed for some other animal, one of those rigid pattern-followers. Perhaps termites. No surprise that such lessons capture our thinking at its worst: paralyzed, plodding, anxious. For this book, forget all that and summon your true nature, your inner body.
What are the rules?
- This book tackles a specific and uniquely human kind of play: games, also known as "play with rules." They range from those with myriad rules (like Monopoly) to those with just one (like "the floor is lava"), from occasions of merciless competition (like Monopoly) to ones of profound collaboration (like "the floor is lava"), from the worst artifacts of human culture (like Monopoly) to the best (like "the floor is lava"). For this book, I have sought games with simple, elegant rules that give rise to rich, complex play. You know the saying: "a minute to learn, a lifetime to master."
- What do you mean by "mathematical games?" Good question. I first fielded it from Vito Sauro, one of Minnesota's friendliest experts in tabletop gaming. Almost every board game, he pointed out, consists of a thematic skin over a mathematical skeleton. Would my book attempt to cover all games that have ever existed? No, no, I told Vito. A mathematical game, by my definition, is one that makes you go Mmm-mmm, that's mathy. Vito considered this (1) a total non-answer, and (2) fairly satisfactory. In any case, I've tried to compile timeless games of logic, strategy, and spatial reasoning. My three criteria were: (1) fun, (2) easy to play, and (3) mathematically thought-provoking.
- The book has five sections:
- Spatial Games
- Number Games
- Combination Games
- Games of Risk and Reward
- Information Games.
There's an element of whimsy to these classifications. Don't think of them as a perfect taxonomy, with each specimen filed away. They're more a kind of mood lighting, highlighting distinctive features. For example, chess could comfortably belong to any of the five categories- though it'd look a little different under each light. Each section begins with a playful essay on the relevant branch of mathematics. After that come five featured games, roughly increasing in complexity (though each new section is a fresh start). Last is a chapter of miscellaneous games described in brief (including some of my very favorites).
4. Each featured game follows the same structure.
- First, in How To PLAY, I'll lay out the mechanics, including what you need, the goal, and the rules.
- Second, in TASTING NOTES, I'll elaborate on the flavor of the gameplay, the elusive je ne sais quoi. You may glean a few strategic tips, but that's not my aim. I'm focused on teasing out the subtle hues and shades of mathematical play, a variety so exquisite that it makes wine look like sad old grape juice.
- Third, in WHERE IT COMES FROM, I'll tell you what I know of the game's origins. Some are ancient and timeless, some are silly and novel, and some are both at once (don't ask me how that works, it just does).
- Fourth, in WHY IT MATTERS, I'll tell you why this game brings out the best in human thought. Maybe it models the quantum structure of matter. Maybe it reveals the austere beauty of topology or the cold logic of gerrymandering. Maybe it unlocks your inner genius or, better yet, your inner chimpanzee. In any case, I view this as the chapter's crux, and the book's driving purpose.
TASTING NOTES FOR THIS BOOK
You can, if you like, treat this as an ordinary work of nonfiction. Turn the pages. Smile politely at the jokes. Mutter to yourself, "Wow, what bad drawings. I sure am getting my money's worth." Moving chapter to chapter, front to back, game to game, you'll have a perfectly pleasant time. You'll also miss all the fun. This book is meant to be played. A human playing with math is like an elephant playing with its trunk, a bird with its wings, or a Batman with its fancy car. It's a creature doing what it was born to do. Your capacity for mathematical thinking is a gift of extraordinary dimensions, a force unmatched in the animal kingdom, except perhaps by a cat's capacity for contempt. Please don't leave this evolutionary present in its wrapping paper. Take it out. Play with it. Or at least, like a cat, play with the wrapping paper. Most of the games are multiplayer. I hope you find a playmate who shares your curiosity, with whom you can prod and poke the games, analyzing as you go. "Only dead mathematics can be taught where competition prevails," said the mathematician Mary Everest Boole. "Living mathematics must always be a communal possession." The way I see it, even competitive games are collaborative projects, in which minds unite to build extraordinary chains of logic and strategy. David Bronstein called this "thinking for two." Karl Menninger called it "a progressive interpenetration of minds." I call it "play." All that said, this is a book, and I do hope you read it. Each game shines light on a deep truth about mathematics, from the combinatorial explosion to information theory. Those mathematical truths, in turn, shine light on the games. If that all sounds blindingly bright, don't worry, your eyes will adjust. As the cleric Charles Caleb Colton once wrote, "The study of mathematics, like the Nile, begins in minuteness but ends in magnificence."
WHY MATHEMATICAL GAMES MATTER
Because they bring out the best in human thought. In 1654, a gambler wrote to two mathematicians to share a puzzle. It goes like this: Imagine two people playing a simple 50/50 game of chance, scoring 1 point per round. The first to 7 points wins $100. But partway through, with one player leading 6 to 4, the game is interrupted. What's the fair way to split the prize?
The two mathematicians, Blaise Pascal and Pierre de Fermat, solved the problem, and better yet, their solution helped to birth the mathematical study of uncertainty, known today as probability theory.
A fundamental tool of modernity, born from a simple puzzle about a game of chance.
Here's another true story. On Sunday afternoons in the 1700s, the citizens of Königsberg (known today as Kaliningrad) enjoyed wandering the four regions of their riverside city, aiming to cross the seven bridges the Blacksmiths, the Connecting, the Green, the Merchant's, the Wooden, the High, and the Honey exactly once each. No one could manage it. Then, in 1735, the mathematician Leonhard Euler proved why: It was impossible. No such path could exist. Today, we recognize his proof as the dawn of graph theory, the study of networks, which underlies everything from social media to internet search algorithms to epidemiology. Google, Facebook, and the fight against
Need another example? How about John Horton Conway, a larger-than-life mathematician who passed away while I was writing this book. He explored math in all its staggering variety, from cellular automata to abstract algebra. Yet he kept returning, again and again, to games. His favorite discovery was the surreal numbers, which encoded the structure of two-player games into a numerical system. His most heralded (and thus least favorite) discovery showed how worldlike complexity could arise from a few simple rules; it was called the Game of Life.
"I was struck," writes mathematician and admirer Jim Propp, "by the way his ideas about games turned out to play a role in his work on lattices, codes, and packings... What are the chances that a mathematician who loved games would have the luck to find that games secretly underlie other subjects he studies?"
I could go on a weekly poker night inspired John von Neumann to develop game theory, whose strategic insights now penetrate ecology. diplomacy, and economics - but I'm not here to worship applications. I don't really care that mathematical play has helped to mint billionaires or to create trillions of dollars of wealth. My point is that mathematical play does this as an accidental by-product.
See my post Learning To Appreciate Learning , Khe Hy’s telic vs atelic activities, or this section by Venkatesh Rao contrasting appreciative vs instrumental modes: Appreciative worldviews, which are at the heart of guru factors, emerging via accumulation of appreciative knowledge, a term due to urbanist John Friedman, who defines it in his book Planning in the Public Domain as follows: The social validation of knowledge through mastery of the world puts the stress on manipulative knowledge. But knowledge can also serve another purpose, which is the construction of satisfying images of the world. Such knowledge, which is pursued primarily for the worldview that it opens up, may be called appreciative knowledge. Contemplation and creation of symbolic forms continue to be pursued as ways of knowing about the world, but because they are not immediately useful, they are not validated socially and are treated as merely private concerns or entertainment. Friedman uses the term manipulative knowledge in opposition to appreciative, but he doesn’t mean manipulative in a Machiavellian sense. He simply means knowledge of how to actually do things to drive change in the world, accumulated by actually trying to do those things.
When you look up from your game and realize you've inadvertently changed the course of human history, you know you're playing with a special kind of fire. "All good thinking is play," writes Mason Hartman. She means that our best thoughts explore ideas the way a baby chimp explores the forest, with a kind of freedom and abandon. It's not a game of Parcheesi, with every move geared toward victory; rather, it's a game of make-believe, a game of "yes, and ...," a game passed from generation to generation, a torch that never goes dark. "A finite game is played for the purpose of winning," wrote James Carse, "an infinite game for the purpose of continuing to play."
We often see mathematics as a series of finite games. Questions to answer. Puzzles to solve. Theorems to prove. But taken together, they form a vast and never-ending game, encompassing the thoughts of every sentient ape. "I love mathematics," said mathematician Rózsa Péter, "because man has breathed his spirit of play into it, and because it has given him his greatest game-the encompassing of the infinite."
Personally, I say man's greatest game is "the floor is lava," but I get a out of encompassing the infinite now and then. I cordially invite you to join me.
A Game of Risk and Reward
AN UNCERTAIN TRIVIA GAME FOR AN UNCERTAIN WORLD
I enjoy trivia games: the camaraderie, the tension, the chips, the salsa...all of it, really, except the peaky part where I need to know things. Outrangeous is a game for folks like me.
You answer each question: ie “How many apostles did Jesus have?" not with a specific number, but with a range. Miss the truth (e.g., "50 to 100"), and you score no points (hence, "out-range-ous"). Capture the truth, and you score more points based on how narrow your range is (so "10 to 13" beats "11 to 18"). In the end, the game isn't about how much you know. It's about recognizing what you don't.
HOW TO PLAY
- What do you need?
Four to eight players (though you can make do with three.) Also, pencils, paper, and at least for the first few minutes the internet. Before beginning, have everybody take five minutes to come up with a few trivia questions whose answers are (a) numbers and (b) easily googled. So, in this game we just sit around on our phones? Only at the start.
- What's the goal?
Each answer is a number. You'll guess a range of values. trying to make it as narrow as possible while still including the true answer.
- What are the rules?
- One player — the judge for the round — announces the trivia question. The other players act as guessers, each secretly writing down a range of values.
- When everyone has committed their answer to paper, the guesses are revealed. The goal is to capture the true value, while keeping your range as narrow as possible.
- The judge reveals the true answer. Anyone who missed the answer- no matter how painfully close they came-receives 0 points. Instead, the judge receives 1 point per wrong guess, as a reward.
- Then, among the players with the correct answer, order them from the narrowest range (i.e., most impressive guess) to the widest range (i.e. least impressive guess.)
- These players receive 1 point per guesser that they beat. Note they all beat anyone who missed the answer
- Play enough rounds so that each person has an equal number of turns as judge. In the end, whoever has the most points is the winner.
WHERE IT COMES FROM
In Douglas Hubbard's How to Measure Anything, I came across 10 Outrangeous-style questions, along with an instruction: Make each range wide enough that you're 90% confident of capturing the true answer. That's 90% exactly: no more, no less. As a math teacher, a probability aficionado, and (as my siblings describe me) "a robot," I felt positive I'd nail it. I'd be 90% accurate, missing one of the 10. Maybe zero or two, depending on my luck. Instead, I missed four. Watching my surefire A- pale into a D- prompted a small crisis of confidence. As it should have, because my confidence was the whole problem. My confidence had slipped its leash and was now running amok, barking at squirrels, chasing traffic. How could I trust myself to calculate life's risks and rewards knowing I had such an inflated sense of my own powers? Inspired and chastened, I developed Outrageous as a classroom game. Other folks have independently developed the same concept.
[Kris: this was one of the first games I played as a trainee at SIG and I still use it today in teaching contexts around trading. It is the essence of market-making where you often don’t know the right answer but need to have strong meta-knowledge of what it could be. If you make markets too tight, you will get all the market share and be sad. If you make them too wide, you’ll never trade and never profit.]
WHY IT MATTERS
Because to take calculated risks, you must know the limits of your own calculations. Humans are not perfect. Your view of the world, just like mine, is a simmering mix of fact and fiction, history and myth, "tomato is a fruit” and "who am I kidding; you can't put tomato in a fruit salad." The question isn’t whether my beliefs are true or false. I have true and false beliefs, both in abundance. The question is whether I can tell the two apart and the sorry reality is most of us can't. We carry all of our opinions, right and wrong alike, with a swashbuckling, wholly unearned confidence.
In a classic study, psychologists Pauline Adams and Joe Adams quizzed subjects on how to spell some tricky words and asked them to rate their confidence in each. Occasionally, folks would say "100%." That means deadlock certainty. Total guarantee. If I created a YouTube supercut of every time you've ever claimed 100% certainty, it should include exactly zero cases in which you were wrong.
Instead, on such 100% answers, the study found a 20% rate of error. "I'm absolutely positive and would bet my cat's life on it" translates to "Eh, call it four out of five."
A little overconfidence isn't a crime, at least not in most jurisdictions. It can even help, by giving us the courage to start an ambitious yet likely-to-fail project, such as writing a novel, running for political office, or zero. Still, whenever humans work together, we need to share our knowledge. That's a doomed endeavor if nobody can distinguish their knowledge from their ignorance. What's the point of pooling our money if we can't tell the real bills from the counterfeits?
Luckily, a noble few have learned to navigate these dark tunnels of uncertainty. They are called statisticians, and they will tell you, in no uncertain terms, that nothing is truly certain.
Imagine a study that finds the average American thinks about cheese 14.2 times per day. No matter how careful the researchers, or how tantalizing the Gruyère, there remains a modicum of doubt. Perhaps the true answer is a little lower (because we polled an unusually cheese-loving sample) or a little higher (because our subjects were unusually cheese-averse). The solution is a confidence interval. Or better yet, a collection of them.
Such intervals embody an inherent trade-off. You can give a narrow, precise range. Or you can give a wide range that's almost certain to capture the truth. But you can't do both at once.
The tighter the range, the greater the risk of missing the mark. Outrangeous demands the same trade-off. You can give a narrow range, which might garner a lot of points. Or you can give a wide range, upping your chances to score at least some points. You just can't do both at once. To execute either strategy, you need to pursue a rarefied psychological state: good calibration. This means that your confidence matches your accuracy. When you feel 90% confident, you're right 90% of the time. When you feel 50% confident, you're right 50% of the time. You say what you mean and mean what you say. Subjective feeling aligns with objective success.
[Kris: see my post Calibrated Confidence which explores this topic and how to improve. You can also take a calibration test!]
To be clear, good calibration is a narrow virtue. If you're 50% confident that sharks are fish (true) and 50% confident that prairie dogs are fish (not so much), then you're well-calibrated, but a fool. Meanwhile, if you're 5% confident that testing your bomb will extinguish life on earth, yet you shrug and start the countdown, then you may or may not be well calibrated, but you're definitely a monster.
Good calibration isn't sufficient for good judgment be necessary. Games like Outrangeous offer a unique window into your calibration and a training ground for improving it. But it may very well be necessary.
When my wife was in grad school for math, we'd team up with some friends from her program to play bar trivia on Thursday nights. Our team won every week, and usually in the same way: by hanging close in the themed rounds (sports, geography, music, etc.), then surging to victory in the final general knowledge round. This posed bit of a mystery. If another team outperformed us on say, history, science, and film, then shouldn't they beat us on general knowledge, too?
I eventually developed a theory of our strange success. During themed rounds, a team can hand their answer sheet to the relevant specialist-the sports fan, the music expert, the geography whiz-and defer to them. But in general knowledge, no expert reigns. Everyone weighs in. You'll soon have four or five suggestions, one of them probably right. How do you know which? How can the group settle on the correct answer, rather than the most overconfident?
This is where the mathematicians shone. Mathematical research forces you to distinguish carefully between airtight knowledge, credible belief, plausible hunch, and blind guesswork. Our teammates never fought for an answer just because it was their own. Instead, the truth would rise to the top. The mathematicians were well-calibrated.
That's my belief, anyway. It's also possible that by the final round, everyone else was drunk, while the mathematicians held their liquor better. As with anything, I'll never be 100% sure.
VARIATIONS AND RELATED GAMES
- RATIO SCORING Say we're guessing the distance to the moon. I put "3,000 miles to 300,000 miles," while you put "100,000 to 400,000 miles." We both get it right (the truth being 239,000). And, per the rules, my range is a bit narrower. But was mine really the better guess? My lower bound suggests that the moon and Earth might be closer together than New York and London. Your guess seems far more sensible. Shouldn't it score more highly? The solution: divide rather than subtract. That is, calculate a ratio, rather than a width. Here, my ratio is 100 (that's 300,000 divided by 3,000), while yours is just 4 (from 400,000 divided by 100,000). Your guess is far more precise. I recommend this scoring system for questions where ranges may span several orders of magnitude (e.g., "number of slot machines in Las Vegas"). For more restricted ranges (e.g., the age of a particular celebrity), the original scoring system works fine.
- THE KNOW-NOTHING TRIVIA GAME Years ago, in the course of a long plane flight, the mathematician Jim Propp and two friends invented this strange jewel of a game. It's almost a contradiction in terms: a trivia game you can play without ever finding out the answers. It works for any odd number of players. Take turns coming up with a numerical trivia question (e.g., "How many home runs did Barry Bonds hit in his career?"); then all of you (including the question asker) write down a secret guess. When the guesses are revealed, the winner is whoever's guess is in the middle. For example, if the three guesses are 900, 790, and 2,000, then the person who guessed 900 is the winner. Never mind that the truth is 762. You're not trying to guess the right answer, but the answer that will land between your friends' answers (though in practice that usually means just giving it your best guess).
- Clusters around how deep respondents took the logic
- First order: guess of 33 (based on avg guess of 50)
- Infinite Regress: guess of zero!]
[Kris: This reminds me of the Keynes Beauty Contest game. MobLab’s version of the beauty contest has simple rules. Students pick an integer from 0-100 with the goal of picking the number that is closest to 67% (2/3) of the average of all guesses.
Superforecasting talks about how the average was 18.9 (for a winning answer of about 13). We confirmed this in the Stock Slam sample of about 50 people
ADVICE FOR WRITING QUESTIONS
- Spend 10 minutes on Google and/or Wikipedia before the game begins, so that by the time it's your turn to judge, you've got two or three questions ready to roll.
- Play to your audience. Absurdly hard questions are no good; everyone just shrugs and gives a very wide range. The best questions are you don't know the answer, but feel like you should.
- Phrase questions as precisely as possible. Where relevant, specify tantalizing: units ("distance in miles"), dates ("population as of 2019), and sources ("the film's budget according to Wikipedia").
- Here are some suggestions. You can also use these to inspire other ideas just swap in a different celebrity/place/world record/piece of pop culture.
- Age of Jamie Foxx
- Age at which Abraham Lincoln died
- Age of the oldest-ever manatee
- Amount of money Judge Judy makes per year
- Current day of the month (without looking)
- Distance to the moon in miles -Distance from NYC to LA (as crow flies)
- Height of the tallest-ever ice cream cone
- Height of the tallest-ever
- WNBA player
- Hottest land temperature ever recorded
- Length of "Bohemian Rhapsody"
- Length of Canada's coastline
- Length of every Simpsons episode ever, if watched back to back to back
- Length of Nelson Mandela's prison term
- Length of the longest fingernails ever
- NBA season record for rebounds per game
- NFL single-season record for most interceptions thrown
- Number of episodes of Sesame Street
- Number of in-ground pools in Texas
- Number of lakes in Minnesota
- Number of goldfish crackers (out of 10) that I will successfully toss into this bowl from a distance of six feet
- Number of novels by Agatha Christie
- Number of species of penguin
- Number of bird species that can fly backward
- Number of studio albums by Jennifer Lopez
- Number of US states with wild alligators
- Number of words in Hamlet's "to be or not to be" soliloquy
- Percentage of the presidential vote won by Ross Perot in 1992
- Percentage of US adults that believe chocolate milk comes from brown cows
- Percentage of the US that identifies as male
- Population of Atlantic puffins worldwide
- Population of South America
Games Of Information
Bullseyes and Close Calls
A CLASSIC CODE-BREAKING GAME
Under the name Mastermind, this became one of the biggest board games of the 1970s, selling as many units as The Godfather sold tickets (about 50 million, if you're keeping score). But the game didn't begin with those colorful plastic pegs. For a century beforehand, it was played using pen and paper, under the earthy name Bulls and Cows. Now, as an ardent bovine feminist, I reject the idea that bulls are better than cows, so I've renamed the former as Bullseyes and the latter as Close Calls. But feel free to use whatever words you wish. This code game, under any code name, remains a stone-code classic.
[Kris: I’ve recommended Mastermind before. See Fun Ways To Teach Your Kids Encryption]
HOW TO PLAY
- What do you need? Two players, pens, and paper.
- What's the goal? Guess your opponent's secret number before they guess yours.
- What are the rules?
- Each player writes down a secret four-digit number. All of the digits must be different
- Take turns guessing four-digit numbers. (Again, no repeating digits in a number.) Your opponent will report how many of your digits were bullseyes (right digit, right position), and how many were close calls (right digit, wrong position). However, you will not be told which digits were which.
- The winner is whoever scores four bullseyes in the fewest guesses.
WHY IT MATTERS
Because life is a hunt for information, and humans are lazy hunters. You know this already. Either you're human yourself, or you're conversant enough in human culture to enjoy human books like this one. Either way, you've seen Homo sapiens spend hours gorging on information, then somehow emerge from the feast without an ounce of nourishment.
Take a wretched and typical specimen: me. I subscribe to 77 podcasts, follow 600 people on Twitter, and long ago maxed out the number of open in my Wikipedia phone app. Given all this information, how informed am I? The other day my young daughter picked up a pinecone. "That's a pinecone," I volunteered. "It comes from a pine tree. It's... some kind of big seed, I think?" This wasn't a tough pitch to hit. My daughter had not picked up a quasar, a Tom Stoppard play, or the hard problem of consciousness. The truth about pinecones is definitely out there. I just didn't know it. Nine words in, I had exhausted my knowledge.
As a rule, humans don't seek information in the right places. In a classic psychology study, subjects were shown four cards, each with a letter on one side and a number on the other. Then they learned a rule: A card with a vowel must also have an even number.
The question: Which cards do you need to flip over to see if the rule has been violated?
Before reading further, think it over. What would you flip? If you prefer to copy off of other people's homework, here are the most common answers from a typical iteration of the study, conducted in 1971:
So the consensus: definitely A... and maybe 4. It's pretty clear that we need to flip A. But after that, controversy begins. Most people want to flip 4, presumably in order to check for a vowel. But suppose you find J or W or P-who cares? No violation there. The rule said that a vowel must have an even number; it didn't forbid a consonant from having one, too.
Meanwhile, most folks decline to flip 7. It's not even, so the rule is irrelevant, right? Wrong. If you flip 7, and find an E or U, then the rule has been broken.
The study highlights a pattern called confirmation bias. Instead of seeking examples that could challenge our theories, we seek examples to confirm them. Confirmation bias is often blamed on emotion. If I believe Demublicans to be paragons of civic virtue and Republocrats to be monstrous hypocrites, then confirmatory examples will make me feel righteous and superior, while counterexamples will make me feel anxious and persecuted. No surprise which one I seek out.
Still, though emotion plays a role, confirmation bias runs deeper. In the four-card study, folks have no emotional stake in an abstract rule about vowels. There's no benefit to being wrong. Still, 96% fail to give the logically correct answer. By habit, we look for information in the wrong places, like a space program sending probes to the wrong planets.
Most wrong beliefs cost you nothing. Flat earthers can still buy plane tickets; doubters of the moon landing can still stargaze; you can dislike Outkast and still live an arguably happy life. By contrast, Bullseyes and Close Calls will call you on your crap. Ask a worthless question, get a worthless answer. Instead of letting information wash over us, we must seek it out, probing the world with a sense of purpose-and maybe, when the over, looking up what pinecones actually are. is game.
AN AUCTION GAME
I'm afraid I can't teach you how to win at Caveat Emptor. But I can easily tell you the best way to lose: Just win every auction. I mean it. Play a few rounds, and you'll find that overbidding is all too common. It's a game marked by Pyrrhic victories, with winners forced to take home prizes for more than they're worth. This phenomenon-losing your shirt on a winning bid is so pervasive that auction economists have dubbed it "the winner's curse."
Lucky for you, Caveat Emptor offers a lot more information than the typical auction. Will that be enough for you to escape the curse?
HOW TO PLAY
- What do you need? Two to eight players; it's best with four to six. Spend a few minutes gathering five random household objects to auction off. Each person also needs six cards, numbered 1 to 6. (Scraps of paper will work.) These cards are not used in bidding; instead, they're used to determine each item's secret "true value." On another sheet, set up a table on which to track each player's score and the cards that they've used.
- What's the goal? Win household items at auction, but don't pay more than they're worth.
- What are the rules?
- Each round, one of the players the auctioneer-picks an item and gives a little speech about how delightful and valuable it is.
- Now it's time to determine the true value of the item. To do this, every player (including the auctioneer) secretly chooses a value from 1 to 6. The sum of these values-which no one yet knows is the true value of the item on auction.
- Next, it's time for bidding. Each player hopes to buy the item for less than its true value. Bidding begins with the player to the left of the auctioneer, who states a price they'd be willing to pay for the item.
- Bidding continues to the left. On your turn, you must either raise the bid or drop out of the auction. When you drop out, reveal the value you selected. Thus, as players drop out, the remaining players gain information about the item's true value.
- If you are the last player remaining, then you win the auction at the price of your last bid. Reveal your own number. The item's true value is now known to all.
- Subtract the price paid from the item's true value, and score this many points, which may be negative, for the "winner" of the auction. Also, whatever value you selected, you cannot use again. Discard those scraps of paper and cross out the corresponding numbers on the scoring table. (You can also accomplish the same result by keeping discarded scraps faceup and visible to everyone.)
- Play five rounds, changing the auctioneer each time. It's okay if not everyone gets the same number of chances as auctioneer. In the end, the highest total score wins.
My favorite part of the game is the speeches. I have been roused to the admiration of a broken pencil and brought to the brink of tears by a peanut-buttered cracker. It seems that everyone becomes a poet when asked to eulogize a coupon for 20% off at Bed Bath & Beyond.
WHY IT MATTERS
Because everything has a price, and auction winners often overshoot it.
We live in a world on auction. Photographs have been auctioned for $5 million, watches for $25 million, cars for $50 million, and (thanks to the advent of non-fungible tokens) jpegs for $69 million. Google auctions off ads on search terms, the US government auctions off bands of the electromagnetic spectrum, and in 2017, a painting of Jesus crossing his fingers fetched $450 million at auction. Before we dub this the worst-ever use of half a billion dollars, remember two things: (1) The human race spent $528 million on tickets to The Boss Baby, and (2) it's a notorious truth about auctions that the winner often overpays.
Why does this winner's curse exist? After all, under the right conditions, we're pretty sharp at estimation. Case in point: In the early history of statistics, 787 people at a county fair attempted to guess the weight of an ox. These were not oxen experts. They were not master weight guessers. They were ordinary, fair going folks. Yet somehow their average guess (1,207 pounds) came within 1% of the truth (1,198 pounds). Impressive stuff. Did you catch the key word, though? Average. Individual guesses landed all over the map, some wildly high, some absurdly low. It took aggregating the data into a single numerical average to reveal the wisdom of the crowd.
[Kris: From Superforcasting]
How The Wisdom of Crowds Works
- Bits of useful and useless information are distributed throughout a crowd. The useful information all points to a reasonably accurate consensus while the useless information sometimes overshoots and sometime undershoots but critically…cancels out.
- Aggregation works best when the people making judgments have a lot of knowledge about many things.
- Aggregations of aggregations or “polls of polls” can also yield impressive results. That's how foxes think. They pull together information from diverse sources. The metaphor Tetlock uses is they see with a multi-faceted dragonfly's eye.
Now, when you bid at an auction — specifically, on an item desired for its exchange value not for sentimental or personal reasons — you are in effect estimating its value. So is every other bidder. Thus, the true value ought to fall pretty close to the average bid. Here's the thing: Average bids don't win. Items go to the highest bidder, at a price of $1 more than whatever the second highest bidder was willing to pay. The second-highest bidder probably overbid, just as the second-highest guesser probably overestimated the ox's weight.
To be sure, not all winners are cursed. In many cases, your bid isn't an estimate of an unknown value but a declaration of the item's personal value to you. In that light, the winner is simply the one who values the item most highly. No curse there.
But other occasions come much closer to Caveat Emptor: The item has a single true value which no one knows precisely and everyone is trying to estimate.
[Kris: In Recipe For Overpaying I note investor Chris Schindler's intuitive explanation for why high volatility assets exhibit lower forward returns: a large dispersion of opinion leads to overpaying. He points to private markets where you cannot short a company. The most optimistic opinion of a company’s prospects will set the price.]
- Liar’s Dice
- Liar’s Poker