by Haim Shapira
One of the suggested solutions involves considering geometric mean as opposed to arithmetic mean. Geometric mean is the square root of the product of two numbers. For example, the geometric mean of 4 and 9 is the square root of their 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 we have in our hands. The logic behind using geometric mean is the fact that we speak of multiplication (‘twice as much’) and not of addition. If we said that one envelope 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 the other 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 define uniform 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 best version 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.
Confused? So am I.
Either way, many maintain pessimistically that there’s no paradox here, that such is life, and no matter what you do or wherever you go, the opposite will always be better. For example, if you’re married, perhaps you should have stayed 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 first time the phrase ‘not good’ appears in the Bible is in Genesis 2:18, ‘It is not good for 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 the remaining two players need to negotiate on how to split a given sum of money between them. Each player has two balls with stickers on them: one that says SPLIT and one that reads STEAL. If both players choose SPLIT, the money is divided 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.
SPLIT STEAL
SPLIT (X/2, X/2) (0, X)
STEAL (X, 0) (0, 0)
A quick glance at the chart above, based on the rules of the game, clearly shows that STEAL is better than SPLIT, if each player thinks only about his or her own gains. The problem is that if both players do that, both lose. (Yes, it’s quite similar to the Prisoner’s Dilemma, which you may already know. We’ll discuss this famous dilemma later.)
In most cases, players try to convince each other to choose SPLIT, and this sometimes works. Many YouTube videos of the game contain quite a few heart-breaking scenes of players who trusted their opponent, chose SPLIT, and found out they’d been cheated.
One day, a player named Nick came along with an unexpected approach. Nick told 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: Nick repeatedly promised he’d cheat while insisting that saying so in advance showed his basic honesty: Ibrahim could be confident of getting half the money. ‘You can’t lose if you choose SPLIT,’
Nick told him. ‘You can only gain.’ At that point the players were asked to stop talking 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 to save 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 the Year’.
This game is not only about negotiation strategies, but also about trust between players.
GAME 7
THE INTRICACIES OF CHESS
(The following is for chess and maths lovers only)
Many believe Game Theory came to life in 1944, the year of publication of the canonical book Theory of Games and Economic Behaviour, authored by the great mathematician John von Neumann (1903–1957) and the economist Oscar Morgenstern (1902–1977). (The problems that Game Theory addresses, however, have more or less existed since the beginning of time. Early examples 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 1913 when 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 a win, or both sides can force at least a draw.’ In other words, he stated there are only three options:
1 White has a strategy that, when followed, will always win.
2 Black has a strategy that, when followed, will always win.
3 Black and White have a combination of strategies that, when followed, will always end in a tie.
When I first read this theorem, I remember thinking (with my usual sarcasm), ‘Wow! This is clever … and new … This German thinker is telling me that either White wins, or Black wins, or their game ends in a tie. And there I was, believing there are so many more options …’ It was only when I started reading his proof that I understood what the theorem was about.
In fact, Zermelo proved that the game of chess is not different from finite (3 x 3) tic-tac-toe: as already mentioned, if both players of tic-tac-toe are not temporarily insane (which sometimes happens), all games will always end in a tie. There’s no other option. Even players who lose tic-tac-toe games one after the other at first will eventually find a way to never lose, which would make the already unexciting tic-tactoe game as boring as reading a book with white pages and no type.
Zermelo managed to prove that chess (and many other games) is almost exactly the same game as tic-tac-toe, the difference being not qualitative but quantitative.
In the game of chess, ‘strategy’ is a set of responses to any situation that may materialize on the board. Clearly, two players can have a huge number of strategies 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 three options:
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 always take the game to a tie (just like with tictac- toe):
If that’s the case, why do people keep playing chess? Why is it even interesting? The truth is that when we play or watch a game of chess, we can’t know which of the three cases we’re facing. Supercomputers may perhaps be able 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 American mathematician and cryptographer Claude Shannon (known as the ‘father of information theory’), there are more than 10 to the power of 43 legal positions in chess. 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 a computer to check all the chess possibilities is beyond the limits even of the most modern technology.
Human-versus-human chess games are much more interesting.
In our own time, when chess is played by Grandmasters, sometimes the player who starts the game wins, sometimes the responding player wins, and sometimes the game ends in a draw. Chess players and theorists generally agree that the White player who makes the first move has a slight advantage. Statistics support this view: White consistently wins a little more often than Black, around 55 per cent of the matches.
Chess players have long debated whether – if both players have a perfect game – White will always win or the game will end in a draw. They don’t believe that there’s a winning strategy for Black (although, contrary to this popular 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 chess players played it right, it would always end in a draw (just like tic-tac-toe). In the future, computers will be able to check all the relevant options and decide whether I am right about my draw assertion.
Interestingly enough, scientists still can’t agree on the true meaning of the Zermelo theorem. It was written in German originally, and if you’ve ever read scientific or philosophical texts in German (Hegel would be a fine example) you too wouldn’t be surprised that the meaning is vague (how lucky we are that the current 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.
Spotlight
THE KEYNESIAN BEAUTY CONTEST
Imagine a fictional newspaper contest in which participants are asked to choose the most attractive face from 20 photographs. Those who pick the most popular face are then eligible to a prize – a lifetime subscription to the newspaper, a coffee 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 the subscription, the machine and the badge.
The great English economist John Maynard Keynes (1883– 1946) described a version 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 of readers. This is the first degree of sophistication. Yet if we are even more sophisticated, 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 average opinion expects the average opinion to be’. Naturally, we can go on to the next level, and onwards.
Keynes, of course, was not speaking about photographs, but about playing on the 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 acting poorly. It would be wiser to keep the money under the mattress or in a savings account. The value of shares rises not when they are good, but when enough people believe they are, or when enough people believe that enough people believe they are.
The Amazon share price is a good example. In 2001 Amazon’s shares were worth more than all the other American booksellers combined – even before Amazon earned a single dollar. That happened because many believed that many believe 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.
ALAIN LEDOUX’S GUESSING GAME
A group of people in a room are asked to choose one number each – between 0 and 100. After they do, the game organizer finds the average of the selected numbers and multiplies it by 0.6. The result is the target number, and whoever is closest to that number wins a Mercedes (they were available at a good discount).
What number would you choose? Take a moment to think about that.
There are two ways to make your selection: normatively or positively.
In the normative version, which assumes that all the other players are wise and rational, we should choose zero. Here’s why. If we assume that people select these numbers randomly, the expected average would be 50; so to win this game, 50 x 0.6 = 30 – which is the number to choose. But wait! What if everyone figures that out? Then the average would be 30, and thus we should choose 18 (the result of 30 x 0.6). And what if everyone figured that out? The average would be 18, and thus we should choose 10.8 (the result of 18 x 0.6). The story, of course, doesn’t end here, and if we keep going in this direction we’ll eventually reach zero.
The strategy of selecting 0 is the Nash Equilibrium (we’re going to meet this mega-famous concept in the following chapter), meaning that once I realize that everyone has opted for 0, there’s no reason for me to do otherwise.
Choosing 0 is the normative recommendation: that is, it’s the rational choice if we believe that everyone else is wise and rational. But what should we do if they aren’t?
The positive approach to this game is based on the fact that it would be very hard to guess the distribution of numbers ordinary people would choose, and that psychology and intuition play a more important role than mathematics.
In some cases, people simply don’t understand the game. For example, a faculty member in one of the world’s leading universities chose 95. Why did he do that? I mean, even if for some odd reason you believe that everyone has chosen 100, the average would be 100, which means that the highest winning number conceivable is 60. Yet this strange choice (95) could still win the game if all the other players opted for an even stranger strategy and choose 100.
Once, a professor of physics explained to me that he chose 100 in order to raise the average and to punish all his superclever colleagues who chose low numbers. ‘They have to learn that life isn’t a picnic.’
Incidentally, I’ve tried this game more than 400 times now, and 0 won only once (in a small group of children with extraordinary mathematical skills). When a group opts for low numbers, this means that its members have thought about the problem more than other groups, and have thought that other members can think too.
Clearly, many diverse factors determine the number chosen by the experiment’s participants. In certain economy classes I gave, my students kept scoring poorly until one day I realized: they just weren’t motivated enough! Then, because I couldn’t give them a little Mercedes each time we played the game, I told them that I’d give the winning student a 5-point bonus on top of his or her grade. Their scores improved immediately.
Try this game with your friends. Beware of disappointments.
Chapter 5
THE MARRIAGE BROKER
(A Little on the Connections between the
Nash Equilibrium, Buffaloes, Matchmaking and the Nobel Prize)
In this quite long chapter we’ll learn about the legendary Nash Equilibrium and see how it’s manifested in different situations – ranging from matchmaking strategies to the struggles between lionesses and buffaloes. We’ll also find out how the algorithm for finding matches for two equally sized groups of men and women that absolutely rule out infidelity won the Nobel Prize in Econ
omics.
BLONDES IN BARS
John Nash, the great mathematician and a Nobel Prize laureate, and his wife Alicia were killed on 23 May 2015, in a car accident on their way home after visiting Norway, where Nash had received the prestigious Abel Prize.
In the first half of A Beautiful Mind – a film based (quite loosely) on John Nash’s biography – we witness the following scene. Nash and some of his friends are sitting in a pub as one blonde woman and a few brunettes enter. Film director Ron Howard didn’t really trust his viewers’ intelligence and made it plain that the blonde was the prettiest while the other women were … well, brunettes (forgive me: this is what’s in the movie). Nash and crew decide to make a pass at the blonde, but Nash stops everyone after a moment’s thought and reels off a strategic argument. ‘Our strategy is faulty,’ he says (I’m paraphrasing). ‘If we all go for the blonde, we’ll end up hindering each other. Since it’s generally unacceptable for one girl to leave a pub with five guys, certainly not on a first date, ‘not a single one of us is going to get her. So then we go for her friends, and they’ll all give us the cold shoulder because no one likes to be second choice. But what if none of us goes for the blonde? We won’t get in each other’s way and we won’t insult the other girls. It’s the only way to win. It’s the only way we all get laid.’
Thus spake Nash.
After he’d convinced his friends that approaching her would be poor strategy, the blonde was left suitorless and Nash won her, which was his plan all along. While his friends remained seated in the corner of the pub, angry, embittered, and not understanding how they’d fallen for this trick, Nash approached the belle, talked to her, and even thanked her for something (possibly a mathematical idea that just popped into his mind?), but he left her there soon after that. It seems that the filmmakers’ idea was to present Nash as a scatterbrain scientist who is interested in formulae and equations more than in women. Some claim that a mathematician is a person who’s found something more interesting than sex. Oh well.