Ramble, Part 32: What Do You Expect?
Sep. 21st, 2007 07:47 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowIn a previous post, I mentioned the origins of modern probability theory in the joint work of Pascal and Fermat on the "problem of the points". In the next generation or two, great strides were taken in the field, at the hands of (among others) the Dutch polymath Christiaan Huygens and several members of the Bernoulli family. I'm not going to go into much detail on this, but there are a couple of points of interest that I'd like to discuss.
Christiaan Huygens probably can't be ranked among the very best mathematicians of the 17th century, but this is no great shame: the competition includes Pascal, Fermat, Descartes, Newton, Leibniz, and (late in the century) the Bernoulli brothers. Still, he was a great and versatile thinker; in addition to his mathematical work, he made major contributions to optics, where he challenged Newton's particle theory of light (he preferred a wave theory), and to other branches of physics. He also came up with the idea of the pendulum clock. For our purposes, though, his importance is as the writer of the first formal treatise on probability, based on and extending the ideas exchanged by Pascal and Fermat. Most importantly, he developed the notion of mathematical expectation.
Suppose you and a friend wager on the outcome of a coin toss; whoever wins the toss receives a dollar from the other. Assuming the coin is fair, in, say, a hundred tosses you will win the toss about fifty times (thus winning about $50), and lose about the same number of times (thus losing about $50). Your net gain, on average, is $0. But suppose the coin is not fair - that it comes up "heads" 55% of the time - and that you call "heads" every time. Then, in that same hundred tosses, you will win about fifty-five times and lose about forty-five, for a net gain of about $10, or about $.10 per toss. If your friend catches on and then demands that you pay $1.10 for each win while he pays only $.90, your fifty-five or so wins will net you about $49.50, and his forty five or so wins will give him about the same amount.
Considerations like these lead to Huygens' notion of "mathematical expectation". Suppose there are a number of possible, and mutually exclusive, outcomes to a certain event (like a coin toss), and that each outcome has a certain value to you. If you multiply the value of each outcome by its probability and add them up, this is the mathematical expectation (for you) of the event. For another example, suppose you are to roll a single die, and that you will win $2 if you roll a 1 and $1 if you roll a 2 or 3, but lose $1 on any other roll. The outcome "1" has a probability of 1/6; the outcome "2 or 3" has a probability of 1/3; the outcome "any other roll" has a probability of 1/2. Your mathematical expectation is 1/6 * $2 + 1/3 * $1 + 1/2 * (-$1) = $1/6, and on average you will win about 1/6 of a dollar per roll.
One thing you may note about mathematical expectation is that it is at most equal to the value of the best possible outcome, and at least equal to the value of the worst. In particular, some outcomes will usually be more valuable than the expectation, and some will be less. This seems obvious. However....
Jakob Bernoulli, one of the Bernoulli brothers, also made major contributions to probability theory, but two of his nephews took the notion of mathematical expectation in new directions. The older of the two, Nicolaus, discovered what is known as the Petersburg paradox. Consider the following "game". I toss a fair coin repeatedly. If it comes up heads on the first toss, you win $1. If the first head occurs on the second toss, you win $2; if on the third, $4; if on the fourth, $8; and so on. What is the expectation?
Another way of considering this problem is as follows. If your mathematical expectation from a game is $x, then, in fairness, you ought to pay $x for the privilege of playing. But in the Petersburg game, you would have to pay an infinite amount, in fairness - even though you could not possibly recoup that amount!
One of the problems with mathematical expectation as a guide (descriptive or normative) to human behavior is precisely the fact that it is probabilistic; it makes predictions about the results of many-times-repeated events. But in practical terms, this may be implausible. Consider the game strategy of "doubling up", in which, on the first play, you bet $x; you do the same on the next play after a win; on the next play after a loss, you bet twice the previous bet. In probabilistic terms, this is a reasonable strategy, as, provided that it is at least possible to win, you will eventually recoup any losses. Unfortunately, it requires unlimited funds, or at least unlimited credit; the bets, during a losing streak, are literally growing exponentially, and will consume any finite reserves frighteningly quickly. A more reasonable approach would have to take into account the player's existing wealth in making decisions and formulating strategy.
Nicolaus' brother Daniel took a stab at this last. The fundamental idea is that a given amount of money is more valuable to a poor person than to a rich one; where a Dives can shrug off a loss of a denarius, a Lazarus could be devastated by the same loss. Daniel Bernoulli suggested, then, that we consider the value of an outcome as the ratio between the person's wealth after and before that outcome. After a rather involved argument - see this article for details - this leads to the notion of "moral expectation". If a particular outcome has value (in Bernoulli's sense) v and probability p, its contribution to the moral expectation is vp; the moral expectation is, not the sum, but the product of these contributions. If the moral expectation is greater than 1, the event is deemed a profitable one; if not, then not. It can be shown that the moral value of the event - that is, the product of the moral expectation with the person's initial wealth - is never greater than the mathematical value - the sum of the mathematical expectation and the initial wealth, and the disparity is greater for smaller initial wealth; that is, the rich are in a better position to gamble than are the poor.
Daniel Bernoulli's notion of moral expectation is one of the earliest attempts to develop a mathematical theory of economic behavior. The assumptions underlying it are a bit simplistic, but it does provide explanations for some otherwise puzzling phenomena. (The article linked above gives an example of such a phenomenon, the practice of buying and selling insurance.) Other attempts at mathematical analysis of social behavior were to follow, and I'll mention some of them in future posts.
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Christiaan Huygens probably can't be ranked among the very best mathematicians of the 17th century, but this is no great shame: the competition includes Pascal, Fermat, Descartes, Newton, Leibniz, and (late in the century) the Bernoulli brothers. Still, he was a great and versatile thinker; in addition to his mathematical work, he made major contributions to optics, where he challenged Newton's particle theory of light (he preferred a wave theory), and to other branches of physics. He also came up with the idea of the pendulum clock. For our purposes, though, his importance is as the writer of the first formal treatise on probability, based on and extending the ideas exchanged by Pascal and Fermat. Most importantly, he developed the notion of mathematical expectation.
Suppose you and a friend wager on the outcome of a coin toss; whoever wins the toss receives a dollar from the other. Assuming the coin is fair, in, say, a hundred tosses you will win the toss about fifty times (thus winning about $50), and lose about the same number of times (thus losing about $50). Your net gain, on average, is $0. But suppose the coin is not fair - that it comes up "heads" 55% of the time - and that you call "heads" every time. Then, in that same hundred tosses, you will win about fifty-five times and lose about forty-five, for a net gain of about $10, or about $.10 per toss. If your friend catches on and then demands that you pay $1.10 for each win while he pays only $.90, your fifty-five or so wins will net you about $49.50, and his forty five or so wins will give him about the same amount.
Considerations like these lead to Huygens' notion of "mathematical expectation". Suppose there are a number of possible, and mutually exclusive, outcomes to a certain event (like a coin toss), and that each outcome has a certain value to you. If you multiply the value of each outcome by its probability and add them up, this is the mathematical expectation (for you) of the event. For another example, suppose you are to roll a single die, and that you will win $2 if you roll a 1 and $1 if you roll a 2 or 3, but lose $1 on any other roll. The outcome "1" has a probability of 1/6; the outcome "2 or 3" has a probability of 1/3; the outcome "any other roll" has a probability of 1/2. Your mathematical expectation is 1/6 * $2 + 1/3 * $1 + 1/2 * (-$1) = $1/6, and on average you will win about 1/6 of a dollar per roll.
One thing you may note about mathematical expectation is that it is at most equal to the value of the best possible outcome, and at least equal to the value of the worst. In particular, some outcomes will usually be more valuable than the expectation, and some will be less. This seems obvious. However....
Jakob Bernoulli, one of the Bernoulli brothers, also made major contributions to probability theory, but two of his nephews took the notion of mathematical expectation in new directions. The older of the two, Nicolaus, discovered what is known as the Petersburg paradox. Consider the following "game". I toss a fair coin repeatedly. If it comes up heads on the first toss, you win $1. If the first head occurs on the second toss, you win $2; if on the third, $4; if on the fourth, $8; and so on. What is the expectation?
The first possible event (Heads) has probability 1/2, and contributes 1/2 * $1 or $.50 to the expectation. The second (Tails-Heads) has probability 1/4, and contributes 1/4 * $2 or $.50; the third (Tails-Tails-Heads) has probability 1/8, and contributes 1/8 * $4 or, again, $.50. This continues indefinitely, and the expectation is the sum of infinitely many $.50's.The expectation is infinite - even though every possible payoff is finite.
Another way of considering this problem is as follows. If your mathematical expectation from a game is $x, then, in fairness, you ought to pay $x for the privilege of playing. But in the Petersburg game, you would have to pay an infinite amount, in fairness - even though you could not possibly recoup that amount!
One of the problems with mathematical expectation as a guide (descriptive or normative) to human behavior is precisely the fact that it is probabilistic; it makes predictions about the results of many-times-repeated events. But in practical terms, this may be implausible. Consider the game strategy of "doubling up", in which, on the first play, you bet $x; you do the same on the next play after a win; on the next play after a loss, you bet twice the previous bet. In probabilistic terms, this is a reasonable strategy, as, provided that it is at least possible to win, you will eventually recoup any losses. Unfortunately, it requires unlimited funds, or at least unlimited credit; the bets, during a losing streak, are literally growing exponentially, and will consume any finite reserves frighteningly quickly. A more reasonable approach would have to take into account the player's existing wealth in making decisions and formulating strategy.
Nicolaus' brother Daniel took a stab at this last. The fundamental idea is that a given amount of money is more valuable to a poor person than to a rich one; where a Dives can shrug off a loss of a denarius, a Lazarus could be devastated by the same loss. Daniel Bernoulli suggested, then, that we consider the value of an outcome as the ratio between the person's wealth after and before that outcome. After a rather involved argument - see this article for details - this leads to the notion of "moral expectation". If a particular outcome has value (in Bernoulli's sense) v and probability p, its contribution to the moral expectation is vp; the moral expectation is, not the sum, but the product of these contributions. If the moral expectation is greater than 1, the event is deemed a profitable one; if not, then not. It can be shown that the moral value of the event - that is, the product of the moral expectation with the person's initial wealth - is never greater than the mathematical value - the sum of the mathematical expectation and the initial wealth, and the disparity is greater for smaller initial wealth; that is, the rich are in a better position to gamble than are the poor.
Daniel Bernoulli's notion of moral expectation is one of the earliest attempts to develop a mathematical theory of economic behavior. The assumptions underlying it are a bit simplistic, but it does provide explanations for some otherwise puzzling phenomena. (The article linked above gives an example of such a phenomenon, the practice of buying and selling insurance.) Other attempts at mathematical analysis of social behavior were to follow, and I'll mention some of them in future posts.
Previous Next
Ramble Contents
![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
no subject
Date: 2007-09-22 06:00 pm (UTC)I am too ignorant to understand a lot of this, but one must chip away at that ignorance no matter how small the shards one knapps off the boulder.
Love, C.
no subject
Date: 2007-09-22 06:01 pm (UTC)Love, C.
no subject
Date: 2007-09-22 06:09 pm (UTC)no subject
Date: 2007-09-22 06:09 pm (UTC)*goes off to read the link*
link
Date: 2007-09-22 06:12 pm (UTC)Re: link
Date: 2007-09-22 06:18 pm (UTC)Re: link
Date: 2007-09-22 06:31 pm (UTC)no subject
Date: 2007-09-22 07:11 pm (UTC)no subject
Date: 2007-09-22 07:56 pm (UTC)no subject
Date: 2007-09-22 08:07 pm (UTC)no subject
Date: 2007-09-22 08:12 pm (UTC)In any case, it's not true that "almost all the time you'll come out even"; you'll never come out even - but the probability-weighted outcomes balance out. (Compare, e.g., a lottery; if you buy a $1 ticket on a million-to-one chance at a million dollars, you'll either wind up out a dollar, or up $999999.)