behavior—the presence of *feedbacks*—is characteristic of non-linear systems.

Let us consider a more realistic concrete example by way of illustration: the relationship between material wealth and subjective utility. On the face of it, we might assume that the relationship between these two quantities is linear, at least in most cases. It seems reasonable, that is, to think that getting $10 would not only leave you with more utility--make you happier--than getting $5 would, but also that it would leave you with *twice* as much utility. Empirical investigation has not supported this idea, though, and contemporary economic theory generally holds that the relationship between wealth and utility is non-linear.

This principle, called the principle of diminishing marginal utility, was originally developed as a response to the St. Petersburg Paradox of decision theory. Consider a casino game in which the pot begins at a single dollar, and a fair coin is tossed repeatedly. After each toss, if the coin comes up heads the quantity of money in the pot is doubled. If the coin comes up tails, the game ends and the player wins whatever quantity is in the pot (i.e. a single dollar if the first toss comes up tails, two dollars if the second toss comes up tails, four if the third toss comes up tails, &c.). The problem asks us to consider what a rational gambler ought to be willing to pay for the privilege of playing the game. On the face of it, it seems as if a rational player ought to be willing to pay anything less than the expected value of a session of the game--that is, if the player wants a shot at actually making some money, she should be willing to pay the casino anything less than the sum of all the possible amounts of money she could win, each multiplied by the probability of winning that amount. The problem is that the value of this sum grows without bound: there is a probability of one-half that she will win one dollar, probability one-fourth that she’ll win two dollars, probability one-eighth that she’ll win four dollars, &c.

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