Sam’s envelope stall.
A seasoned gambler named Sam with axe to grind sets up a stall near Harvard university. He has thought up a scam to make money out of clever academics from the business school. He offers any punter a choice between two envelopes. Inside each envelope is written a sum of money on a ticket. He promises that the sum in one envelope is twice the sum in the other. If questioned he will tell you that he will round the lower amount down to the first decimal, so if the higher says $1.56.3, the lower will say $0.78.1. The punter pays nothing until he has picked one of the two envelopes. He now has a choice: walk away and pay nothing, or pay the price on the ticket, plus a commission of 5%, in exchange for the sum of money written on the other envelope. The rich smart students from the business school are attracted by the stall, and many of them try it out. At first they assume it’s a trick of some kind, but Sam never tampers with the envelopes and always allows a complete free choice. The envelopes are completely identical. What really convinces the students that there is no trick is that one statistician does an exit poll and discovers that half the punters double their money, and half the punters come away with half their original stake, both minus the commission of course. So it is clear that Sam is playing fair. There is a ½ probability of choosing the envelope with the lowest amount and thus doubling your money. Once word gets around, then more students roll up to play Sam’s game.
Some guys from the business school decide that Sam must be a fool. They reason like this: when you see the price on the ticket you know you have a ½ probability of doubling your money, and a ½ probability of halving your money if you buy the ticket, minus the commission. Let us say you have a ticket with $X written on it. The expected utility is ½ of $2X + ½ of $ ½X which is $1.25X. Since you only had to pay $1.05X including the commission, you should make a profit in the long run of $0.2X a pop. Some of them run back to their rich fathers telling tales of a unique business opportunity and getting early releases of their inheritances. They will buy as many tickets as Sam will give them.
Now Sam, being a seasoned gambler, knows how to be random. The amounts in the envelopes vary enormously. In one pair there will be $8843.33.1, and $17686.66. In the next there will be $4 and $2. There is no limit on how high the tickets will go. As his fortune increases he increases the range of amounts he puts on the tickets, so even the range of possibilities is not constant. But what infuriates and befuddles the students is that Sam’s fortune does increase. And although some students do come away richer, most do not, and the worst off are those who blew their inheritances on as many tickets as they could buy.
People from the maths department begin to ask Sam all kinds of strange questions. How does he select the amounts to put in the envelope? What is the prior probability distribution? Does he toss a coin at any point? Sam won’t give straight answers to these questions. He looks at tea leaves, he says, and the way the birds fly. He uses Archimedes principle and counts the bubbles in his bath.
Even the philosophy department gets involved. Which envelope is held fixed? They ask. Do you select both envelopes in the same possible world? Do you rigidly designate one sum and double it, or rigidly designate the other and halve it? Would you ever put an infinite amount in one envelope, and in which case, how much would you put in the other envelope? Sam smiles at these questions and taps his ever burgeoning wallet.
So what is the solution? How does Sam make his living? The answer is surprisingly simple. Although it is true that half the people who buy a ticket double their money and the other half halve their money, if the exit pole had been a little more detailed it would have revealed that the average price of a winning ticket was half the average price of a losing ticket. So although in pure frequency terms the chance of doubling is 0.5, in financial terms the losing tickets count as double since they are twice as expensive. This means that only a third of the money is doubled, whereas 2/3 of the money is halved. So, aside from the commission, the amount of money going in to Sam’s capital is on average equal to the amount going out. Of course, Sam should only rarely approach his total capital since a series of big losses could put him out of business. But he has total control of the stake size, so he doesn’t need to take risks. As long as the students keep buying tickets, he will make a steady profit.
So what fooled the students? They neglected to realise that the losing envelopes cost twice as much as the winning envelopes. They should have seen that it was a zero sum game minus the commission and walked away. Had they realised that being indifferent was the correct attitude they could have calculated the correct probabilities using a clever formula worked out by Frank Plumpton Ramsey in 1926. The formula says that if a Subject is indifferent between options:
1. A for certain or
2. B if p and C if ~p,
Then the subject’s degree of belief that p is equal to
A – C/B – C.
Suppose that the ticket price is X, and p is the proposition that S will double his money on swapping, then S should be indifferent between:
1. $0 for certain. (He can just walk away with what he started with) or
2. $X if p and – $1/2X if not p (because he wins his stake back twice if he wins and loses half his stake if he loses)
Then S’s degree of belief that p = 0.5X/1.5X = 1/3.
What this shows is that given that indifference is the correct attitude, then the probability one should assign to doubling your money in Sam’s stall is 1/3 and the probability of losing half your stake is 2/3. This number does not represent the frequency of events fitting this description, but the expected utitlity. It takes into account the initial stake size. It calculates the frequency of units of value that get doubled, rather than the frequency of transactions that get doubled. When calculating expected utility it is the former which is of prime importance and therefore it is probability so interpreted that should be used in expected utility calculations.
What confuses matters is that the amounts in the envelopes are random. This focuses attention unduly on how this process is randomised which is a distraction. The only function the randomising serves is to introduce ignorance into the situation. If, as the classic two envelope paradox is properly presented, you are presented with just two envelopes, one with double the other, it is hard to see why the selection procedure is relevant at all. We could represent the ignorance more simply by imagining a thousand students being presented with the same to envelopes without being able to communicate. Lets say the envelopes contain £10 and £20. Probabilistically, around 500 will pick £10 and the other 500 will pick £20 first. If they all elect to swap we can record the results in two ways. We can either say that 50% doubled their money and 50% lost half their stake. This would be enumerating the transactions. Or we could say that £10 000 worth of deals made a loss of 50% and yet only £5 000 worth of deals doubled in value. So the relative frequency of loss to gain is 2:1, making the probability of doubling your money 1/3 and halving it 2/3. Of the thousand students, five hundred lost £10 and five hundred won £10. Sam makes his money from the commission and the insensitivity of decision theory to risk.
Labels: expected utility, paradox, two envelope problem