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 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 demonstrates 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 that the players receive aren’t fixed, but rather determined by their chosen strategies), people regularly reject the (logical) choice of $5 and very often choose the $100 option In fact, this Indian economist stated, when players are lacking in relevant formal knowledge, they ignore the economic approach and actually attain better results Giving up on economic thinking and simply trusting the other player is the reasonable thing to do All this boils down to the simple question, can we trust Game Theory? Another interesting finding about this game is that the players’ actions depend on the size of the bonus When it’s very low, recurring games lead to the highest sum possible being called Yet when potential profit is significant enough, the sums offered converge towards the Nash Equilibrium – that is, the lowest possible declarable sum This finding was further corroborated by a study of various cultures conducted by Professor Ariel Rubinstein, who won the Israeli Prize for economics in 2002 ‘All men make mistakes, but only wise men learn from their mistakes.’ Winston Churchill Kaushik Basu believes that moral qualities such as honesty, integrity, trust and caring are essential for a sound economy and a healthy society Although I totally agree with him, I seriously doubt that world leaders and economic policymakers are endowed with such qualities More often than not, integrity and trust are qualities that give you no edge whatsoever in political races, and thus it would be a real miracle if individuals guided by such moral standards were indeed to assume key political or economic positions STAG, RABBIT, START-UPS AND THE PHILOSOPHER Below is the matrix of a game known as the Stag Hunt: Stag Rabbit Stag 2,2 0,1 Rabbit 1,0 1,1 Two friends go hunting in a forest populated by stags and rabbits, where rabbit stands for the smallest trophy a hunter can find, and stag represents the largest possible gain Hunters can catch rabbits on their own, but they must cooperate to catch a stag There are two equilibrium points in this game: the two hunters can go after either a rabbit or a stag They’ll be better off choosing the larger target, but will they do that? It’s a question of trust They might both commit to staghunting if each feels the other would be a reliable and cooperative partner This is a situation where two players must choose between, on the one hand, the certain but less favourable result (rabbit); and, on the other, the larger and more promising result (stag), which requires trust and cooperation Even if the two hunters were to shake their (empty) hands and decide to hunt for stag together, one of them might break the deal for fear that the other might do the same A rabbit in the hand is worth more than a stag no one helped you hunt Similar situations can be found, of course, outside the forest A veteran employee of a hi-tech company considers quitting his job and initiating a start-up with a friend Just before he gives notice to his boss, he begins to worry that his friend wouldn’t quit his job, with the result that he’d be left hanging, without either his current job (rabbit) or the dream start-up (stag) Many years before Game Theory was even born, philosophers David Hume and Jean-Jacques Rousseau used a verbal version of this game in their discussions of cooperation and trust It may be interesting to point out that while the Prisoner’s Dilemma is usually considered the game that best exemplifies the problem of trust and social cooperation, some Game Theory experts believe that the Stag Hunt game represents an even more interesting context in which to study trust and cooperation CAN I TRUST YOU? Sally is given $500 and told she may give Betty as much of that as she wants to (even nothing) The sum Sally chooses to give will be multiplied by 10 before Betty receives it Thus, if Sally gives Betty $200, the latter will actually receive $2,000 In the second stage of this game, Betty may, if she wishes, pay Sally back from the actual sum she has received (if at all) What do you think will happen here? Note that the value of the game (that is, the maximum total sum the two players between them could gain) is $5,000 Suppose Sally gives Betty $100, which means that Betty actually receives $1,000 What would count as a logical move by Betty? What’s the honest thing to do? Does she have to give Sally her $100 back? She could do that and throw in a reward for Sally’s trust, or she might be upset that Sally didn’t trust her enough to gave her at least $400 What would you in each role? From experiments performed with my students I’ve seen that there’s a variety of possible behaviours: some students gave half of the sum, some didn’t give a red cent, some trusted the other player in full and gave them all the money, some of the generous students were rewarded in return and some not … So it goes in the world Chapter 14 HOW TO GAMBLE IF YOU MUST The title says it all … I’m about to give you a mathematical tip that will greatly improve your chances of winning at the roulette tables But before I that, and before you book a flight to Las Vegas, I must insist that the best tip I can give you is: if you can avoid it, it’s never a good idea to gamble in a casino I hope you understand that it’s no accident that casinos are built, and that people are flown in, fed delicacies and given expensive shows on the house Nobody should imagine that casino managers only want their clients to have a good time Yet, if gamble you must, here’s an example to get things going for you Imagine a man in a casino who has only $4 in hand, but badly needs $10 (If you must have a sob story, that man entered the casino with $10,000 in his pocket but lost it all except for the last $4 He now needs $10 for the bus fare home.) That man won’t quit before he wins another $6 – unless, that is, he loses his last dime and has to walk home in pouring rain and freezing wind (Are you weeping yet?) Standing in front of a roulette table, he must decide how to play I can mathematically and precisely prove that the best strategy for maximizing his chances of turning his $4 into $10 is to bet on a single colour the smaller amount: either all that he has, or the sum he needs to reach $10 Let me explain: He has $4 and needs $10, so he bets the entire $4 on red Of course, the house might swallow that $4 and the gambler will go home on foot, but if red does come up, his fortune is doubled Now that he has $8, he doesn’t want to be again staking the entire sum, because he only needs another $2 So he should bet only $2 and, if he’s lucky again, he’ll have his desired $10 If he loses the $2, he’ll still have $6, and should bet $4 of that He’ll play in that manner until he loses all his money or reaches the desired $10 The optimum strategy is to opt for this ‘bold play’ – that is, to bet all your money, or the sum you’re short That may seem like an odd strategy, because most people would think that they’d do better betting a dollar or two at a time They are wrong Bold play is the best move because if you are the ‘lesser player’, you should play as few games as possible Who is the lesser player? It’s the player whose chances of winning a bet are smaller than his opponent’s (even if only by a fraction), or the player who has less money (and fewer opportunities to correct losses) than the other player When you play against the house, your place on that scale is quite clear The house always has the edge (that’s what the single zero and the double zero on the roulette wheel are for), the experience, and the money you don’t have Let me warn you again, though Do not gamble! This is perhaps the best mathematical advice I can give you (except if you’re doing it for fun and don’t mind paying for it by losing – and, if that’s the case, I’d suggest you decide on the price you’re willing to pay before you play, and stick to it) Perhaps you’ll be surprised to know that I can intuitively explain why the bold play is the optimum strategy that would maximize your chances of making that $10 To simplify the explanation, let me present another problem that will shed a clear, bright light on the roulette table question Imagine that I happen to come across the basketball genius Michael Jordan and he agrees to shoot some hoops with me At the moment in question, neither of us is an active NBA player and we both have plenty of free time Certain of his skills, MJ generously lets me call the score we aim for What would you suggest? I hope that the answer is clear The best thing for me to do is to call the whole thing off, give Michael a hug and call it a tie (though it would be very foolish to pass up the chance to play against my hero) The second-best solution is to play for a single point I mean, miracles do happen I could shoot, and the ball could be kind to me and swoosh right in, while MJ might miss his shot (it happens to the best of us) If I choose to play for two or three points, my chances of winning would drop depressingly low; and if we play on, I’ll most certainly lose The ‘law of large numbers’ predicts that, in the long run, what’s expected will happen If we play for one point, I can at least fantasize about beating the Michael Jordan in basketball Dreaming is free If we return to the casino question, let me remind you that the roulette contains zeroes, and they tip the scales in the house’s favour and make the whole game unfair (to me) On the qualitative level, betting against the house is the same as playing basketball against Michael Jordan The house is the better player, so it would be advisable for me to play as few times as possible, because in the long run the house always wins Casino experts or reasonable mathematicians may wonder what happens if, having $4, we bet a single dollar first and then play the following strategy: if we win and have $5, we bet on $5, and if we lose and are down to $3, we switch to the aforementioned ‘bold game’ strategy The answer is that this strategy gives the same winning probability as playing the bold game right from the start In any event, this remark is for experts only On the other hand, if your goal is to spend some quality time in the fancy casino, the bold game isn’t your best option, because it might motivate the house detective to show you to the door after just one game If spending time in the casino is your ultimate goal, I’d suggest playing cautiously – bet a single dollar each time and take long breaks This isn’t the brightest strategy, but it’s highly effective as a way to spin out your time and money Let me sum up this chapter with an insight attributed to the British statesman David Lloyd George: ‘There is nothing more dangerous than to leap a chasm in two jumps.’ Conclusion GAME THEORY GUIDELINES Game Theory deals with formalizing the reciprocity between rational players, assuming that each player’s goal is to maximize his or her benefit – in terms of such benefits as money, fame, clients, more ‘likes’ on Facebook, pride and so on Players may be friends, foes, political parties, states or any other kinds of entity with which you can be interactive When you’re about to make a decision, you should assume that, in most cases, the other players are as smart and as egotistical as you are When entering negotiations, you must take three key points into consideration You must be prepared to take into account the possibility of ending the talks without an agreement; you must realize that the game may be repeated; and you must deeply believe in your own stands and stick by them Playing rationally against an irrational opponent is often irrational, and playing irrationally against an irrational opponent is often rational Try as much as you can to guess what your opponent will do by trying to walk in their shoes You are not him or her, however, and you can never know exactly what makes them tick: you’ll never have a complete handle on what they’ll do and why Remember that to explain is much easier than to predict Most things are more complicated than you think, even if you think you understand this sentence Always take into account the human unwillingness to accept injustice, as well as the significance of honour Beware! The mathematical solution of a game often ignores such important things as envy (every time a friend succeeds, I die a little), insult, schadenfreude, self-respect and moral indignation Motivation may improve strategic skills Before making any decision, ask yourself what would happen if everyone shared your views … and remember that not everyone does share your views Sometimes ‘ignorance is bliss’: it may happen that the least-knowledgeable player makes the highest profit when competing against extremely clever, allknowing players When each player plays his or her own best choice and takes no care at all of the consequences of their action on other players, this may result in a catastrophe for all In many situations egoistical behaviour is not only morally problematic but also strategically unwise Contrary to the popular belief that having more options is a better option, it may happen that narrowing down the number of choices will improve the result People tend to cooperate when faced with the ‘shadow of the future’ – when further encounters are expected, we change the way we think When the game is played over and over again, stick to the following: ‘Play nice Never be the first to betray, but always react to betrayals Avoid the pitfall of blind optimism Be forgiving Once your opponent stops betraying, stop betraying too.’ Bear in mind the words of Abba Eban: ‘History teaches us that men and nations behave wisely once they have exhausted all other alternatives.’ Study the possible permutations of success and failure that result from particular moves in the game in question Learn the consequences of both honesty and duplicity, and the risks involved in trust Don’t get sidetracked by the fascination of complexity if your simple aim is to win As Winston Churchill said, ‘However beautiful the strategy, you should occasionally look at the results.’ Giving up on economic/strategic thinking and simply trusting the other player is time and again the reasonable thing to do Moral qualities such as honesty, integrity, trust and caring are essential for a sound economy and a healthy society There’s a question as to whether world leaders and economic policymakers are endowed with such qualities, which give you no edge whatsoever in political races If you are the ‘lesser player’, you should play as few games as possible Trying to avoid risk is a very risky course of action REFERENCE NOTES Chapter 3: The Ultimatum Game Page 15: An extensive review of the Ultimatum Game can be found in Colin F Camerer, Behavioural Game Theory, Princeton University Press, NJ, 2003 Page 15: The article by Werner Guth, Rolf Shmittberger, and Bernd Schwarze is ‘An Experimental Analysis of Ultimatum Bargaining’, Journal of Economic Behaviour and Organization, 3:4 (December), pp 367–88 Page 22: Maurice Schweitzer and Sara Solnik wrote up their study on the impact of beauty on the Ultimatum Game in ‘The Influence of Physical Attractiveness and Gender on Ultimatum Game Decisions’, Organizational Behaviour and Human Decision Processes, 79:3, September 1991, pp 199–215 Page 23: The Jane Austen quotation is from Pride and Prejudice, volume 1, chapter 6 Chapter 4: Games People Play Page 39: Martin Gardner’s thoughts on Game 5 can be found in Martin Gardner, Aha! Gotcha: Paradoxes to Puzzle and Delight, W H Freeman & Co Ltd, New York, 1982 Page 40: See Raymond M Smullyan, Satan, Cantor, and Infinity: And Other Mind-boggling Puzzles, Alfred A Knopf, New York, 1992; and Dover Publications, 2009 Chapter 6: The Godfather and the Prisoner’s Dilemma Page 88: Robert Axelrod’s The Evolution of Cooperation, Basic Books, 1985; revised edition 2006 Chapter 7: Penguin Mathematics Page 99: A note on Strategy 1, War of Attrition: In 2000 I published a paper, with Professor Ilan Eshel, that deals with the mathematical aspects of volunteering and altruism The mathematics is not easy, but if you’re interested the paper is free on the Internet: just Google ‘On the Volunteer Dilemma I: Continuous-time Decision Selection 1(2000)1– 3, 57–66’ Page 103: See John Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, Cambridge, 1982 Chapter 8: Going, Going … Gone! Page 107: The article referred to here, published in 1971, is Martin Shubik, ‘The Dollar Auction Game: A Paradox in Noncooperative Behaviour and Escalation’, Journal of Conflict Resolution, 15:1, pp.109–11 Page 115: The article cited on the ‘Winner’s Curse’ is Ed Capen, Bob Clapp and Bill Campbell, ‘Competitive Bidding in High-risk Situation’, Journal of Petroleum Technology, 23, pp 641–53 Chapter 10: Lies, Damned Lies, and Statistics Page 136: Edward H Simpson’s 1951 paper ‘The Interpretation of Interaction in Contingency Tables’ was published in the Journal of the Royal Statistical Society, Series B 13, pp 238–41 Page 146: The two books I’m recommending here are: John Allen Paulos, A Mathematician Reads the Newspaper: Making Sense of the Numbers in the Headlines, Penguin, London, 1996; Basic Civitas Books, 2013; and Darrell Huff, How to Lie with Statistics, revised edition, Penguin, London, 1991 Chapter 11: Against All Odds Page 152: Stuart Sutherland’s 1992 book Irrationality: The Enemy Within was published in a 21st anniversary edition (with a foreword by Ben Goldacre) by Pinter & Martin, London, 2013 Chapter 12: On Fairly Sharing a Burden Page 157: The Airport Problem was first presented by S C Littlechild and G Owen in a 1973 paper, ‘A Simple Expression for the Shapely Value in a Special Case’, Management Science, 20:3, Theory Series (Nov 1973), pp 370–2 Chapter 14: How to Gamble if You Must Page 170: ‘Bold play’ is the phrase used in the Bible of roulette games: Lester E Dubbins and Leonard J Savage, How to Gamble if You Must, Dover Publications, New York, reprint edition, 2014; first published 1976 as Inequalities for Stochastic Processes BIBLIOGRAPHY Chapter 1 Gneezy, Uri, Haruvy, Ernan and Yafe, Hadas, ‘The Inefficiency of Splitting the Bill’, Economic Journal, 114:495 (April 2004), pp 265–80 Chapter 2 Aumann, Robert, The Blackmailer Paradox: Game Theory and Negotiations with Arab Countries, available at www.aish.com/jw/me/97755479.html Chapter 3 Camerer, Colin, Behavioral Game Theory: Experiments in Strategic Interaction, Roundtable Series in Behavioral Economics, Princeton University Press, 2003 Chapter 4 Davis, Morton, Game Theory: A Nontechnical Introduction, Dover Publications, reprint edition, 1997 Chapter 5 Gale, D and Shapley, L S, ‘College Admissions and the Stability of Marriage’, American Mathematical Monthly, 69 (1962), pp 9–14 Intermezzo: The Gladiators Game Kaminsky, K S, Luks, E M and Nelson, P I, ‘Strategy, Nontransitive Dominance and the Exponential Distribution’, Austral J Statist, 26 (1984), pp 111–18 Chapter 6 & Chapter 9 Poundstone, William, Prisoner’s Dilemma, Anchor, reprint edition, 1993 Chapter 7 Sigmund, Karl,The Calculus of Selfishness, Princeton Series in Theoretical and Computational Biology, Princeton University Press, 2010 Intermezzo: The Raven Paradox Hempel, C G, ‘Studies in the Logic of Confirmation’, Mind, 54 (1945), pp 1–26 Chapter 8 Milgrom, Paul, Putting Auction Theory to Work (Churchill Lectures in Economics), Cambridge University Press, 2004 Chapter 10 Huff, Darrell, How to Lie with Statistics, W W Norton & Company, reissue edition, 1993 Chapter 11 Morin, David J, Probability: For the Enthusiastic Beginner, CreateSpace Independent Publishing Platform, 2016 Chapter 12 Littlechild, S C and Owen, G, ‘A Simple Expression for the Shapely Value in a Special Case’, Management Science 20:3 (1973), pp 370–2 Chapter 13 Basu, Kaushik, ‘The Traveler’s Dilemma’, Scientific American, June 2007 Chapter 14 Dubins, Lester E and Savage, Leonard J, How to Gamble If You Must: Inequalities for Stochastic Processes, Dover Publications, reprint edition, 2014 Karlin, Anna R and Peres, Yuval, ‘Game Theory Alive’, American Mathematical Society (2017) The story of Watkins dates back to 1893, when the scholar of esotericism John Watkins founded a bookshop, inspired by the lament of his friend and teacher Madame Blavatsky that there was nowhere in London to buy books on mysticism, occultism or metaphysics ~ at moment marked the birth of Watkins, soon to become the home of many of the leading lights of spiritual literature, including Carl Jung, Rudolf Steiner, Alice Bailey and Chögyam Trungpa Today, the passion at Watkins Publishing for vigorous questioning is still resolute Our wide-ranging and stimulating list reflects the development of spiritual thinking and new science over the past 120 years We remain at the cutting edge, committed to publishing books that change lives DISCOVER MORE Read our blog Watch and listen to our authors Sign up to our in action mailing list JOIN IN THE CONVERSATION WatkinsPublishing @watkinswisdom watkinsbooks watkinswidom watkins-media Our books celebrate conscious, passionate, wise and happy living Be part of the community by visiting www.watkinspublishing.com This edition published in the UK and USA 2017 by Watkins, an imprint of Watkins Media Limited 19 Cecil Court, London WC2N 4EZ enquiries@watkinspublishing.com Design and typography copyright © Watkins Media Limited 2017 Text copyright © Haim Shapira 2017 Haim Shapira has asserted his right under the Copyright, Designs and Patents Act 1988 to be identified as the author of this work All rights reserved No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, without prior permission in writing from the Publishers 1 3 5 7 9 10 8 6 4 2 Typeset by JCS Publishing Services Ltd, www.jcs-publishing.co.uk Printed and bound in Finland A CIP record for this book is available from the British Library ISBN: 978-178678-010-2 www.watkinspublishing.com ... Chapter 4 GAMES PEOPLE PLAY In the following chapter we learn about several games that can be both fun and enlightening We will expand our games vocabulary, gain some insights and improve our strategic... 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... ‘Strategist of the Year’ Let’s play! GAME 1 THE PIRATES GAME ‘You can always trust the untrustworthy because you can always trust that they will be untrustworthy It is the trustworthy you can’t trust. ’