Gladiators, pirates and games of trust how game theory, strategy and probability rule our lives

When we’re about to make a decision while playing a game whose resultdepends on the decisions of others, we should assume that, in most cases, theother players are as smart and as egotis

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 GameTheory and another PhD for his research on the mathematical and philosophicalapproaches to infinity He now teaches mathematics, psychology, philosophy andliterature He is an author of seven bestselling books His stated mission as awriter is not to try to make his readers agree with him, but simply to encouragethem to enjoy thinking One of Israel’s most popular and soughtafter speakers,

he lectures on creativity and strategic thinking, existential philosophy andphilosophy in children’s literature, happiness and optimism, nonsense andinsanity, imagination and the meaning of meaning, as well as friendship andlove He is also an accomplished pianist and an avid collector of anythingbeautiful

Bibliography

This book deals with Game Theory, introducing some important ideas aboutprobabilities and statistics These three fields of thought constitute the scientificfoundation of the way we make decisions in life Although these topics are quiteserious, I’ve made a tremendous effort not to be boring and to write a book that’srigorous 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 withhis celebrated equilibrium

• Find optimal strategies for playing at roulette tables

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 themenu, 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 expensivetoo Suppose it costs $200 Now Tom must decide: to order or not to order Thismay sound very dramatic, Shakespearean even, but not really a hard decision tomake 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 todifferent

people For a street beggar, it’s a fortune; but if you were to put $200 into BillGates’s account, it wouldn’t make any kind of difference In any event, this is arelatively 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 ninefriends, 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 orderedtheir 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: hetreats himself to Rossini’s gourmet opera and pays just over 10 per cent of itsadvertised price

is ordered instead All of Tom’s friends suddenly appear to be great connoisseursand order from the expensive part of the menu It’s an avalanche, an economicdisaster, accompanied by several expensive bottles of wine When the checkfinally 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’renot surprised by that

Tom realizes he’s made a terrible mistake, but is he the only one? Fighting fortheir pride and attempting to avoid being fooled by Tom in this way, everyoneends up paying much more than they’d initially intended for food they nevermeant 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? Youdecide In any event, that was the last time this group of friends went outtogether

This scene in the restaurant demonstrates the interaction between severaldecision-makers and is a practical example of issues that Game Theoryaddresses

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

Please, don’t panic! In this book I shall try to refrain from using numbers andformulae Many excellent books do that anyway I will try to present the moreamusing 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 thatmay be

Players may be friends, foes, political parties, states or anything that behavesinteractively really One of the problems with game analysis is the fact that, as aplayer, 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 wouldbenefit us

This is the right place to point out, I guess, that the participants’ reward is notonly measured in money The reward is the satisfaction players receive from theresults 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 resultdepends on the decisions of others, we should assume that, in most cases, theother players are as smart and as egotistical as we are In other words, don’texpect to enjoy your Tournedos Rossini while others sip their sodas, pay theirshare and happily share your joy

There are many ways to apply Game Theory to life situations: business orpolitical negotiations; designing an auction (choosing between the Englishmodel, where the price continually rises, and the Dutch model where the initialprice is high and continually falls); brinkmanship models (the Cuban missilecrisis, the Isis threat to the Western world); product pricing (should Coca-Colalower prices before Christmas or raise them? – how would Pepsi respond?);street peddlers haggling with accidental tourists (what’s the optimal speed oflowering the price of their goods? – going too fast might signal that the productisn’t worth much, whereas going too slow might make the tourist lose patienceand walk away); whaling restrictions (all countries that keep on whaling as usualwant restrictions to apply to others – since without them whales might becomeextinct); finding clever strategies for board games; understanding the evolution

of cooperation; courtship strategies (human and animal); military strategies; theevolution of human and animal behaviour (I’m flagging now and have started togeneralize); and so on (phew!)

The big question is: can Game Theory really help to improve the way peoplemake their daily decisions? This is where opinions vary Certain experts are

convinced of the game theoreticians’ crucial impact on almost everything; yetthere are no lesser experts who believe that Game Theory is nothing more thanhandsome mathematics I believe the truth is somewhere in between … thoughnot really in the middle In any event, it’s a fascinating field of thought thatoffers numerous insights into a wide variety of issues in our lives.

I believe that examples are the best way to teach and learn about GameTheory, or anything else The more examples we see, the better we understandthings Let’s begin

The Blackmailer’s Paradox game was first presented by the before-mentionedRobert Aumann, a great expert on conflict and cooperation through game theoryanalysis Here’s my version:

Jo and Mo walk into a dark room where a tall, dark, mysterious strangerawaits them Wearing a dark suit and tie, he takes off his shades and places abriefcase 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 afew moments, under one condition The two of you must agree on how to dividethe money between you If you reach an agreement, any agreement, the cash isyours If you don’t, it goes back to my boss I’m leaving you alone now Takeyour time considering I’ll be back in an hour.’

The tall man is gone now, so let me guess what you’re thinking, my esteemedreaders: ‘What a simple game! A complete no-brainer There’s no need tonegotiate anything I mean, why should a Nobel Prize winner even worry aboutstuff like that? Did I miss something? Of course not This must be the simplestgame 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 thecash between them and go home, I wouldn’t have written about them in thisbook

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 hefunny? 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?’

‘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 guyslike you for breakfast And I never joke I don’t have the app for it! This is myfinal 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 shouldtake 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 upperlip visibly twitching ‘One more word from you, and I’ll lower my generousoffer to $50,000 All you gotta say now is “OK, let’s do it”, or we walk awaywith nothing.’

Mo, but about real-life negotiations Professor Aumann, under whom I wasprivileged to study many years ago, believed that this story is closely related tothe 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 innegotiations held at the Paris Peace Conference of 1919 (leading to the Treaty ofVersailles), the Molotov–Ribbentrop Pact of 1939, the Moscow theatre hostagecrisis of 2002, and the recent talks on nuclear development between the IslamicRepublic of Iran and a group of world powers – to name but a few instances

Aumann argued that, entering negotiations with its neighbours, Israel musttake 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 thenegotiations room empty-handed, it’s strategically crippled because then thegame 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 painfulconcessions and accept humiliating terms for the sake of an agreement, thatstand will affect future talks, because when the players meet again Mo mightoffer 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 unfairoffer Mo insists, Jo tries again, but the clock’s ticking … and then there’s aknock 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 andBlackmailer 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 whosevalue might be eroded with time without even being used Let’s call this thePopsicle 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, whoused 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 everyday that went by Let’s suppose that he makes an offer to the Israeli andJordanian governments, saying that he’s willing to pay $100 billion for the DeadSea (a lake that shrinks daily and might really die one day) and that the offer willdrop a billion lower every day If eventually, owing to bureaucratic red tape orpolitical discord, the states should take their sweet time answering, they justmight 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.

Let me tell you now about my conclusions from the blackmailer story:

1 Playing rationally against an irrational opponent is often irrational.

3 When you think about this game (and similar situations in life) a bit more

deeply, the rational way to play is not always clear (even the meaning of theword ‘rational’ isn’t clear – after all, Mo wins the game and is the one whowalks away with $900,000)

4 Be very careful trying to guess what your opponent would do by trying to

walk in their shoes You are not him, and you can never know what makeshim tick and why It is hard-to-impossible to predict how others would act

in a given situation

Of course, there are plenty of examples to demonstrate my point I’ve chosen

a few randomly In 2006, Professor Grigory Perelman declined the Fields Medal

(a Nobel Prize equivalent for mathematicians) saying, ‘I’m not interested in

money or fame.’ In 2010, he won a million dollars for proving the Poincaré Conjecture, but again refused to take the money You see – some people don’t

love money In World War Two, Joseph Stalin rejected a POW-exchange offerand wouldn’t give away Friedrich Paulus, a German field marshal the Sovietscaptured in the Battle of Stalingrad, for his own son Yakov Dzhugashvili, who’dbeen in German captivity since 1941 ‘You don’t trade a Marshal for aLieutenant,’ Stalin declared At the same time, some people give their kidneys toperfect strangers Why? Your guess is as good as mine And Vladimir Putinwoke up one morning and decided that the Crimean Peninsula belongs to MotherRussia: I wouldn’t have even started guessing that

Incidentally, after the fact, political pundits offered clever explanations ofPutin’s act (you’re welcome to Google it) The only problem is that none of themhad predicted the move, which proves they didn’t have the faintest idea aboutwhat went on in Putin’s head

And now, the most important insight:

5 While studying Game Theory models is important and helpful, we must

remember that, more often than not, real issues in life are much more

complicated than they initially appear (and they don’t become simpler whenexamined for the second and third time), and no mathematical model cancapture their entire full-scale complexity Maths is better at studying therules of nature than the nature of humankind

***

Even if we aren’t fully aware of the fact, conducting negotiations is an essentialpart of our lives We do this all the time with our spouses, children, partners,

bosses, subordinates and even total strangers Of course, negotiations are akeystone of inter-state diplomatic relations or the conduct of political bodies (forexample, when coalitions are formed) It’s therefore quite surprising that notonly ordinary people but also major political and economic figures can at times

be so unskilled in negotiating techniques and philosophies

In the following chapter we’ll look at a famous game pertaining to aspects ofnegotiation

THE ULTIMATUM GAME

In this chapter I’ll focus on an economic experiment that provides insights into human behaviour, undermines standard economics assumptions, illustrates the human unwillingness to accept injustice, and clearly shows the huge difference between Homo economicus and real human beings We’ll also study different negotiation strategies in a recurring Ultimatum Game version

In 1982 three German scientists, Werner Güth, Rolf Schmittberger and BerndSchwarze, wrote an article about an experiment they’d conducted whose resultssurprised economists (but no one else) Known as the Ultimatum Game, theexperiment has since become one of the most famous and most studied games inthe world

The game is similar to the Blackmailer’s Paradox, but the differences arecrucial The main dissimilarity is the asymmetry of the Ultimatum Game

The game goes like this Two players who don’t know each other are in aroom Let’s call them Maurice and Boris Boris (let’s call him the proposer) isgiven $1,000 and instructed to share it with Maurice (let’s call him theresponder) in whatever way he sees fit The only condition here is that Mauricemust agree to Boris’s method of division: if he doesn’t, the $1,000 will be takenaway and both players will end with nothing

It should be noted that this is a game of two fully informed players Thus, ifBoris offers $10 and Maurice accepts, Boris ends up walking away with $990.Yet if Maurice is displeased with this offer (remember, he knows Boris has

$1,000), both will remain empty-handed

What do you think will happen? Will Maurice accept Boris’s ‘generous’ $10offer? How much would you propose if you were playing the game? Why?What’s the smallest sum you would take as a responder? Why?

MATHS VS PSYCHOLOGY

I believe that this game points to the huge tension that often exists between adecision based on mathematical principles (a ‘normative’ decision) and onebased on intuitive principles and psychology (a ‘positive’ decision).

Mathematically, this game is easily resolved, but the wonderful easy solutionisn’t exactly wise If Boris wants to maximize his personal gain, he shouldpropose one dollar (assuming that we play with whole dollars, not cents).Presented with this proposal, Maurice faces a Shakespearean dilemma: ‘To take

or not to take, that is the question.’ If Maurice is an ordinary Homo economicus mathematicus statisticus – that is, a maths buff and sworn rationalist – he would

ask himself just one question: ‘Which is more: $1 or $0?’ In just a few moments,he’ll remember that his kindergarten teacher used to say that ‘One is better thannone’ and he’ll take the dollar, leaving Boris with $999 There’s just one littleproblem: surely an actual game would never go this way It really doesn’t makesense for Maurice to accept the single dollar, unless he truly loves Boris andwants to be his benefactor It’s much more likely that the proposal would upsetand even insult Maurice After all, Maurice isn’t such an extreme rationalist Hehas human feelings – known as anger, honesty, jealousy Knowing that, what doyou think Boris should offer to make the entire deal happen?

We may well ask why some people refuse to accept sums that are offered tothem – often large sums too – merely because they have heard or insist onknowing how much the other guy gets How can we factor insult into

mathematical calculations? How can it be quantified? How much are people

willing to lose to avoid feeling like fools?

This game has been tried in various places, including the USA and Japan,Indonesia and Mongolia, Bangladesh and Israel, and such games have involvednot only the distribution of money but also jewelry (in Papua New Guinea) andcandy (when children played it) This game has been played between economystudents and Buddhist meditators, and even between chimpanzees

I have always found this game irresistibly appealing and have made severalexperiments with it As in many real-life situations, I’ve seen people turn downinsulting offers, many refusing to accept, for example, less than 20 per cent ofthe total (a phenomenon observed in many different cultures) Naturally, the 20per cent barrier applies only when the game is played for relatively small sums,

where ‘relative’ is very relative I mean, if Bill Gates offered me even 0.01 per

cent of his fortune, I wouldn’t be offended

As always, nothing is simple, and there are no unequivocal conclusions to bedrawn In Indonesia, for example, players were given a total sum of $100 – arelatively large sum of money there – and yet some players refused $30proposals (two weeks’ wages)! Yes, people are strange and some are stranger

than most, regardless of our expectations In Israel too we saw people who weredispleased when offered 150 shekels out of 500: deciding between 150 or 0, theychose zero! This seems like a great moment to reveal a recent major discovery inrelation in value: 150 is more than 0 This being the case, why do people makesuch choices? The respondent knows that the proposer keeps 350 and will not

accept the situation, believing it to be unfair and insulting Zero is better for his

nerves In the past, mathematicians didn’t pay enough respect to people’s sense

of justice They do now

The Ultimatum Game is fascinating from a sociological standpoint, because itillustrates the human unwillingness to accept injustice, as well as highlightingthe significance of honour The psychologist and anthropologist Francisco Gil-White from the University of Pennsylvania found that in small-scale societies inMongolia the proposers tended to offer honourable even splits regardless ofknowing that unequal splits are almost always accepted Maybe a goodreputation is more valuable than economic reward?

in the air until, at a most crucial moment, it realized it was in the air, and onlythen did the creature fall like a rock What made it fall suddenly? Oz asked Andthe answer was: knowledge If it were not aware of having no support under itspaws, a cat could just walk in the air all the way to China

How then should we play this game? What would be an optimal proposal?Well, that depends on numerous variables – including the limits of my own

appetite for risk Clearly, there’s no universal answer, since this is a personalmatter Another important question at this point relates to the number of timesthis game is played In a one-shot game, the reasonable strategy would be to takewhatever we’re offered (except if we find it too insulting), and buy a book, go tothe movies, get a sandwich, buy a funny hat, or give the cash to charity –something is better than nothing Yet when the Ultimatum Game is repeatedseveral times, that’s an entirely different story.

FALSE THREATS AND TRUE SIGNALS

In a recurring Ultimatum Game, it actually makes sense to refuse even largesums Why? To teach the other guy a lesson and give out a clear signal: ‘I’m notthat cheap! Look, you proposed $200 and I turned you down Next time, youbetter improve your offer I’d even suggest you consider splitting evenly, oryou’ll walk away with nothing.’ Alas, nothing is ever as simple as it seems atfirst glance If the responder refuses $200 in the first round, what should beproposed next? In this situation, there are several responses to consider

One idea suggests that the proposer should offer $500 as soon as the secondround starts so as not to upset the responder After all, he already blew up onedeal and it would be a shame to repeat that The problem is that going from 200

to 500 in a single leap might be viewed as weakness on the proposer’s part Theresponder could try to squeeze more by rejecting the proposal again, thinking heshould take nothing this time, but force the proposer to give him 600, 700 oreven 800 in the coming rounds

Another possible solution (the Vladimir Putin approach) is to go the otherway If the responder rejected the $200 offer, the proposer should offer $190.Where’s the logic in that? Well, such a move signals to the responder: ‘You’replaying tough? I’m tougher still Every time you refuse an offer, I’ll propose $10less I’m economically solid, and you’re welcome to refuse offers till you’re blue

in the face You’ll lose too much and I don’t care.’

What strategy should the responder follow in such a case? If he believes thatthe proposer really is tough, perhaps he should compromise The apparentruthlessness, however, could be an empty threat, so And now we have aproblem because we’re dealing with psychology and mind games Psychology isnothing like mathematics There are no certainties

In any event, it’s clear that a one-shot game and repeated games should betreated differently, and players should use different strategies Yet in some cases,players turn down large sums because they seem not to be aware that the game is

only played once In a one-shot game signalling to the other player is pointless –there is no learning curve As always (I have to repeat myself ), nothing is assimple as it seems.

THE PLEASURE OF GLOATING

In September 2006 I gave a workshop on Game Theory at Harvard A scientistwho attended told me that it’s presently known that certain people who turndown lucrative proposals in single-round ultimatum games do so for biologicaland chemical reasons It so happens that when we turn down unfair offers, ourglands secrete a large quantity of dopamine, producing an effect similar to sexualpleasure In other words, punishing rivals for being unfair is great fun When weenjoy rejecting so much, who needs those lousy $20 gains anyway?

beauty on the Ultimatum Game They had men playing against women, and vice versa It was a one-time game for $10, and both genders rated the members of

the other gender by their beauty before play started

The key result was that the men were not more generous to beautiful women(which is quite surprising), but the women offered a lot more to men they foundattractive Some even went as far as to offer $8 of the $10 allocated for theirgame! In fact, that was the only known experiment of this kind in the Westernworld where the average proposal was more than half! How can we explain that?

I believe that even though they were explicitly told that this was a one-roundgame, these women had recurring games in mind; and although men are not toogood at understanding hints, they do understand the meaning of a ‘one-timeencounter’ Apparently, the women were trying to signal to the handsome men:

‘Look, I gave you everything I got Why don’t you buy me a cup of coffee later?’They were actually attempting to develop a single game into a series Thatwonderful writer Jane Austen was on to something when she said, ‘A lady’simagination is very rapid; it jumps from admiration to love, from love to

I believe that by stepping outside the boundaries of the game, womendemonstrated a strategic and creative edge over the male participants Awoman’s concern with the long-term consequences of her conduct is animportant and most welcome quality in decision-making processes, which iswhy it’s no surprise that a huge recent study by the Peterson Institute forInternational Economics found that companies with more women leaders aremore profitable Gender equality isn’t just about fairness: it’s also the key toimproved business results

$750 overnight.) In such cases, people who wish to use the patent may ask thecourt to grant them a compulsory licence to do so without first obtaining theinventor’s permission Inventors, who fear that others might obtain a compulsorylicence, will not set unreasonable prices They will seek a deal in which theymay not keep the full profit they imagined they could make from the invention,but they will keep the licence Like the players in the Ultimatum game, inventorsalso have to remember that sometimes you have to compromise for a lesser gain,which is still better than none

WHEN REALITY AND MATHS MERGE

In another version of the Ultimatum Game, there are several proposers who offervarious ways of dividing the sum they play for and a single respondent who maychoose one proposal, granting the remainder to its proposer Here, reality andmaths merge into one In the mathematical solution, the proposer offers the

entire sum on the table because this would be the Nash Equilibrium (we’ll talkabout this later, but briefly it means that if the sum played is 100 and oneproposer offers that, no other player could fare better by offering less because theresponder would naturally reject it) In reality, willing their offer to be chosenand out of fear that other proposers may offer a higher sum, proposers tend to

offer to the responders almost the entire sum.

THE DICTATOR GAME

This is yet another version of the Ultimatum Game Here, there are only twoplayers, where the proposer, named the ‘Dictator’, has full control and theresponder must accept anything that is offered – and is, in fact, an ‘idle’ player.According to the mathematical solution, the proposer should pocket the entiresum played for and go home As you must have guessed by now, the standardeconomic assumptions are inaccurate predictors of actual behaviour Very oftenthe entire sum is not withheld: ‘dictators’ tends to give some of the money(sometimes they give substantial sums and sometimes they split the sum evenly)

to the responder Why do they do that? What does this teach us about humannature? What does it have to do with altruism, kindness, fairness and self-respect? Your guess is as good as mine

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 skills While we’re about it, we’ll become acquainted with someone I believe should be known as ‘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.’

Captain Jack Sparrow, Pirates of the Caribbean

A gang of pirates returns home from a hard day in the office, carrying 100 golddoubloons that are to be divided among the top five pirates: Abe, Ben, Cal, Donand Ern – Abe being the leader and Ern the lowliest member of the crew

Although there’s a hierarchy of rank, the group is democratic, which is whythe following principle is decided upon to determine the distribution of thebooty Abe suggests some distribution formula and all the pirates (includingAbe) vote on it If that formula wins the support of the majority of pirates, Abe’sidea is implemented and this is the end of the game; if not, he’s tossed into theocean (even democratic pirates are unruly) If Abe is no longer with us, it’s nowBen’s turn to place a motion on the table They vote again Note that now there’s

a possibility of a tie We’ll assume that in the event of a tie vote, the proposition

is dropped and the proposer will be tossed into ocean (though there’s anotherversion in which in the event of a tie the proposer has the casting vote) If Ben’sproposition wins the support of the majority of pirates, his idea is implemented;

if not, he’s tossed into the ocean and Cal will put an offer on the (shrinking)table And so on

The game continues until some suggestion is accepted by a majority vote If

Before you go on reading, please stop and think for a moment how this gameshould end, assuming that the pirates are greedy and smart

The Mathematical Solution

Mathematicians resolve such question by ‘backward induction’, going from theend to the beginning Let’s assume that we’re now at a point where Abe made asuggestion and failed, Ben’s motion was rejected and he’s no longer with us, andCal didn’t fare any better Don and Ern are the only two pirates left, and now thesolution is quite obvious: D must suggest that E take the 100 doubloons, or else

D might find himself swimming with the sharks (remember that a tied votemeans the proposition fails), which shouldn’t last long Being a clever pirate,Don suggests that Ern take the whole bag

Don Ern

Pirate Cal, who is just as clever, knows that the above will be the final stage ofthe game (if it ever gets that far, which Cal hopes to prevent at all cost).Furthermore, Cal knows he has nothing to offer Ern because Ern’s interest is to

get to that next stage no matter what However, Cal can help Don improve his

situation, compared with what would happen if he were left alone with E, andcan make Don vote for him by offering him a single doubloon (in which case,Don will vote with Cal and together they are the majority) Thus, when we havethree players, the coin distribution is 99 for Cal, 1 for Don, and 0 for Ern

Cal Don Ern

Ben is naturally aware of these calculations He knows there’s nothing he canoffer that would improve Cal’s situation, but he could make Don and Ern offersthey can’t refuse, going about it thus: Cal gets nothing, Ern ends up with onecoin, Don takes two, and Ben lands the remaining 97 coins:

Ben Cal Don Ern

Now we arrive at the point where it should be easy to see how Abe should act(being the senior pirate, he’s very experienced in matters of loot distribution).Abe makes the following suggestion He takes 97 coins; he doesn’t give a penny

to Ben (who can’t be bought in any event); he gives 1 coin to Cal (which isbetter than the 0 coins he’d receive if Abe swims and it is Ben’s turn); Don getsnothing too; and Ern is given 2 coins (Ern’s vote is cheaper to buy than Don’s).This offer will be endorsed by a majority of 3 against 2, and the pirates will go

Are the players allowed to form coalitions and make deals? If so, what wouldthis game look like? The mathematical solution always assumes that all theplayers are wise and rational, but is it wise to make this assumption? Is itrational? (I watched this game played a number of times, and never saw theparticipants reaching the mathematical solution What does this mean?) Themathematical solution ignores important emotions such as envy, insult or

schadenfreude Can feelings change the mathematical calculation?

In any event, though Abe’s distribution 97, 0, 1, 0, 2 is mathematically sound,

I advise him to show how magnanimous he is by offering his fellow pirates thedistribution 57, 10, 11, 10, 12 (i.e an extra 10 from his pot of 97) This shouldhopefully result in relative contentment among the crew and prevent a mutiny

If you feel that the Pirates Game, which is actually a multi-player version ofthe Ultimatum Game, is odd, what will you say about the following game?

The father’s lawyer opens the envelope and reads out the peculiar document

It turns out that the father has left his sons one million and ten thousand dollarsand a set of possible distribution outcomes

In the first option, Sam, the elder, may take $100 for himself right away, give

a dollar bill to his younger brother, and give the rest to charity (which would beindeed be quite charitable)

Sam Dave

100 1

Sam is under no obligation to accept that instruction, and may instead pass thelead to his younger brother Dave If Dave handles the money, he takes $1,000,Sam gets $10, and the remainder goes to charity This is the second option

Sam Dave

100 1

10 1,000

But now it is Dave’s turn to decline, if he wishes He may let Sam decide on abetter distribution method in which he takes $10,000, gives Dave $100, and therest … you guessed it

10,000 100

1,000 100,000

This, of course, is not carved in stone Dave may decide to let Sam divide themoney again, but in the following manner: $1 million for himself, $10,000 forhis hated brother, with zero dollars going to charity

of the third round Keep going now and see that the game will not reach the thirdround either … and not the second one It’s very surprising, but under the

assumption that the two brothers are of the same species, Homo economicus statisticus (that they are both calculating humans who look after only

themselves), the game should have ended with the very first step – with Samcollecting $100 and giving Dave $1 and lots of money to charity (bad intentionsmay lead to a generous outcome and the brothers winning a heavenly rewardperhaps) This is the mathematical solution – $100 for Sam and $1 for Dave, and

The opening player marks an X in one of the boxes He can choose any box hewishes

You’re welcome to try playing this on a 7 x 4 board (7 rows and 4 columns, or

vice versa).

If the game is played on a square (equal number of rows and columns), there’s

Solution: Let’s assume that the game is played between Joan and Jill If Joan

is the opening player, she should stick to the following strategy and win As herfirst move she must choose the box right above Poison, diagonally

GAME 4

NO GAME FOR OLD MEN

One of the most precious skills I acquired at grammar school, back in myhometown of Vilnius, Lithuania, was playing strategic games, on paper, in class,without being caught by my teachers I was fond of the ‘infinite’ version of tic-tac-toe (or ‘noughts and crosses’) The game often helped me to survive boringclasses I had to attend

3 x 3 grid, which is fascinating up to the age of six Older children (and adults)normally end this game with a tie, unless one of the players falls asleep halfwaythrough (which makes sense: it’s a boring game after all)

In the infinite version, however, the game is played on a board with a limitlessgrid and the goal is to create a sequence of five Xs or Os which, as in theoriginal game, can be vertical, horizontal or diagonal Players take it in turns tomark the grid with either an X or an O (according to prior agreement) and thefirst to complete a quintet wins

In the drawing on the left the X player has already won the game

In the drawing on the right it’s the O player’s turn, but she can do nothing toprevent the X player from winning Do you see why?

When I retire and have plenty of time, I’ll try to find the winning strategy for thestarting player

Still, to be completely honest, I must say I have not played it for decades and

was reminded of it while writing this book Since my plans to revisit its strategicaspects are very long-term, you are welcome to go ahead, find it, and save metime and effort.

GAME 5

THE ENVELOPE IS ALWAYS GREENER ON THE OTHER SIDE

Imagine the following I’m presented with two cash envelopes and told that one

of them contains twice as much as the other I may choose and take for myselfwhichever envelope I want

Suppose I choose an envelope, open it, and find $1,000 inside I’m pleased atfirst, but then I start wondering about the content of that other envelope, theunchosen one Of course, I don’t know what’s in it It could be $2,000, whichmeans I made a bad choice, or it could be $500 I’m sure you can see theproblem Reflecting on this a while, I reach the following conclusion: ‘I’m nothappy because the average of the potential money in the unchosen envelope islarger than the sum in my hands After all, if it contains either $2,000 or $500with equal probabilities, the average is $1,250, which is more than $1,000 Iknow my maths!’

In truth, anything I find in my envelope will prove Murphy’s Law which states

that ‘Anything that can go wrong, will go wrong.’ The unchosen envelope will

always be better than mine, on average If I found $400 in mine, the other couldcontain $800 or $200, and the mean would be $500 Thinking like that, I cannever choose right The unchosen gain will forever be 25 per cent better thanmine So would I change my mind if that option is offered to me before Iexamine the contents of the other envelope? If I do that, I start a never-endingloop So why did such a simple choice get so complicated?

In truth, the story I just told you is a famous paradox that was first presented

by a Belgian mathematician named Maurice Kraitchik (1882–1957), except thathis story was about neckties Two men argued about whose tie was nicer Theyasked a third person, the leading Belgian expert on neckties, to referee and heagreed to the request, but only on condition that the winner would give his tie tothe loser as a consolation prize The two tie owners considered this idea brieflyand agreed to it, because they both thought: ‘I don’t know whether my necktie isnicer I may lose my necktie, but I may win a better one, so the game is to myadvantage Therefore the wager is to my advantage.’ How can both competitorsbelieve they have the advantage?

In 1953 Kraitchik offered another version of the story, involving two other

quarrelsome Belgians They didn’t wear ties, because they were so stuffed withBelgian chocolate that they couldn’t breathe Instead they challenged each otherabout the contents of their wallets and decided that the one who turns out richerand happier would give his wallet to his poorer rival If they are tied, they goback to their chocolates.

Again, both believed they had the upper hand If they lost the bet, they wouldreceive more money than they had to give if they won Is this a great game orwhat? Try playing it with people you don’t know on the street, and see what

happens In 1982 Martin Gardner made the story popular in Aha! Gotcha, one of

the finest, simplest and most amusingone of the finest, simplest and mostamusing books ever written about smart thinking

Barry Nalebuf (Milton Steinbach Professor of Management at Yale School ofManagement), a leading Game Theory expert, offered the envelope version ofthis story in a 1989 article You may find it surprising, but even today this gamehas no solution that all statisticians agree on

One of the suggested solutions involves considering geometric mean asopposed to arithmetic mean Geometric mean is the square root of the product oftwo numbers For example, the geometric mean of 4 and 9 is the square root oftheir product (that is, the two numbers multiplied together) – namely, 6 Now, if

we found X dollars in our envelope and knew that the other contained 2X or ½X,the geometric mean of the other envelope would be X, which is exactly what wehave in our hands The logic behind using geometric mean is the fact that wespeak of multiplication (‘twice as much’) and not of addition If we said that oneenvelope contains $10 more than the other, we would use the arithmetic mean,find it and end up with no paradox, because if our envelope contains X and theother holds X+10 or X-10, then the mean of the unchosen envelope is X

Students who take classes in probabilities would say that you ‘cannot defineuniform distribution for a set of rational numbers’ How impressive is that?

If you don’t understand what this means, that’s perfectly fine because the bestversion of this paradox has nothing to do with probabilities This last version

appears in Satan, Cantor and Infinity, a brilliant book (with a brilliant title, don’t

you think?) by Raymond M Smullyan, an American mathematician, philosopher,classical pianist and magician Smullyan presents two versions of this paradox:

1 If there are B notes in your envelope, you’ll either gain B or lose ½B if

you replace this envelope with the other Therefore you should switch

2 If the envelopes contain C and 2C respectively and you choose to switch

one for another, you will either gain C or lose C, so the chances are even,and you can gain as much as you can lose

Either way, many maintain pessimistically that there’s no paradox here, thatsuch is life, and no matter what you do or wherever you go, the opposite willalways be better For example, if you’re married, perhaps you should havestayed single After all, Anton Chekhov wrote: ‘If you are afraid of loneliness,

do not marry.’ Yet if you chose to remain single, you’re wrong again The firsttime the phrase ‘not good’ appears in the Bible is in Genesis 2:18, ‘It is not goodfor man to be alone.’ God said that, not me

GAME 6

GOLDEN BALLS

Golden Balls is a British TV game show, which aired from 2007 to 2009 We

won’t elaborate on its rules and moves, but in the final stage of the game theremaining two players need to negotiate on how to split a given sum of moneybetween them Each player has two balls with stickers on them: one that saysSPLIT and one that reads STEAL If both players choose SPLIT, the money isdivided between them; if both choose STEAL, they both end up with nothing;and if they choose a different ball each, the one who chose STEAL takes the pot.The players may discuss their situation before they choose

In most cases, players try to convince each other to choose SPLIT, and thissometimes works Many YouTube videos of the game contain quite a few heart-breaking scenes of players who trusted their opponent, chose SPLIT, and foundout they’d been cheated

One day, a player named Nick came along with an unexpected approach Nicktold his opponent, Ibrahim, that he was going for STEAL and begged Ibrahim to

go for SPLIT, promising he’d split the money (a pot of £13,600 in this instance)

between them after the game was over Ibrahim couldn’t believe his ears: Nickrepeatedly promised he’d cheat while insisting that saying so in advance showedhis basic honesty: Ibrahim could be confident of getting half the money ‘Youcan’t lose if you choose SPLIT,’

Nick told him ‘You can only gain.’ At that point the players were asked to stoptalking and grab a ball

Ibrahim chose SPLIT, but Nick took the SPLIT ball too! Why did he do that?Nick was so certain he’d talked Ibrahim into cooperating that he chose SPLIT tosave himself the trouble of dividing the money at the end of the game

You have to admit that Nick probably deserved the title of ‘Strategist of theYear’

This game is not only about negotiation strategies, but also about trustbetween players

can be found in the Talmud, in Sun Tzu’s Art of War and in Plato’s writings.)

Yet some believe that the Game Theory discipline was conceived in 1913when German mathematician Ernst Zermelo (1871–1953) presented his theorem

on chess, the ‘game of kings’: ‘Either white can force a win, or black can force awin, or both sides can force at least a draw.’ In other words, he stated there areonly three options:

In fact, Zermelo proved that the game of chess is not different from finite (3 x3) tic-tac-toe: as already mentioned, if both players of tic-tac-toe are nottemporarily insane (which sometimes happens), all games will always end in atie There’s no other option Even players who lose tic-tac-toe games one afterthe other at first will eventually find a way to never lose, which would make thealready unexciting tic-tactoe game as boring as reading a book with white pagesand no type

Zermelo managed to prove that chess (and many other games) is almostexactly the same game as tic-tac-toe, the difference being not qualitative butquantitative

In the game of chess, ‘strategy’ is a set of responses to any situation that maymaterialize on the board Clearly, two players can have a huge number ofstrategies between them Let’s mark the strategies of the White (first) player with

S and his rival’s with T The Zermelo theorem, as noted, speaks of only threeoptions:

Either White has a strategy (let’s call it S4) with which he’ll always win,regardless of what Black does;

(W = White wins; B = Black wins; X = tie)

Or Black has a strategy (let’s call it T3) with which he’ll always win, regardless

of what White does:

Or both players have a combination of strategies that, if followed, will alwaystake the game to a tie (just like with tictac- toe):

If that’s the case, why do people keep playing chess? Why is it eveninteresting? The truth is that when we play or watch a game of chess, we can’tknow which of the three cases we’re facing Supercomputers may perhaps beable to find the right strategies in the future, but we’re nowhere near that stage,which is why the game remains so intriguing According to the Americanmathematician and cryptographer Claude Shannon (known as the ‘father ofinformation theory’), there are more than 10 to the power of 43 legal positions inchess Take a look at that number: 10,000,000,000,000,000,000,000,000,000,000,000,000,000,000 Wow! Many think that the time frame required for acomputer to check all the chess possibilities is beyond the limits even of themost modern technology

Once I had lunch with Boris Gelfand, finalist of the World ChessChampionship in 2012, and told him that not so many years ago, I, a lousy chessplayer, could beat any computer program but that today the computers beat mewith such an ease that it’s embarrassing He remarked that the gap betweenhuman players and computers is getting larger every day, and not in our favour.Today, he added, computer programs are able to easily beat the strongest humanplayers; the gap is so large that humanversus- computer matches are no longer ofany interest In the game of chess humans have suffered a stinging defeat Today,

concluded Grandmaster Gelfand, for a human being to play chess against strongcomputer programs (known as ‘engines’) is very much like wrestling against agrizzly bear – not advisable.

Human-versus-human chess games are much more interesting

In our own time, when chess is played by Grandmasters, sometimes the playerwho starts the game wins, sometimes the responding player wins, and sometimesthe game ends in a draw Chess players and theorists generally agree that theWhite player who makes the first move has a slight advantage Statistics supportthis view: White consistently wins a little more often than Black, around 55 percent of the matches

Chess players have long debated whether – if both players have a perfectgame – White will always win or the game will end in a draw They don’tbelieve that there’s a winning strategy for Black (although, contrary to thispopular opinion, the Hungarian Grandmaster András Adorján thinks that the idea

of White’s advantage is a delusion)

My guess, as a retired and unsuccessful chess player, is that if both chessplayers played it right, it would always end in a draw (just like tic-tac-toe) In thefuture, computers will be able to check all the relevant options and decidewhether I am right about my draw assertion

Interestingly enough, scientists still can’t agree on the true meaning of theZermelo theorem It was written in German originally, and if you’ve ever readscientific or philosophical texts in German (Hegel would be a fine example) youtoo wouldn’t be surprised that the meaning is vague (how lucky we are that thecurrent language of science is English)

* This story is based on the famous game known as Centipede that was first introduced in 1981 by Robert Rosenthal.

THE KEYNESIAN BEAUTY CONTEST

Imagine a fictional newspaper contest in which participants are asked to choosethe most attractive face from 20 photographs Those who pick the most popularface are then eligible to a prize – a lifetime subscription to the newspaper, acoffee machine and a badge of honour

How should we play this game? Let’s suppose that my favourite photo is #2.Should I give it my vote? Yes – if I want my opinion known No – if I want thesubscription, the machine and the badge

The great English economist John Maynard Keynes (1883– 1946) described aversion of this contest in chapter 12 of his book The General Theory of Employment, Interest and Money (1936), saying that if we want to win the prize,

we need to guess which of the photographs will be favoured by the majority ofreaders This is the first degree of sophistication Yet if we are even moresophisticated, we should jump to the second degree and try to guess which of the

photos other players will think that others will choose as the most beautiful As

Keynes put it, we need to ‘devote our intelligences to anticipating what averageopinion expects the average opinion to be’ Naturally, we can go on to the nextlevel, and onwards

Keynes, of course, was not speaking about photographs, but about playing onthe stock market, where in his opinion a similar behaviour was at work After all,

if we intend to buy a share because we think it’s a good one, we’d be actingpoorly It would be wiser to keep the money under the mattress or in a savingsaccount The value of shares rises not when they are good, but when enoughpeople believe they are, or when enough people believe that enough peoplebelieve they are

The Amazon share price is a good example In 2001 Amazon’s shares wereworth more than all the other American booksellers combined – even beforeAmazon earned a single dollar That happened because many believed that manybelieve that many believe that Amazon is going to be Amazon

The following game is a fine example of Keynes’s idea Alain Ledoux did

much to popularize this version, having published it in his French magazine Jeux

et Stratégie in 1981.