ABSTRACT
However, additional factors enter when the stakes are high. Most of us would be willing to pay up to the expected value of $2.50 for a lottery ticket giving us equal chances of winning $10.00 or losing $5.00. On the other hand, most of us would not be willing to pay as much as $25,000 for a lottery ticket with equal chances of win ning $100,000 or losing $50,000 even though $25,000 is the expected value of this lottery. This is because a few $50,000 losses will leave most of us without the re sources to continue. We cannot #/play the averages'' over a series of decisions where the stakes are this large, and thus consider ations of long-run average returns are less relevant to our decision making. (A vicepresident of a Fortune 500 company once commented to me, "Most of the decisions we analyze are for a few million dollars. It is adequate to use expected value for these." Whether this is true or not depends on your asset position.)
A decision maker's attitude toward risk taking is addressed in decision analysis us ing the concept of the certainty or certain equivalent, which is the certain amount that is equally preferred to an uncertain al ternative. If certainty equivalents are known for the alternatives in a decision, it is easy to find the most preferred alterna tive: It is the one with the highest (lowest) certainty equivalent if we are considering profit (cost). Someone who prefers to re ceive the expected value of an uncertain alternative for certain rather than the un
certain alternative is called risk averse. Certainty equivalents for such a person can be determined using a utility function u(x). This function is defined in such a way that alternatives can be ranked on the ba sis of their expected utilities, and hence the certainty equivalent CE for an uncertain al ternative is determined by u{CE) = E[u(x)] where E[u(x)] is the expectation of the util ity for the uncertain situation. Further de tails on basic decision analysis are given in Clemen [1990], McNamee and Celona [1990], and Samson [1988]. Related topics are covered by Watson and Buede [1987], and von Winterfeldt and Edwards [1986]. Model Formulation
Decision trees are well-known tools for modeling decision problems under uncer tainty. Their use is illustrated by an actual application [Crawford, Huntzinger, and Kirkwood 1978]. An electric utility is plan ning a high-voltage transmission line across its service area. The general charac teristics of the transmission line have been selected, but the specific conductor size is still to be determined. The possible con ductors differ in both construction and op erating costs, and these costs are uncertain. The primary component of the operating cost is the cost of power lost due to heat ing of the conductor. For a small conduc tor, this loss is relatively large while for a large conductor it is smaller. Conversely, the construction cost of a small conductor is lower than that of a large conductor. The cost of the lost power is uncertain be cause the utility company's power genera tion plants use oil, and its price can vary over time. A sequential alternative is avail able that hedges against the uncertainties; the transmission towers can be prebuilt so
that they could hold a larger conductor than that actually installed. Then, if the price of oil turns out to be high, a larger conductor can be installed later without building new towers.
