Gladiators, Pirates and Games of Trust

by Haim Shapira

  Evolutionary Game Theory offers a nice definition that expands the Nash Equilibrium. It was first suggested in 1967 by the British evolutionary biologist W D Hamilton (1936–2000), though it’s often attributed to another British evolutionary biologist, John Maynard Smith (1920–2004), who expanded and developed it. With these pioneers we enter the realm of the Evolutionary Game Theory equivalent of the Nash Equilibrium, known as ESS: Evolutionary Stable Strategy.

  Without going into the ‘epsilon–delta’ language that mathematicians are so fond of, which is so difficult that ordinary people would rather study old Chinese idioms, we could say that ESS is a Nash Equilibrium plus another stability condition: if a small number of the players suddenly change their strategy, those players who adhere to the original strategy have the advantage.

  For further reading about the association between evolution and games theory, I suggest Maynard Smith’s canonical Evolution and the Theory of Games.

  Intermezzo

  THE RAVEN PARADOX

  Carl Gustav Hempel (1905–1997) was an important philosopher of German origin who gave plenty of thought to the philosophy of science, but gained international fame when he published the Raven Paradox in 1940 (when he lived in New York and taught at City College). His paradox deals with logic, intuition, induction and deduction – and does all of that at the expense of ravens. This is my version:

  One cold and rainy morning Professor Smartson took one look out the window and decided he didn’t want to go to the university that day. ‘I’m a logic expert,’ he thought, ‘so all that I need to do my work are paper, pencils and an eraser, and I can find them all right here at home.’ He sat by the window, sipping his oolong (Black Dragon) tea and thinking, ‘What should I study today?’ Suddenly he saw two black ravens in a tree. ‘Are all ravens black?’ he wondered, after suddenly catching sight of a third raven and observing that it too was black. ‘It does seem so.’ This claim should either be refuted or confirmed – but how? Clearly, every new black raven he sees will increase the probability of the ‘all ravens are black’ assertion, but it’s impossible to observe all the ravens in the world. Nevertheless, Professor Smartson decided to start observing ravens, hoping they’d all be black.

  So he sat by the window and waited, but there were no more ravens in sight. ‘I guess I’ll have to get out and look for ravens,’ he thought, but the idea didn’t appeal to him. After all, he’d stayed home for a reason, and the rain had just turned into a hail storm. Suddenly he had a brilliant idea. He remembered that the assertion ‘All ravens are black’ is logically equivalent to the argument that ‘Everything that is not black is not a raven.’ Remember, he was a logic professor. Wise and logical people (such as you, my reader) are welcome to ponder that and realize that the two assertions are equivalent. Simple logic.

  Thus, instead of trying to prove that ‘All ravens are black,’ Prof Smartson decided to confirm that ‘Everything that is not black is not a raven’ and – lo and behold – he didn’t have to leave home to do so. All he had to do was find all kinds of things that weren’t black and make sure they weren’t ravens. Now, that was a cushy job.

  Our professor looked outside his window again and soon found countless examples. He saw a green lawn, yellow and red leaves falling, a purple car, a man with a red nose, an orange sign with white letters, blue skies, and grey smoke coming out of a chimney. Suddenly he saw a black umbrella. That startled him for a moment, but he soon recovered and reminded himself that his claim didn’t argue that all things black are ravens (that would have been silly) but that ‘Everything that is not black is not a raven’ – that was all.

  Now that he was all relaxed, home and dry, he kept observing the street under his window and found an endless array of things that were neither black nor ravens. Satisfied with the work he’d done, he turned to his notebook and wrote: ‘Based on my extensive research, I can state with almost full certainty that all ravens are black.’ QED.

  Can you point out Professor Smartson’s mistake? Did he make one?

  Chapter 8

  GOING, GOING … GONE!

  (A Brief Introduction to Auction Theory)

  I start this chapter by showing how to sell a $100 bill for $200. Next we’ll take a very brief course in Auction Theory, a prolific branch of Game Theory. We’ll examine different kinds of auctions, try to understand the nature of the Winner’s Curse phenomenon, and find out which auction won the Noble Prize.

  HOW MUCH FOR $100?

  Originally, this sequential game was called ‘Dollar Auction’, but to make it a bit more interesting (after all, with inflation, the dollar ain’t what it used to be), let’s speak of a $100 bill. Opinions vary as to who really authored this game. Some say that Martin Shubik, Lloyd Shapley and John Nash invented it in 1950. In any event, Shubik, an American economist teaching at Yale University, wrote the article that discusses the game in 1971.

  The rules of the game are very simple. A $100 bill is auctioned and goes to the highest bidder. At the same time, the second-highest bidder also has to pay the sum he or she offered, but gets nothing. Sounds simple, doesn’t it?

  I play the game quite often in classes I give. I used to come to class and put a $100 bill on auction, as described. I promise to give the bill to the highest bidder, even if the price offered is very low. Sounds excellent. There’s always a student in the room who offers $1 and thinks he’s made the deal of his life. What then? Well, if the class kept silent, that student would have really gained handsomely. The problem, however, is that this never happens. The minute they notice that someone is about to grab a $100 bill for a buck, there’ll always be someone who offers to pay $2; after all, why should they be a sucker and let someone else win? – some people just squirm at the very thought.

  Once a $2 bid is made, the first student will lose his $1 because he must pay his bid and walk away empty-handed. Naturally, the second-highest bidder must now offer $3. Once a second player joins the game, the die is cast. I, the seller, will profit and the players will lose, no matter what. There are no two ways about it. Suppose, for example, that a player offers to buy my $100 for his $99 after another has offered $98. The $98 bidder had better offer $100, because he stands, quite simply, to part with $98. The best deal for him is to offer a full $100 and leave the game neither gaining nor losing. Alas, after he makes the $100 bid, the player who offers $99 is about to take a real blow and so (even though it seems like a total absurdity) must offer $101 so that he comes out $1 and not $99 short. By the way, I, the seller, just pocketed $201 minus $100 (the value of the sold bill), making a net profit of $101.

  When does this game end? Mathematically, never. In practical terms, it ends when one of the following happens: (1) the players run out of money; (2) the bell rings and the class ends; (3) one of the players gets wise and quits, losing his bid.

  This game, by the way, nicely shows how excellent tactics might turn into a terrible strategy. Mathematical logic says that bidders should raise the price at each stage, but how far will this kind of logic take us? Isn’t it smarter to lose $4 and quit than pay $300 and win a $100 bill?

  Once, when I conducted this game in a strategic-thinking workshop, it took two minutes before I had a $290 offer for my auctioned $100 (bids jumped up in $10 units). I noticed that the players soon forgot what the auction was about and simply competed against each other. Winning, and not letting the other guy win, was all they cared about.

  People sometimes behave quite oddly indeed. In another incident when I ran this experiment, one of the players remained uninvolved until the bidding reached $150 and then surprised everyone by offering $160! Why did he do that? He could have simply walked into a bank and bought an endless amount of $100 bills for $160 each. Why did he even choose to participate?

  A friend of mine who took part in a seminar for senior businessmen at Harvard University told me that the host collected $500 for his auctioned $100. Were the game participants simply irrational? Not necessarily. It’s quite p
ossible that $500 was not so much for the winning businessman, if we consider that by offering this sum he signalled to the other players that he was determined to go all the way. This is a very important signal in our day and age when there’s always a chance that businessmen will meet each other again (and it stands to reason that he can write off the $500 he invested as expenses).

  This kind of behaviour, whereby the reason people won’t quit a game because they’ve made a considerable investment in it, happens in daily life all the time, in small things and big. For example, this is what happens when you call the cable company and wait for a customer service officer to answer. You hold the line for a very long time, sweet music helping you pass the time, but no one answers. What do you usually think? ‘OK, I’ve been waiting here for so long already, it would be a shame to hang up now.’ So you wait some more. You wait and wait, and that music has become profoundly irritating; but the more you wait, the sillier it seems to quit, because you’ve already invested so much time.

  Following the same kind of logical dynamic, we can see that the same happens when a state funding body, having invested $200 million in a project initiated by a businessman, decides to give him a further $100 million to rescue the project after it fails. It’s the same kind of mistake.

  The best thing to do when you come across a ‘$100 Sale Game’ is not to participate at all; and if you happen to join in by mistake, the best thing would be to quit at once. Someone once suggested a ‘safe’ strategy for winning the game. Your first bid should be $99. Sure, you won’t make an impressive profit, but it’s nice to win. Personally, I wouldn’t recommend that strategy. There’s always the possibility that someone might suddenly offer $100 for the $100. Why would he do that? Just because: ‘because’ is an answer too sometimes.

  In any event, staying in the game just because we’d already lost a lot of money is never a good idea. As in many other things, the ancient Greeks knew that – it’s implied in their time-honoured notion that ‘Not even the gods can change the past.’

  I’ll end this disquisition on money auctions with a little mind game for you:

  A similar game was played in a prestigious military academy. A $20 bill was auctioned in the manner described above, with bids of at least $1. Two officers reached a stage where one of them offered $20 and the other raised his bid to $21. At this point, the $20 bidder suddenly made a surprise move and offered $41. Thus the game ended. Why? (Cover up the answer in the next paragraph.)

  (If he offers $42, he will lose $22. Better to lose $21 than $22.)

  ***

  Auctions are probably the oldest branch of Game Theory. It’s been said that the first auction took place when Joseph’s brothers sold him and his multi-colour coat to slave traders. Herodotus, the 5th-century bc Greek historian, wrote about auctions held in his time where women were sold for marriage. The auction would start with the prettiest woman, and after the seller had received a handsome price for her, he’d auction the rest in decreasing order of beauty, and the asking price dropped accordingly. Very unattractive women had to pay for a husband, which means that these auctions had negative bidding too.

  Auctions were popular in the Roman Empire as well, so much so that in ad 193, the entire empire was auctioned! Didius Julianus won, but was assassinated some two months later, which goes to show that winning an auction isn’t always cause for celebration.

  There are countless auction methods, the main templates being: English, Dutch, First-Price Sealed-Bid, and Vickrey Auction.

  ENGLISH AUCTION

  In an English auction an item is auctioned at a basic price that keeps rising with the level of demand, and goes to the highest bidder. Bids may be made by phone. It’s well known that famous and rich people usually stay away from bidding halls, because their very presence there might send the prices soaring.

  In one version of the English auction, the price rises continuously, bidders quit when the price becomes too high for them, and the last one remaining wins the item. This method provides the participants with information about how all their competitors estimate the price of the item.

  DUTCH AUCTION

  In this method, items are offered at a highest price, which keeps dropping until a buyer finds the price that is right for him or her and stops the decline by accepting it. The system is named ‘Dutch’ because this is how they sell flowers in Holland.

  I once saw an interesting Dutch auction played in a Boston antique shop. Every item in the shop has an attached price tag that also reports the date the item was first placed in the store. The price you pay for items is the figure on the tag minus a discount that depends on the duration of time the item has spent in the shop – the longer it lingers, the cheaper it is; and discounts may go as far as 80 per cent of the original price. When a Bostonian sees a chair he likes that presently costs $400, he may sensibly conclude that the price will fall in a month and that it’s best to wait. He’s right, of course – provided no one else buys that chair in the meantime.

  ENGLAND VS HOLLAND

  An interesting question now arises: which is better, the English or the Dutch auction method?

  Suppose we want to put a really special book (say, Ulysses, autographed by James Joyce) on sale in a Dutch auction, and suppose we set the opening price at $10,000 and let it drop $100 every 10 seconds. This selling method might greatly distress potential buyers, because the moment someone stops the clock, the sale is done. It stands to reason that a person who believes that the amount of pleasure he will derive from this book is worth $9,000 of his money will wait until the price falls to that level and then make his bid (if the book is not sold by then).

  In an English auction, however, you can sometimes obtain items for much less than you originally intended to pay for them. If, for example, no one else cares for the original autographed edition and our bid of $700 is the highest made, good for us: we paid $700 for an item we were willing to buy for $9,000. On the other hand, this method encourages people to increase their bid again and again, and the main reason for that is the human competitive streak. Suppose you’re willing to pay $9,000 for the book but find that someone else offers that price. Will you call $9,100? You probably will, because it’s only $100 more than your original intention; but then the other bidder, whose intention is similar, will raise to $9,200, which you will have to call and then raise to $9,300 … and so it goes, and who knows where it will end?

  Another reason why prices keep rising in English auctions is information obtained during the sale. Let me explain. Suppose one of the participants feels that the book might be worth $9,000 but is far from certain. Could this in fact be an exaggerated price? Perhaps the book is barely worth half as much? He’ll become more sure of his assessment if he sees that someone has offered, for example, $8,500. This will indicate to him that his view is not completely unrealistic. Auctioneers often use this to their advantage, bringing in false bidders whose role is to push the prices up.

  There are serious disagreements as to which of the two auction methods is better, but clearly the English auction is more popular (I’ve witnessed auctions that started out with the Dutch method, reached a certain price, and went on from there using the English method).

  FIRST-PRICE SEALED-BID AUCTION

  Seriously big-ticket items (oil fields, banks, airlines companies …) are usually auctioned in the following manner: potential bidders are given a bidding period during which they prepare their bid and hand it in in a sealed envelope. On the designated date, the envelopes are opened and the winner is declared. This is the resolution phase. These auctions are usually subject to a long and tedious book of rules, but reading this may be worthwhile for the bidder, because they may find some surprises there. For example, the book may sometimes say that sellers are not obligated to select the highest bidder (and my intelligent readers can certainly figure out why).

  Since we’ve touched upon oil fields, now is a good time to learn about the ‘Winner’s Curse’ phenomenon.

&n
bsp; This was first documented in 1971 by three petroleum engineers – Ed Capen, Bob Clapp and Bill Campbell – in a seminal article. The trio showed that if you win an auction, you should ask yourself: ‘How come the other bidders didn’t feel that the oil field I just bought was worth more than the price I offered?’ Statistically, the idea is very simple.

  Suppose the owner of an oil company went bankrupt and the field is auctioned out. Ten companies made the following sealed bids (in billions of dollars): 8, 7.2, 7, 13, 11.3, 6, 8, 9.9, 12 and 8.7.

  Who knows what the oil field is really worth? Who can guess oil prices even in the near future? No one! Still, it would be safe to assume that the bidding companies hired experts to study the issue before they made their bids. At the same time, it would be completely logical to estimate the price of the field at around the average of the bids made. There’s no reason to assume that the highest bid ($13 billion) is closest to the field’s real yield prospects, but it will (almost) certainly win. The winner, however, is advised to hold back on the champagne and the parties and take time to reflect.

  VICKREY AUCTION

  (OR SECOND PRICE AND NOBEL PRIZE)

  The second-price Vickrey auction, named after 1996 Nobel Prize laureate, the Canadian-born, Colombia University professor of economics William Vickrey, is held thus: contestants offer sealed bids for an item they wish to purchase and the winner is the highest bidder, with the twist that, unlike in ordinary auctions, he or she pays the second highest bid, not the highest.