Outcome Probability Calculation Weighted Value Win $20 1/6 $20 X 1/6 = $3.33
Win $40 1/6 $40 X 1/6 = $6.67 Lose $30 1/6 -$30 X 1/6 = -$5.00 No Change 3/6 $0 X 3/6 = $0.00
Expected Value = $5.00
The example depicted in Exhibit 2.1 illustrates the general methodology for computing an expected value. The first column identifies all of the possible results and their associated payoffs. The second column is the probability of the outcome, commonly expressed as a decimal fraction but sometimes described as a percent or a common fraction. In any event, the sum of the probabilities in column 2 must be unity, or 100%, if all of the possibilities have indeed been accounted for. The third column is the product of the payoff times its associated probability. The method of computing this third column illustrates how the final results will depend equally on the size of an event’s payoff and on the
magnitude of its probability. Finally, adding up the third column yields the weighted average or “expected value.”
The valuation of future prospects is sometimes depicted as if decision-makers compared prospects exclusively in terms of their expected values. As a rough description, this practice is quite useful and has a considerable measure of plausibility. It does seem sensible, after all, that the alternative consequences should have “weights” approximately equal to their likelihoods. The actual process whereby individuals reduce prospects to certainty equivalents is nonetheless much more complicated. For instance, it is also plausible to think that such a decision-making process would pay some attention to the dispersion of the possible outcomes.
For instance, the same $5 expected value would result if the Fairy Godmother were about to flip a coin and give a person $50,010 for a head but extract $50,000 if a tail comes up:
½ X +$50,010 = +$25,005 ½ X - $50,000 = -$25,000 Expected Value = +$5
The expected value computation above implies that this prospect is a “good” one, that it is worth $5. But would you pay $5 to have this prospect? Would you be indifferent between this prospect and the Fairy Godmother’s die-tossing prospect described above? (Remember, both of these prospects have the same $5 expected value.) Do you think that some people might even pay something to not be exposed to the prospect of flipping a coin for $1000? These questions, when answered in the way most people do, betray the fact that typical decision-makers do not care only about the average results that the expected value describes; they are also concerned with some measure of the degree of inherent uncertainty, the range or dispersion of the results around that average. Since expected values depend exclusively upon an averaging concept and do not indicate the dispersion around that summary measure, it should be apparent that expected values are not perfectly satisfactory descriptions of the way most people compare uncertain prospects and convert them to certainty equivalents. Hence, we shall have to complicate the story a bit in order to formulate even a very simple theory of decision making under uncertainty.
3. RISK PREFERENCE
When a decision-making entity cares about the dispersion of prospects in addition to their expected values, “risk preference” is said to be present. The existence of risk preference, its strength, and even its direction are a matter of each individual’s subjective preferences. If a particular person really did not care about the inherent uncertainty or dispersion in his prospects, then expected values would be a totally accurate way of describing that person’s assessment of prospects. Such a person, who would always rank different prospects in the exact order of their expected values, is called “risk neutral.” Many of us, however, would be prepared at times to reject an opportunity that has a higher expected value in favor of one with a lower expected value, provided that the alternative with the higher expected value also had a sufficiently higher level of risk attached. Although higher expected value is a “good,” additional risk in the form of result-dispersion is a “bad,” and we are prepared to trade off losses of expected value in exchange for reductions of risk, at least to some extent. Thus, in arriving at certainty equivalents, one might want to “discount” a prospect’s expected value in order to reflect dislike of its magnitude of risk. Individuals who attach such a negative value to risk are called “risk averse.” [Do not risk provoking horselaughs from the cognoscenti by saying risk adverse rather than risk averse; the proper sense of the term is aversion or avoidance, not opposition.] Finally, there are some people, epitomized by the “gambler” type, who may actually place a positive value on risk and who would adjust expected values upward to reflect a premium for risk. The latter are described as risk-seekers or “risk prone.”
What accounts for the existence of different sets of risk preferences in individuals? One explanation derives from the fact that the payoffs of prospects are commonly denominated in terms of things like money rather than in units of satisfaction. Would a person’s decline in satisfaction from a $1000 loss be
exactly the same magnitude as the increase in satisfaction from a $1000 gift? Suppose that the answer is “no,” that the relevant numbers are a -100 and a +60, respectively, as measured on a particular person’s internal pleasure meter. Economists call such satisfaction measures “utility” indexes. For now, we shall assume that we are working with a utility index that is “cardinal,” i.e., whose numbers have a meaning analogous to those on a thermometer, units of utility measuring satisfaction in the same way that degrees on the thermometer measure quantities of heat. (So-called “ordinal” utility indices, which measure directions of preference change but not magnitudes, will be treated in the “indifference curve” models of that we will develop in the “Topographic Models” materials below.) Then, on the above facts, where a $1000 loss implies a 100-unit utility reduction but a $1000 gain produces only a 60-unit utility increase, the coin-flip for $1000 is not an attractive prospect in utility terms: (-100 X .5) + (+60 X .5) = (-50 + 30) = -20. Even though the pecuniary expected value of the prospect is zero, corresponding to what is sometimes called a “fair bet,” it would not be surprising if a person with the hypothesized characteristics were willing to pay out a small sum of money rather than be forced to play the coin-flip game for those stakes.
The operative principle in the above example is “diminishing marginal utility of money,” i.e., each successive increment of money generates a smaller increase in satisfaction than the one before it. If this condition is true, equal monetary prospects of gains and losses do not counterbalance each other exactly; a dollar’s worth of loss is more important than a dollar’s worth of gain. A rudimentary “utility function” will make this clear. A utility function is just a mathematical way of describing the relationship between money (or some other good or bad) and a person’s utility index. Here is a simple example:
U= l00M½
In words, this says that the utility index U rises proportionately to one hundred times the square root of the quantity of money M that the person has to spend. (A number raised to the l/2 power is a square root.) In general, utility functions are hypothesized or “made up” in order to exemplify some specific point in much the same way that illustrative facts are supplied in law school hypothetical cases. The example given above deliberately incorporates the principle of declining marginal utility of income. Based on the function specified, one could construct a table that shows various income levels and their resultant utility levels: