by Haim Shapira
Well, it doesn’t matter what Jane had to say. You can imagine how such disputes go on and on. So how do you divide the cost of installing an elevator when tenants live on different floors?
I could tell you how, but elevators are rather boring. Instead, I’d like to tell you a story about an airport.
THE AIRPORT PROBLEM
Once upon a time, there were four good friends – Abe, Brian, Calvin and Dan. They all did so well in their lives that they decided to buy themselves airplanes. They further agreed that they would jointly build a private landing strip that would serve only themselves. Being the poorest of the four, Dan bought a two-seater Cessna. Cal chose to spend a little more and purchased a four-seater jet. Brian, who was slightly richer, acquired a Learjet 85. Abe, who’d recently made a fortune, got carried away completely. He bought a double-deck Airbus A380 and spoiled himself with an on-board swimming pool, a state-of-the-art gym, an Indonesian spa and a holographic screening room. He also hired a former astronaut as his pilot and a group of top models as stewardesses. The whole thing only cost him $444 million.
Then came the time to start building the airstrip that could serve Abe’s Airbus. The price tag on that was $200,000. Clearly, the other three would be able to use it with their smaller planes. The airstrip that Brian actually needed, however, should have cost only $120,000; Calvin could have managed with a $100,000 strip; and pauper Dan needed only $40,000 for his tiny plane.
How should the four friends divide the $200,000 between them and build an airstrip that would serve them all without paying another person’s share?
Being the richest and the oldest, Abe suggested a relativistproportional scheme: he should pay twice as much as Cal (200/100) and five times Dan’s share (200/40); Brian should pay Dan’s sum times three (120/40), and so on. If you want to try solving this maths quiz for the sixth grade, you’re welcome to check my figures: that airstrip will set Abe back $86,956, Brian will have to pay $52,175, Cal must give $43,478, and Dan needs to contribute $17,391 (the figures were slightly rounded up to make a total of $200,000).
Three of the four felt that was a fair deal, but Dan, who’d incurred new debts (partly because he’d bought the plane), had a different idea. ‘If all of you had bought nice little planes like I did, we could all make do with a $40,000 airstrip. Abe bought the most expensive and largest jet, so he should offer to build an airstrip at his own expense. He’d have paid $200,000 anyway, if we weren’t in on the deal with him. In fact, we’re only helping him with his own project. I know we are friends and all, and I don’t expect people who are super-rich to make grand gestures. I only want a fairer and more logical distribution. If you studied Game Theory, and if you specialized in the Shapely value, you’d know it has its advantages.
Lloyd Shapley, as you, my reader, already know, won a Nobel Prize in Economics in 2012, and so I believe he deserves our attention. For my part I find the method Dan suggested, based on Shapley, a lot fairer than the proportional division that Abe suggested. ‘We’ll all use the airstrip segment that my jet needs,’ said Dan. ‘That segment should cost some $40,000 and thus should be equally divided between us so that each of us pays $10,000.
‘I don’t need the next segment, but Cal does and Abe and Brian will use it too. It should cost another $60,000 (placing the airstrip cost at $100,000), which the three of you should split. Thus, each of you will pay $20,000. Similarly, Abe and Brian should go Dutch on the $20,000 segment that they both need, and Abe should pay another $80,000 for the segment that will only serve him (and drives the cost up to a total of $200,000).’
Here’s a chart summarizing that proposal:
The table below compares the proposal of ‘poorest’ Dan with that of ‘richest’ Abe:
Dan’s Abe’s
Dan 10,000 17,391
Cal 30,000 43,478
Brian 40,000 52,175
Abe 120,000 86,956
Clearly, Dan’s offer is good for him, but also for Cal and Brian. It was put to democratic vote and Dan’s suggestion was endorsed by a majority of 3 against 1. This is social justice at its best: the mogul will pay more than half the cost.
That would be Shapley’s solution. But even though he’s a Nobel laureate and all, Shapley’s solution, like any other in Game Theory, is merely a non-binding recommendation.
This could have been the happy ending of the story, but Abe announced he wouldn’t pay – obviously not minding turning the other three into his former friends. He threatened that if his suggestion for proportional distribution of the cost was rejected, he’d quit the quartet and let his three poor friends pay for their own strip, if they wanted to. ‘If I have to pay more than half,’ stated the Airbus owner, ‘I might as well pay for the whole damn thing and have my own private airfield. You know I can afford it.’
The three friends asked Abe to give them a minute, in which they realized that if Abe were to quit, they’d have to pay $120,000 for the airstrip and somehow divide that between them. Yet if they accepted Abe’s suggestion, the three would have to pay only $113,044 (the total $200,000 minus Abe’s share, $86,956), which is less than the $120,000 they’d have to come up with if Abe quit the deal.
Did they surrender to the mogul, or did they insist on their independence? Can you guess? Hint: Brian is the new oligarch.
Let’s calculate the costs (with new oligarch Brian’s figure rounded up for the sake of social justice):
Segment cost 40,000 60,000 20,000 Total per perso
Dan 13,333 0 0 13,333
Cal 13,333 30,000 0 43,333
Brian 13,334 30,000 20,000 63,334
If the cost is divided according to Dan’s new model, it’s better for both him and Cal than Abe’s plan (though for Cal the difference is minute), but Brian stands to lose, being the richest of the three.
Will Brian quit? Will he join Abe? If the group sought arbitration, how would that be settled? What does this question have to do with the ubiquitous neighbourly disputes over installing elevators in apartment buildings? How is that issue associated with the dilemma that economic leaderships must face when dealing with the fair distribution of infrastructure costs among various population strata? You have the tools to consider these fascinating questions.
Chapter 13
TRUST GAMES
In this chapter we’ll meet the great Indian economist Kaushik Basu, who invented the thought experiment known as the ‘Traveller’s Dilemma’. Professor Basu will show us that in this game, looking after your interests and not trusting others will actually harm you (and others). In this situation the Nash Equilibrium is a poor outcome – players do better by setting strategies aside and just lowering the bucket into their personal well of trust.
THE CHINESE VASE
Two friends, appropriately named X and Y, attended a strategic-thinking workshop at Harvard University. Before they flew back home, they visited Boston’s Charles Street, which is known for its antique shops. In one such shop, they found a pair of identical Chinese vases that were magnificently painted and particularly cheap. Each bought a vase, but as fortune would have it sometimes, the airline lost both their bags, with the vases inside. The airline company decided to compensate X and Y immediately. They were asked to step into the office of the Lost and Found Department manager. After a short conversation, the manager realized they were interested in strategic thinking, and thus she decided to compensate them along the following lines. The two were to go into separate rooms and write down the sums they wanted for the lost vases on a piece of paper. The sums could vary between $5 and $100. If both wrote the same figure, each of them would get it. If they wrote different figures, both would be paid the lower figure. But there was more: the one who wrote the lower figure would also collect a $5 bonus, while the other, who wrote the higher figure, would be fined $5. For example, if X writes 80 and Y writes 95, X will receive 80 + 5 = 85 and Y will collect 80 – 5 = 75.
What would be your figure of choice?
At first glance, it would seem
that both should write 100, because then this is what both will receive. Reasonable people would probably do that. But what if X and Y maintain an economic worldview – an approach that often tends to be narrow-minded? Most people belong to the species Homo economicus – creatures who wish to maximize their wealth whenever the opportunity presents itself. That approach predicts a very different number.
In this game the Nash Equilibrium is 5 – both players choose that low figure and collect their meagre compensation. Let me explain.
If X believes that Y wrote less than 100 (hoping to be the lowest bidder and collect the $5 bonus), he will not write 100– that much is clear. But even if X does think that Y wrote 100, he still won’t write 100: he’ll opt for 99 because then he gets to collect $104 (99 + 5).
Y understands how X’s mind works, and knows that X won’t write more than 99 – and thus, following the same considerations X made before, Y won’t write more than 98. In which case, X won’t write more than 97 … and on and on it goes. Where does this stop? I know: they have to stop at $5. This is the only choice that guarantees that both players will not retroactively regret their choices – hence, this is the Nash Equilibrium.
We’re reminded of Winston Churchill’s view: ‘However beautiful the strategy, you should occasionally look at the results.’
This strategic, non-zero-sum game, known as the Traveller’s Dilemma, was invented in 1994 by Kaushik Basu, an important Indian economist. Professor Basu (who, among other things, invented Dui-doku, a competitive version of Sudoku) is a leading economist and Senior Vice-President at the World Bank.
The Traveller’s Dilemma reflects a situation where the optimal solution is far from the solution obtained through the Nash Equilibrium. In this kind of scenario, looking after your interests actually harms you (and others). An extensive behavioural experiment of the game (with real financial rewards) yielded some interesting insights.
‘One cannot, without empirical evidence, deduce what understandings can be perceived in a non-zero-sum game of manoeuvre any more than one can prove, by purely formal deduction, that a particular joke is bound to be funny.’
Thomas Schelling
In June 2007, Professor Basu published an article about the Traveller’s Dilemma in Scientific American. He reported that when this simple game is played in practice (and I must add that it isn’t a zero-sum game because the sums 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 stag-hunting 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 do 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 do 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 no
w 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.