Gladiators, Pirates and Games of Trust

by Haim Shapira

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

  1 Playing rationally against an irrational opponent is often irrational.

  2 Playing irrationally against an irrational opponent is often rational.

  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 the word ‘rational’ isn’t clear – after all, Mo wins the game and is the one who walks 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 makes him 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 offer and wouldn’t give away Friedrich Paulus, a German field marshal the Soviets captured in the Battle of Stalingrad, for his own son Yakov Dzhugashvili, who’d been in German captivity since 1941. ‘You don’t trade a Marshal for a Lieutenant,’ Stalin declared. At the same time, some people give their kidneys to perfect strangers. Why? Your guess is as good as mine. And Vladimir Putin woke up one morning and decided that the Crimean Peninsula belongs to Mother Russia: I wouldn’t have even started guessing that.

  Incidentally, after the fact, political pundits offered clever explanations of Putin’s act (you’re welcome to Google it). The only problem is that none of them had predicted the move, which proves they didn’t have the faintest idea about what 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 when examined for the second and third time), and no mathematical model can capture their entire full-scale complexity. Maths is better at studying the rules of nature than the nature of humankind.

  ***

  Even if we aren’t fully aware of the fact, conducting negotiations is an essential part of our lives. We do this all the time with our spouses, children, partners, bosses, subordinates and even total strangers. Of course, negotiations are a keystone of inter-state diplomatic relations or the conduct of political bodies (for example, when coalitions are formed). It’s therefore quite surprising that not only 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 of negotiation.

  Chapter 3

  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 Bernd Schwarze, wrote an article about an experiment they’d conducted whose results surprised economists (but no one else). Known as the Ultimatum Game, the experiment has since become one of the most famous and most studied games in the world.

  The game is similar to the Blackmailer’s Paradox, but the differences are crucial. 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 a room. Let’s call them Maurice and Boris. Boris (let’s call him the proposer) is given $1,000 and instructed to share it with Maurice (let’s call him the responder) in whatever way he sees fit. The only condition here is that Maurice must agree to Boris’s method of division: if he doesn’t, the $1,000 will be taken away and both players will end with nothing.

  It should be noted that this is a game of two fully informed players. Thus, if Boris 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’ $10 offer? 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 a decision based on mathematical principles (a ‘normative’ decision) and one based on intuitive principles and psychology (a ‘positive’ decision).

  Mathematically, this game is easily resolved, but the wonderful easy solution isn’t exactly wise. If Boris wants to maximize his personal gain, he should propose 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 than none’ and he’ll take the dollar, leaving Boris with $999. There’s just one little problem: surely an actual game would never go this way. It really doesn’t make sense for Maurice to accept the single dollar, unless he truly loves Boris and wants to be his benefactor. It’s much more likely that the proposal would upset and even insult Maurice. After all, Maurice isn’t such an extreme rationalist. He has human feelings – known as anger, honesty, jealousy. Knowing that, what do you think Boris should offer to make the entire deal happen?

  We may well ask why some people refuse to accept sums that are offered to them – often large sums too – merely because they have heard or insist on knowing 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 involved not only the distribution of money but also jewelry (in Papua New Guinea) and candy (when children played it). This game has been played between economy students and Buddhist meditators, and even between chimpanzees.

  I have always found this game irresistibly appealing and have made several experiments with it. As in many real-life situations, I’ve seen people turn down insulting offers, many refusing to accept, for example, less than 20 per cent of the total (a phenomenon observed in many different cultures). Naturally, the 20 per 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 be drawn. In Indonesia, for example, players were given a total sum of $100 – a relatively large sum of money there – and yet some players refused $30 proposals (two weeks’ wages)! Yes, people are strange and some are stranger than most, regardless of our expectations. In Israel too we saw people who were displeased when offered 150 shekels out of 500: deciding between 150 or 0, they chose zero! This seems like a great moment to reveal a recent major discovery in relation in value: 150 is more than 0. This being the case, why do people make such 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 it illustrates the human unwillingness to accept injustice, as well as highlighting the significance of honour. The psychologist and anthropologist Francisco Gil- White from the University of Pennsylvania found that in small-scale societies in Mongolia the proposers tended to offer honourable even splits regardless of knowing that unequal splits are almost always accepted. Maybe a good reputation is more valuable than economic reward?

  ‘A good name is better than fine perfume.’

  Ecclesiastes 7:1

  IGNORANCE IS BLISS

  Incidentally, none of that strange behaviour (rejecting substantial sums of money in one-shot anonymous games) would have happened if the responder hadn’t known the sum that the proposer gets to keep in the end. Thus, knowledge isn’t always an advantage. If I proposed you accept $100, giving you no additional information (not telling you I’ll get to keep $900 if you accept my offer), you’d probably take the money and buy yourself something nice. Ecclesiastes had good reason to state that ‘in much wisdom is much grief ’ (1:18). Similarly, Israeli writer Amos Oz talked about an American cartoon he once saw, where a cat was running and running until it reached an abyss. What did the cat do? If you ever watched Tom and Jerry, you know the answer: the cat didn’t stop. It kept running in the air until, at a most crucial moment, it realized it was in the air, and only then did the creature fall like a rock. What made it fall suddenly? Oz asked. And the answer was: knowledge. If it were not aware of having no support under its paws, 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 personal matter. Another important question at this point relates to the number of times this game is played. In a one-shot game, the reasonable strategy would be to take whatever we’re offered (except if we find it too insulting), and buy a book, go to the 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 repeated several times, that’s an entirely different story.

  FALSE THREATS AND TRUE SIGNALS

  In a recurring Ultimatum Game, it actually makes sense to refuse even large sums. Why? To teach the other guy a lesson and give out a clear signal: ‘I’m not that cheap! Look, you proposed $200 and I turned you down. Next time, you better improve your offer. I’d even suggest you consider splitting evenly, or you’ll walk away with nothing.’ Alas, nothing is ever as simple as it seems at first glance. If the responder refuses $200 in the first round, what should be proposed next? In this situation, there are several responses to consider.

  One idea suggests that the proposer should offer $500 as soon as the second round starts so as not to upset the responder. After all, he already blew up one deal 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. The responder could try to squeeze more by rejecting the proposal again, thinking he should take nothing this time, but force the proposer to give him 600, 700 or even 800 in the coming rounds.

  Another possible solution (the Vladimir Putin approach) is to go the other way. 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’re playing tough? I’m tougher still. Every time you refuse an offer, I’ll propose $10 less. 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 that the proposer really is tough, perhaps he should compromise. The apparent ruthlessness, however, could be an empty threat, so ... And now we have a problem because we’re dealing with psychology and mind games. Psychology is nothing like mathematics. There are no certainties.

  In any event, it’s clear that a one-shot game and repeated games should be treated 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 as simple as it seems.

  THE PLEASURE OF GLOATING

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

  MEN, WOMEN, BEAUTY AND SIGNALS

  Dostoevsky stated that ‘Beauty will save the world.’ I don’t know about the world, but how about the importance of beauty in the Ultimatum Game? (Beauty is fascinating, even in economic terms. For example, it’s a known fact – the Beauty Premium – that good-looking people earn more than their lessfavoured colleagues.) In 1999 Maurice Schweitzer and Sara Solnik studied the impact of 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 found attractive. Some even went as far as to offer $8 of the $10 allocated for their game! In fact, that was the only known experiment of this kind in the Western world 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-round game, these women had recurring games in mind; and although men are not too good at understanding hints, they do understand the meaning of a ‘one-time encounter’. 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. That wonderful writer Jane Austen was on to something when she said, ‘A lady’s imagination is very rapid; it jumps from admiration to love, from love to matrimony, in a moment.’

  I believe that by stepping outside the boundaries of the game, women demonstrated a strategic and creative edge over the male participants. A woman’s concern with the long-term consequences of her conduct is an important and most welcome quality in decision-making processes, which is why it’s no surprise that a huge recent study by the Peterson Institute for International Economics found that companies with more women leaders are more profitable. Gender equality isn’t just about fairness: it’s also the key to improved business results.

  THE COURTHOUSE ULTIMATUM

  An example of an Ultimatum Game played in a courthouse setting is the case of ‘compulsory licensing’. When someone comes up with an original new idea, he or she may register it as a patent, which in practice is a licensed monopoly. That is, the patent owner may prevent everybody else from using their invention. Though created by the law to encourage people to contribute to society by inventing new and improved things, in fact this monopoly might be abused by proprietors who don’t let others use their patent, or else charge a lot of money for licensing – particularly when the product has wide potential usage. (Recently, Turing Pharmaceuticals CEO Martin Shkreli jacked up the price of Daraprim, an anti-parasitic drug commonly used to treat HIV patients, from $13.50 a pill to $750 overnight.) In such cases, people who wish to use the patent may ask the court to grant them a compulsory licence to do so without first obtaining the inventor’s permission. Inventors, who fear that others might obtain a compulsory licence, will not set unreasonable prices. They wi
ll seek a deal in which they may 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, inventors also 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 offer various ways of dividing the sum they play for and a single respondent who may choose one proposal, granting the remainder to its proposer. Here, reality and maths 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 talk about this later, but briefly it means that if the sum played is 100 and one proposer offers that, no other player could fare better by offering less because the responder would naturally reject it). In reality, willing their offer to be chosen and out of fear that other proposers may offer a higher sum, proposers tend to offer to the responders almost the entire sum.