Gladiators, Pirates and Games of Trust Read online

  Haim Shapira was born in Lithuania in 1962. In 1977 he emigrated to Israel, where he earned a PhD in mathematical genetics for his dissertation on Game Theory and another PhD for his research on the mathematical and philosophical approaches to infinity. He now teaches mathematics, psychology, philosophy and literature. He is an author of seven bestselling books. His stated mission as a writer is not to try to make his readers agree with him, but simply to encourage them to enjoy thinking. One of Israel’s most popular and soughtafter speakers, he lectures on creativity and strategic thinking, existential philosophy and philosophy in children’s literature, happiness and optimism, nonsense and insanity, imagination and the meaning of meaning, as well as friendship and love. He is also an accomplished pianist and an avid collector of anything beautiful.

  FROM THE SAME AUTHOR:

  Conversations on Game Theory

  Things that Matter

  Infinity: The Neverending Story

  Ecclesiastes: The Biblical Philosopher

  Nocturnal Musings

  A Book of Love

  Happiness and Other Small Things of Absolute Importance

  CONTENTS

  Introduction

  Chapter 1 The Diner’s Dilemma

  (How to Lose Many Friends Really Fast)

  Chapter 2 The Blackmailer’s Paradox

  Chapter 3 The Ultimatum Game

  Chapter 4 Games People Play

  Spotlight The Keynesian Beauty Contest

  Chapter 5 The Marriage Broker

  (A Little on the Connections between the

  Nash Equilibrium, Buffaloes, Matchmaking and the Nobel Prize)

  Intermezzo The Gladiators Game

  Chapter 6 he Godfather and the Prisoner’s Dilemma

  Chapter 7 Penguin Mathematics

  Intermezzo The Raven Paradox

  Chapter 8 Going, Going … Gone!

  (A Brief Introduction to Auction Theory)

  Intermezzo The Newcomb Paradox

  Chapter 9 The Chicken Game and the Cuban Missile Crisis

  Chapter 10 Lies, Damned Lies and Statistics

  Chapter 11 Against All Odds

  Chapter 12 On Fairly Sharing a Burden

  Chapter 13 Trust Games

  Chapter 14 How to Gamble If You Must

  Conclusion Game Theory Guidelines

  Reference Notes

  Bibliography

  INTRODUCTION

  This book deals with Game Theory, introducing some important ideas about probabilities and statistics. These three fields of thought constitute the scientific foundation of the way we make decisions in life. Although these topics are quite serious, I’ve made a tremendous effort not to be boring and to write a book that’s rigorous and amusing. After all, enjoying life is just as important as learning.

  And so, in this book we will

  • Meet the Nobel Prize laureate John F Nash and familiarize ourselves with his celebrated equilibrium

  • Learn the basic ideas of the art of negotiation

  • Review every aspect of the Prisoner’s Dilemma and learn about the importance of cooperation

  • Introduce the world champion in strategic thinking

  • Examine the Stable Marriage Problem and find out how it led to a Nobel Prize

  • Visit a gladiators’ ring and apply for a coaching position

  • Bid in a tender at auction and hope to avoid the Winner’s Curse

  • Learn how statistics bolster lies

  • Become acquainted with the presence of probabilities in operating theatres

  • Discover what the game of Chicken had to do with the Cuban missile crisis

  • Build an airport and divide an inheritance

  • Issue ultimatums and learn to trust

  • Partake in John Maynard Keynes’s beauty competition and study its association with stock trading

  • Discuss the concept of justice as seen through the eyes of Game Theory

  • Meet Captain Jack Sparrow and find out how democratic pirates divide their treasures

  • Find optimal strategies for playing at roulette tables

  Chapter 1

  THE DINER’S DILEMMA

  (How to Lose Many Friends Really Fast)

  In this chapter we’ll visit a bistro in order to find out what Game Theory is all about and why it’s so important. I’ll also provide many examples of Game Theory in our daily lives.

  Imagine the following situation: Tom goes to a bistro, sits down, looks at the menu, and realizes that they serve his favourite dish: Tournedos Rossini. Attributed to the great Italian composer Gioachino Rossini, it’s made of beef tournedos (filet mignon) pan-fried in butter, served on a crouton, and topped with a slice of foie gras, garnished with slices of black truffle, and finished with Madeira demi-glace. In short, it has everything you need to help your heart surgeon make a fine living. It’s a very tasty dish indeed, but it’s very expensive too. Suppose it costs $200. Now Tom must decide: to order or not to order. This may sound very dramatic, Shakespearean even, but not really a hard decision to make. All Tom needs to do is decide whether the pleasure the dish will give him is worth the quoted price. Just remember, $200 means different things to different

  people. For a street beggar, it’s a fortune; but if you were to put $200 into Bill Gates’s account, it wouldn’t make any kind of difference. In any event, this is a relatively simple decision to make, and has nothing to do with Game Theory.

  Why, then, am I telling you this story? How does Game Theory fit here?

  This is how. Suppose Tom isn’t alone. He goes to the same bistro with nine friends, making a total of 10 around the table, and they all agree not to go Dutch, but to split the bill evenly. Tom then waits politely until everyone has ordered their simple dishes: home fries; a cheese burger; just coffee; a soda; nothing for me, thanks; hot chocolate; and so on. When they are done, Tom is struck by an ingenious idea and drops the bomb: Tournedos Rossini for me, per favore. His decision seems very simple and both economically and strategically sound: he treats himself to Rossini’s gourmet opera and pays just over 10 per cent of its advertised price.

  Did Tom make the right choice? Was it really such a great idea after all? What do you think will happen next around the table? (Or as mathematicians would ask, What will be the dynamic of the game?)

  FOR EVERY ACTION THERE’S A REACTION

  (THE ABRIDGED VERSION OF NEWTON’S THIRD LAW)

  Knowing Tom’s friends, I can tell you that his move is a declaration of war. The waiter is called back, and everyone suddenly remembers they are very hungry, particularly for the high end of the menu. Home fries are soon replaced by a slice of Robuchon truffle pie. The cheese burger is cancelled, and a two-pound steak is ordered instead. All of Tom’s friends suddenly appear to be great connoisseurs and order from the expensive part of the menu. It’s an avalanche, an economic disaster, accompanied by several expensive bottles of wine. When the check finally comes and the bill is equally divided, each diner has to pay $410!

  Incidentally, scientific studies have shown that when several diners split a bill, or when food is handed out for free, people tend to order more – I’m sure you’re not surprised by that.

  Tom realizes he’s made a terrible mistake, but is he the only one? Fighting for their pride and attempting to avoid being fooled by Tom in this way, everyone ends up paying much more than they’d initially intended for food they never meant to order. And don’t get me started on their caloric intake …

  Should they have paid much less and let Tom enjoy his dream dish? You decide. In any event, that was the last time this group of friends went out together.

  This scene in the restaurant d
emonstrates the interaction between several decision-makers and is a practical example of issues that Game Theory addresses.

  ‘Interactive Decision Theory would perhaps be a more descriptive name for the discipline usually called Game Theory.’

  Robert Aumann (from Collected Papers)

  The Israeli mathematician Professor Robert Aumann received the Nobel Prize in Economics for his pioneering work on Game Theory in 2005. Following his definition, let’s pin down Game Theory as … a mathematical formalization of interactive decision-making.

  Please, don’t panic! In this book I shall try to refrain from using numbers and formulae. Many excellent books do that anyway. I will try to present the more amusing sides of this profession and will focus on insights and bottom lines.

  Game Theory deals with formalizing the reciprocity between rational players, assuming that each player’s goal is to maximize his or her benefit, whatever that may be.

  Players may be friends, foes, political parties, states or anything that behaves interactively really. One of the problems with game analysis is the fact that, as a player, it’s very hard to know what would benefit each of the other players. Furthermore, some of us are not even clear about our own goal or what would benefit us.

  This is the right place to point out, I guess, that the participants’ reward is not only measured in money. The reward is the satisfaction players receive from the results of the game, which could be positive (money, fame, clients, more ‘likes’ on Facebook, pride and so on) or negative (fines, wasted time, ruined property, disillusionment and so on).

  When we’re about to make a decision while playing a game whose result depends on the decisions of others, we should assume that, in most cases, the other players are as smart and as egotistical as we are. In other words, don’t expect to enjoy your Tournedos Rossini while others sip their sodas, pay their share and happily share your joy.

  There are many ways to apply Game Theory to life situations: business or political negotiations; designing an auction (choosing between the English model, where the price continually rises, and the Dutch model where the initial price is high and continually falls); brinkmanship models (the Cuban missile crisis, the Isis threat to the Western world); product pricing (should Coca-Cola lower prices before Christmas or raise them? – how would Pepsi respond?); street peddlers haggling with accidental tourists (what’s the optimal speed of lowering the price of their goods? – going too fast might signal that the product isn’t worth much, whereas going too slow might make the tourist lose patience and walk away); whaling restrictions (all countries that keep on whaling as usual want restrictions to apply to others – since without them whales might become extinct); finding clever strategies for board games; understanding the evolution of cooperation; courtship strategies (human and animal); military strategies; the evolution of human and animal behaviour (I’m flagging now and have started to generalize); and so on (phew!).

  The big question is: can Game Theory really help to improve the way people make their daily decisions? This is where opinions vary. Certain experts are convinced of the game theoreticians’ crucial impact on almost everything; yet there are no lesser experts who believe that Game Theory is nothing more than handsome mathematics. I believe the truth is somewhere in between … though not really in the middle. In any event, it’s a fascinating field of thought that offers numerous insights into a wide variety of issues in our lives.

  I believe that examples are the best way to teach and learn about Game Theory, or anything else. The more examples we see, the better we understand things. Let’s begin.

  Chapter 2

  THE BLACKMAILER’S PARADOX

  ‘Let us never negotiate out of fear, but let us never fear to negotiate.’

  John F Kennedy

  In the following chapter we’ll learn about a game that deals with negotiations, invented by Robert Aumann. The game is very simple but this may be misleading – it conceals some profound insights.

  The Blackmailer’s Paradox game was first presented by the before-mentioned Robert Aumann, a great expert on conflict and cooperation through game theory analysis. Here’s my version:

  Jo and Mo walk into a dark room where a tall, dark, mysterious stranger awaits them. Wearing a dark suit and tie, he takes off his shades and places a briefcase on a table in the middle of the room. ‘In here’, he says authoritatively, pointing at the briefcase, ‘is a million dollars in cash. It can all be yours in just a few moments, under one condition. The two of you must agree on how to divide the money between you. If you reach an agreement, any agreement, the cash is yours. If you don’t, it goes back to my boss. I’m leaving you alone now. Take your time considering. I’ll be back in an hour.’

  The tall man is gone now, so let me guess what you’re thinking, my esteemed readers: ‘What a simple game! A complete no-brainer. There’s no need to negotiate anything. I mean, why should a Nobel Prize winner even worry about stuff like that? Did I miss something? Of course not. This must be the simplest game in the world. All that Jo and Mo need to do now is …’

  Hold your horses, my friends. Don’t rush to conclusions. Remember, nothing is ever as simple as it looks. If all that the two players have to do is to split the cash between them and go home, I wouldn’t have written about them in this book.

  Here’s what really happens next:

  Jo is a nice and decent guy who believes his qualities are universal. Beaming, he turns to Mo and says, rubbing his hands: ‘Can you believe that guy? Isn’t he funny? He just left us with half a million each! We don’t even need to negotiate. Let’s end this silly game, split the cash, and go party, right?’

  ‘So this is just a silly game for you, is it?’ says Mo, sounding ominous. ‘I find it fascinating. So while you were talking nonsense, and suggesting an idiotic split, I came up with a much more reasonable solution. This is my offer: I take $900,000 and you take the remaining $100,000, and you only get this much because I’m in a very good mood today, you understand? Now, this is my final offer. Take it or leave it. If you take it, fine – you just made a hundred grand. If you don’t, that’s fine too, and we both walk away with nothing, which I don’t mind at all.’

  ‘You must be kidding?’ says Jo. He’s beginning to worry.

  ‘Never! Don’t forget that my full name is Mo the Money Monster. I eat guys like you for breakfast. And I never joke. I don’t have the app for it! This is my final offer, negotiation over!’

  ‘What’s the matter with you?’ Jo is almost crying. ‘This is a symmetric game of two fully informed players. There’s no reason in the world why you should take a red cent more than me. It makes no sense and isn’t fair at all.’

  ‘Listen, you talk too much and it’s giving me a headache,’ says Mo, his upper lip visibly twitching. ‘One more word from you, and I’ll lower my generous offer to $50,000. All you gotta say now is “OK, let’s do it”, or we walk away with nothing.’

  And Jo says, ‘OK.’

  End of game.

  How did that happen in such a simple game? Where did Jo go wrong?

  When I wrote about this game in a major economic newspaper, I encountered an array of angry political reactions, from left and right across the whole political spectrum (which, by the way, proves that my article was balanced and fair). This was because the readers understood that the game was not about Jo or Mo, but about real-life negotiations. Professor Aumann, under whom I was privileged to study many years ago, believed that this story is closely related to the Israeli-Arab conflict and can teach us a thing or two about conflict resolution in general. We can also find various aspects of the Blackmailer’s Paradox in negotiations held at the Paris Peace Conference of 1919 (leading to the Treaty of Versailles), the Molotov–Ribbentrop Pact of 1939, the Moscow theatre hostage crisis of 2002, and the recent talks on nuclear development between the Islamic Republic of Iran and a group of world powers – to name but a few instances.

  Aumann argued that, entering negotiations with i
ts neighbours, Israel must take three key points into consideration: it must be prepared to take into account the (sad) possibility of ending the talks (or ‘game’) without an agreement; it must realize that the game may be repeated; and it has to deeply believe in its own red-line positions and stick by them.

  Let’s discuss the first two points. When Israel is not willing to leave the negotiations room empty-handed, it’s strategically crippled because then the game is no longer a symmetric one. The party that’s mentally prepared to fail has a huge advantage. In the same way, when Jo is willing to make painful concessions and accept humiliating terms for the sake of an agreement, that stand will affect future talks, because when the players meet again Mo might offer worse terms each time they play.

  Importantly, in real life, time is of the essence too. Consider this: Mo attempts to blackmail Jo. Jo is taking his time, trying to negotiate a change to the unfair offer. Mo insists, Jo tries again, but the clock’s ticking … and then there’s a knock at the door. The briefcase owner is back.

  ‘Hey, you two. Have you reached an agreement?’ he asks them. ‘Not yet? Well, the money is gone. Goodbye.’ He walks away, and Honest Jo and Blackmailer Mo are left with nothing.

  That’s actually a well-known business-world situation. Every now and then we hear the news about a company that was made a tempting buy-out offer, but it was taken off the table before it was even properly discussed.

  As a general matter, we need to consider the nature of a given resource whose value might be eroded with time without even being used. Let’s call this the Popsicle Model (don’t bother Googling it): a good thing that keeps melting, until it exists no more.

  There’s a modern fable about a businessman who was richer than rich, who used to have a certain way of going about his affairs. He’d make a financial offer to a company he wished to buy, stipulating that the sum would shrink with every day that went by. Let’s suppose that he makes an offer to the Israeli and Jordanian governments, saying that he’s willing to pay $100 billion for the Dead Sea (a lake that shrinks daily and might really die one day) and that the offer will drop a billion lower every day. If eventually, owing to bureaucratic red tape or political discord, the states should take their sweet time answering, they just might end up paying the businessman a fortune to take that Dead Sea off their hands, which would make him a lake owner and even richer.