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Deciding to cast one's uncertainty model in the form of risk may help to narrow the field of alternative decision making criteria but it does not, by itself, provide a final solution to the problem. Even admitting that one's eventual choice is likely to be probabilistic in nature, a plenitude of possibilities -- and problems -- remains. Let us consider, below, a few of the more popular attempts which have been made in this direction. Expected Return Maximization The most obvious of the extensions of orthodox economic theory which embrace the concept of uncertainty is also the most simple. It consists simply of revising the usual certainty model so that expected return or expected profit becomes the object to be maximized when uncertainty enters the picture. It is, of course, a perfectly symmetrical expansion of its classical predecessor, embracing the former as the special case of zero risk, or of expectations which are held with certainty. Maximizing expected return or profit may also be shown, in innumerable situations, to be a highly defensible, investment strategy. The most obvious of its applications, for example, are to the fields of insurance and gambling. Its history in both of these areas is as old as the study itself. Indeed, the development of statistics as a branch of applied mathematics can be traced to its usefulness as a tool for the more accurate estimation of the expected outcomes of various insurance and gambling risks by expected return maximizing buyers of the risks. The usefulness of expected return maximizing is not limited, of course, to insurance and gambling. It is of equal validity in virtually any environment where the experiment involved is repetitive in nature, successive observations are independent of their predecessors, and possible outcomes are not extreme. Its application, however, has not always been limited to such environments. Relying on Jacques Bernoulli's famous Law of Large Numbers, theorists have for nearly two and one-half centuries ( Ars Conjectandi was published in 1713) attempted to justify expected return maximization in virtually any setting through the purely formal argument that, in the long run, "the expected" becomes "the certain." This is true, of course, only in the sense that a sample statistic, if drawn from an appropriate population, may be expected to asymptotically approach its true (or population) counterpart. An entrepreneur's repeated maximization of expected profit, therefore, can be equated, statistically, to his certain maximization of long run total profit (over an infinitely long time horizon). Even acknowledging the truth of Keynes' well known quip that, "In the long run we are all dead" (and especially so, one might add, in the infinitely long run), and the fact that successive investment decisions are seldom (if ever) statistically independent of one another, the argument retains considerable intuitive appeal. Few persons would prefer an investment whose expected return is 3 per cent to one whose expectation is 6 per cent. Similarly, the person who consistently undertakes projects or risks with high expected returns is "likely" to do better than the person whose allegiance belongs to another investment criterion. Why, then, is the expected return decision criterion limited to risks which are repetitive in nature, independent of one another's outcomes, and whose returns are not extreme? The first condition, that the decision be capable of replication, is built directly into Bernoulli's famous bastion for its popularity, that only in the limit of an infinite number of repetitions can a statistic's mean be guaranteed to approach its true (population) value. One's choice of a measure of central tendency, however, should not be looked upon as crucial to the usefulness of the decision model. Should the decision be truly unique, for example, a case can easily be made for substituting the distribution's mode for its mean (should they differ) as the target variable, or maximand. The requirement that successive experiments be truly independent of their predecessors is also essential to a direct application of Bernoulli's theorem. That this requirement might be fulfilled within the environment of a business firm is virtually inconceivable. Once again, dissatisfaction with purely formal attempts to justify the adoption of an expected return criterion need not rob the decision rule either of its intuitive appeal or of its practical usefulness. More serious difficulties arise, however, as extreme outcomes enter the picture. Daniel Bernoulli, some 25 years after his illustrious uncle's Law of Large Numbers had been published, calls both the rationality and the usefulness of expected return maximizing into question. The basis for his objection is the theory's inability to explain the behavior of rational (or, at any rate, reasonable) risk takers when faced with the St. Petersburg paradox. The paradox may be described roughly as follows: A person buys a chance to flip a coin until heads appears. Should it appear on the first throw, he receives $1. Should it appear on the 2nd, 3rd, 4th, . . . , nth throw he receives $2, $4, $8, . . ., $2n-1, respectively. How much should he rationally pay for a chance to play the game? One would expect a rational, expected return maximizing risk taker, therefore, to pay virtually anything for a chance at it. Why, asks the younger Bernoulli, are takers so scarce at twenty (or even at ten) dollars a throw? A number of answers are possible. One is that an infinite series of tails (or even a fairly short series) would break the bank. Should the house's resources be limited to one million dollars, for example, the 20th consecutive tail would break it. Any tails thrown beyond that limit, therefore, are meaningless. The game's meaningful expected value under this constraint, then, becomes V + ¯ = 1/2 (19) = $9. 50 instead of infinity. Many other reasons have been given. D'Alembert, for example, contends that "very long runs are not only very improbable, but do not occur at all." All explanations, however, center their attention on the upper part of the infinite series. Some emphasize the almost zero probability of long series. The probability of successive tails, for instance, P20 = (1/2)20 is equal to .9537 (10-33), if the coin is fair. Others emphasize the finite size of the bank, as mentioned above; whereas still others criticize the assumed proportionality between money and satisfaction. Bernoulli, himself, takes the last of these tacks. He contends that $20 is not equal to but is less than 20 times as valuable (to an ordinary individual) as $1. So, certainly, is $20 million less than 20 times as valuable as $31 million to any but the most extreme of misers. The paradox lies, all critics agree, in the expected return criterion's symmetrical treatment of extreme, possible outcomes. Another extreme, however, is of more constant and pressing concern to the typical investor -- the possibility of serious financial reverse. Is a loss of $10,000 exactly twice as unpleasant as a loss of half that amount? What is the value to an investor of the last dollar that stands between himself and financial ruin? It seems apparent that, unlike Gertrude Stein "A Rose is a Rose is a Rose," a dollar is not a dollar without regard for the number of its fellows. Symmetrical treatment of the value of one's first and last unit of wealth by the expected return criterion can defend an insurance company's sale of a policy (on which it has a positive, expected return) to a policy holder as being a rational act; but it cannot provide the same justification for the policy's buyer. Indeed, it can always justify one who offers (but never one who takes) an unfair bet. Thus, neither the person who refuses to stake his fortune on a try at the St. Petersburg game, nor the one who accepts an unfair bet from an insurance company can be classified, by this criterion, as a rational investor. Dissatisfaction with each of these conclusions leads to similar dissatisfaction with expected return maximizing as a general definition of rational behavior under uncertainty. Recognizing the obvious asymmetry between a risk's extreme and central outcomes, theorists are led to reject the contention that "an even chance of heaven or hell is precisely as much to be desired as the certain attainment of a state of mediocrity," and gives rise, instead, to an a priori preference for the avoidance of risk. Attempts to build such a preference into formal models of decision making under uncertainty are summarized in the following section.