A key assumption of EU is that everything is comparable. That is, given any two objects, a person must prefer one to the other or be indifferent to both; no two objects are incomparable. For example, a person may compare a dime and a nickel or a cheeseburger and a dime. We might compare a job offer in the Midwest to a job offer on the West Coast. Utility theory implies a single, underlying dimension of “satisfaction” associated with everything. We can recall instances in which we refused to make comparisons, which often happens in the case of social or emotional issues such as having to do with marriage and children. However according to utility theory, we need to be able to compare everything to be truly rational. Many people are uncomfortable with this idea, just as people can be in conflict about what is negotiable.

The closure property states that if x and y are available alternatives, then so are all of the gambles of the form (x, p, y) that can be formed with x and y as outcomes. In this formulation, x and y refer to available alternatives; p refers to the probability that x will occur. Therefore, (x, p, y) states that x will occur with probability p, otherwise y will occur. The converse must also be true: (x, p, y) = (y, 1 – p, x), or y will occur with probability (1 – p), otherwise x will occur.

For example, imagine you assess the probability of receiving a raise from your current employer to be about 30%. The closure property states that the situation expressed as a 30% chance of receiving a raise (otherwise, no raise) is identical to the statement that you have a 70% chance of not receiving a raise (otherwise, receiving a raise).

So far, utility theory may seem to be so obvious and simple that it is absurd to spell it out in any detail. However, we will soon see how people violate basic “common sense” all the time and hence, behave irrationally.

Transitivity means that if we prefer x to y and y to z, then we should prefer x to z. Similarly, if we are indifferent between x and y and y and z, then we will be indifferent between x and z.

Suppose your employer offers you one of three options: a transfer to Seattle, a transfer to Pittsburgh, or a raise of $5,000. You prefer a raise of $5,000 over a move to Pittsburgh, and you prefer a move to Seattle more than a $5,000 raise. The transitivity property states that you should therefore prefer a move to Seattle over a move to Pittsburgh. If your preferences were not transitive, you would always want to move somewhere else. Further, a third party could become rich by continuously “selling” your preferred options to you.

The reducibility axiom refers to a person’s attitude toward a compound lottery, in which the prizes may be tickets to other lotteries. According to the reducibility axiom, a person’s attitude toward a compound lottery depends only on the ultimate prizes and the chance of getting them as determined by the laws of probability; the actual gambling mechanism is irrelevant:

Suppose that the dean of admissions at your top choice university tells you that you have a 25% chance of getting accepted to the MBA program. How do you feel about the situation? Now, suppose the dean tells you that you have a 50% chance of not being accepted and a 50% chance you will have to face a lottery-type admission procedure, wherein half the applicants will get accepted and half will not. In which situation do you prefer to be? According to the reducibility axiom, both situations are identical. Your chances of getting admitted into graduate school are the same in each case: exactly 25%. The difference between the two situations is that one involves a compound gamble and the other does not.

Compound gambles differ from simple gamble ones in that their outcomes are themselves gambles rather than pure outcomes. Furthermore, probabilities are the same in both gambles. If people have an aversion or attraction to gambling however, these outcomes may not seem the same. This axiom has important implications for negotiation; the format by which alternatives are presented to negotiators—in other words, in terms of gambles or compound gambles—strongly affects our behavior.

The substitutability axiom states that gambles that have prizes about which people are indifferent are interchangeable. For example, suppose one prize is substituted for another in a lottery but the lottery is left otherwise unchanged. If you are indifferent to both the old and the new prizes, you should be indifferent about the lotteries. If you prefer one prize to the other, you will prefer the lottery that offers the preferred prize.

For example, imagine you work in the finance division of a company and your supervisor asks you how you feel about transferring to either the marketing or sales division in your company. You respond that you are indifferent. Then your supervisor presents you with a choice: You can be transferred to the sales division, or you can move to a finance position in an out-of-state branch of the company. After wrestling with the decision, you decide you prefer to move out of state rather than transfer to the sales division. A few days later, your supervisor surprises you by asking whether you prefer to be transferred to marketing or to be transferred out of state. According to the substitutability axiom, you should prefer to transfer because as you previously indicated, you are indifferent to marketing and sales; they are substitutable choices.

The betweenness axiom asserts that if x is preferred to y, then x must be preferred to any probability mixture of x and y, which in turn must be preferred to y. This principle is certainly not objectionable for monetary outcomes. For example, most of us would rather have a dime than a nickel and would rather have a probability of either a dime or a nickel than the nickel itself. But consider nonmonetary outcomes, such as skydiving, Russian roulette, and bungee jumping. To an outside observer, people who skydive apparently prefer a probability mixture of living and dying over either one of them alone; otherwise, one can easily either stay alive or kill oneself without ever skydiving. People who like to risk their lives appear to contradict the betweenness axiom. A more careful analysis reveals however, that this situation, strange as it may be, is not incompatible with the betweenness axiom. The actual outcomes involved in skydiving are (a) staying alive after skydiving, (b) staying alive without skydiving, or (c) dying while skydiving. In choosing to skydive therefore, a person prefers a probability mix of (a) and (c) over (b). This analysis reveals that “experience” has a utility.

Suppose that of three objects—A, B, and C—you prefer A to B and B to C. Now, consider a lottery in which there is a probability, p, of getting A and a probability of 1 − p of getting C. If p = 0, the lottery is equivalent to C; if p = 1, the lottery is equivalent to A. In the first case, you prefer B to the lottery; in the second case, you prefer the lottery to B. According to the continuity axiom, a value, p, which falls between 0 and 1, indicates your indifference between B and the lottery. This sounds reasonable enough.

Now consider the following example involving three outcomes: receiving a dime, receiving a nickel, and being shot at dawn.⁶ Certainly, most of us prefer a dime to a nickel and a nickel to being shot. The continuity axiom however, states that at some point of inversion, some probability mixture involving receiving a dime and being shot at dawn is equivalent to receiving a nickel. This derivation seems particularly disdainful for most people because no price is equal to risking one’s life. However, the counterintuitive nature of this example stems from an inability to understand very small probabilities. In the abstract, people believe they would never choose to risk their life but in reality, people do so all the time. For example, we cross the street to buy some product for a nickel less, although by doing so we risk getting hit by a car and being killed.

In summary, whenever these axioms hold, a utility function exists that (a) preserves a person’s preferences among options and gambles and (b) satisfies the expectation principle: The utility of a gamble equals the expected utility of its outcomes. This utility scale is uniquely determined except for an origin and a unit of measurement.

Imagine you have a rare opportunity to invest in a highly innovative start-up company with a revolutionary new technology for its product. On the other hand, the technology is new and unproven. Further, the company does not have the resources to compete with the established competitors. Nevertheless if this technology is successful, an investment in the company at this point will have a 30-fold return. Suppose you just inherited $5,000 from your aunt. You could invest the money in the company and possibly earn $150,000—a risky choice. Or you could keep the money and pass up the investment opportunity. You assess the probability of success to be about 20%. (A minimum investment of $5,000 is required.) What do you do?

The dominance principle does not offer a solution to this decision situation because it provides no clearly dominant alternatives. But the situation contains the necessary elements to use the expected value principle, which applies when a decision maker must choose among two or more prospects, as in the previous example. The “expectation” or “expected value” of a prospect is the sum of the objective values of the outcomes multiplied by the probability of their occurrence.

For a variety of reasons, you believe there is a 20% chance the investment could result in a return of $150,000, which mathematically is 0.2 × $150,000 = $30,000. There is an 80% chance the investment will not yield a return, or 0.8 × $0 = $0. Thus, the expected value of this gamble is $30,000 minus the cost, or − $5,000, = $25,000.

The expected value principle dictates that the decision maker should select the prospect with the greatest expected value. In this case, the risky option (with an expected value of $30,000) has a greater expected value than the sure option (expected value of $0).

A related principle applies to evaluation decisions, or situations in which decision makers must state and be willing to act upon, the subjective worth of a given alternative. Suppose you could choose to “sell” to another person your opportunity to invest. What would you consider to be a fair price to do so? According to the expected-value evaluation principle, the evaluation of a prospect should be equal to its expected value. That is, the “fair price” for such a gamble would be $25,000. Similarly, the opposite holds: Suppose your next-door neighbor held the opportunity but was willing to sell the option to you. According to the expected value principle, people would pay up to $25,000 for the opportunity to gamble.

The expected value principle is intuitively appealing, but should we use it to make decisions? To answer this question, it is helpful to examine the rationale for using expected value as a prescription. Let’s consider the short-term and long-term consequences.⁷ Imagine that you will inherit $5,000 every year for the next 50 years. Each year, you must decide whether to invest the inheritance in a start-up company (risky choice) or keep the money (the sure choice). The expected value of a prospect is its long-run average value. This principle is derived from a fundamental principle: the law of large numbers.⁸ The law of large numbers states that the mean return will get closer and closer to its expected value the more times a gamble is repeated. Thus, you can be fairly sure that after 50 years of investing your money, the average return would be about $25,000. Some years you would lose, other years you would make money, but on average your return would be $25,000. When you look at the gamble this way, it seems reasonable to invest.

Now imagine the investment decision is a once-in-a-lifetime opportunity. In this case, the law of large numbers does not apply to the expected-value decision principle. You will either make $150,000, make nothing, or keep $5,000. No in-between options are possible. Under such circumstances, you can often find good reasons to reject the guidance of the expected value principle.⁹ For example, suppose you need a new car. If the gamble is successful, buying a car is no problem. But if the gamble is unsuccessful, you will have no money at all. Therefore, you may decide that buying a used car at or under $5,000 is a more sensible choice.

The expected value concept is the basis for a standard way of labeling risk-taking behavior. For example, in the previous situation you could either take the investment gamble or receive $25,000 from selling the opportunity to someone else. In this case, the “value” of the sure thing (i.e., receiving $25,000) is identical to the expected value of the gamble. Therefore, the “objective worth” of both alternatives is identical. What would you rather do? Your choice reveals your risk attitude. If you are indifferent to the two choices and are content to decide on the basis of a coin flip, you are risk neutral or risk indifferent. If you prefer the sure thing, then your behavior may be described as risk averse. If you choose to gamble, your behavior is classified as risk seeking.

Although some individual differences occur in people’s risk attitudes, people do not exhibit consistent risk-seeking or risk-averse behavior.¹⁰ Rather, risk attitudes are highly context dependent. The fourfold pattern of risk attitudes predicts people will be risk averse for moderate- to high-probability gains and low-probability losses, and risk seeking for low-probability gains and moderate- to high-probability losses.¹¹

The reactions people have to the St. Petersburg game are consistent with the proposition that people decide among prospects not according to their expected objective values but, rather, according to their expected subjective values. In other words, the psychological value of money does not increase proportionally as the objective amount increases. To be sure, virtually all of us like more money rather than less money, but we do not necessarily like $20 twice as much as $10. And the difference in our happiness when our $20,000 salary is raised to $50,000 is not the same as when our $600,000 salary is raised to $630,000. Bernoulli proposed a logarithmic function relating the utility of money, u, to the amount of money, x. This function is concave, meaning that the utility of money decreases marginally. Constant additions to monetary amounts result in less and less increased utility. The principle of diminishing marginal utility is related to a fundamental principle of psychophysics, wherein good things satiate and bad things escalate. The first bite of a pizza is the best; as we get full, each bite brings less and less utility.

The principle of diminishing marginal utility is simple, yet profound. It is known as the everyman’s utility function (EU).¹⁵ According to Bernoulli, a fair price for a gamble should not be determined by its expected (monetary) value but rather by its expected utility. Thus, the logarithmic utility function in Exhibit A1-3 yields a finite price for the gamble.

According to EU, each of the possible outcomes of a prospect has a utility (subjective value) that is represented numerically. The more appealing an outcome is, the higher is its utility. The expected utility of a prospect is the sum of the utilities of the potential outcomes, each weighted by its probability. According to EU, when choosing among two or more prospects, people should select the option with the highest expected utility. Further, in evaluation situations, risky prospects should have an expected utility equal to the corresponding “sure choice” alternative.

EU principles have essentially the same form as expected value (EV) principles. The difference is that expectations are computed using objective (dollar) values in EV models as opposed to subjective values (utility) in EU models.

A person’s utility function for various prospects reveals something about his or her risk-taking tendencies. If a utility function is concave, a decision maker will always choose a sure thing over a prospect whose expected value is identical to that sure thing. The decision maker’s behavior is risk averse (Exhibit A1-4, Panel A). The risk-averse decision maker would prefer a sure $5 over a 50–50 chance of winning $10 or nothing—even though the expected value of the gamble [0.5($10) + 0.5($0) = $5] is equal to that of the sure thing. If a person’s utility function is convex, he or she will choose the risky option (Exhibit  A1-4, Panel B). If the utility function is linear, his or her decisions will be risk neutral and of course, identical to those predicted by expected value maximization (Exhibit  A1-4, Panel C).

If most peoples’ utility for gains are concave (i.e., risk averse), then why would people ever choose to gamble? Bets that offer small probabilities of winning large sums of money (e.g., lotteries, roulette wheels) ought to be especially unattractive given that the concave utility function that drives the worth of the large prize is considerably lower than the value warranting a very small probability of obtaining the prize.

Consider the following two options. Which would you choose?

• Option A: 80% probability of earning $4,000, otherwise $0

• Option B: Earn $3,000 for sure

If you are like most people, you chose B. Only a small number of people (20%) choose A.

Now, consider two different options:

• Option C: 20% probability of earning $4,000, otherwise $0

• Option D: 25% probability of earning $3,000, otherwise $0

Faced with this choice, a clear majority (65%) choose option C over option D (the smaller, more likely payoff).¹⁶

However in this example, the decision maker violates EU, which requires consistency between the A versus B choice and the C versus D choice. In Exhibit A1-5, Branch 1 depicts the C versus D choice [i.e., 25% chance of making $3,000 (otherwise $0) or 20% chance of making $4,000 (otherwise $0)]. In Branch 2 of Exhibit A1-5, another stage has been added to the gamble between A and B, which effectively makes the two-stage gamble in Branch 2 identical to the one-stage gamble in Branch 1. Because Branch 1 is objectively identical to Branch 2, the decision maker should not make different choices. Stated another way, in the A versus B and C versus D choices, the ratio is the same: (0.8/1) = (0.20/0.25). However, people’s preferences usually reverse. According to the certainty effect, people have a tendency to overweight certain outcomes relative to outcomes that are merely probable. The reduction in probability from certainty (1) to a degree of uncertainty (0.8) produces a more pronounced loss in attractiveness than a corresponding reduction from one level of uncertainty (0.25) to another (0.2). People do not think rationally about probabilities. Those close to 1 are often (mistakenly) considered sure things. On the flip side is the possibility effect: the tendency to overweight outcomes that are possible relative to outcomes that are impossible.

People tend to overweight low probabilities and underweight high probabilities.

The crossover probability is the point at which objective probabilities and subjective weights coincide. Prospect theory does not pinpoint where the crossover occurs, but it is definitely lower than 50%.¹⁷

Adding two probabilities, p ₁ and p ₂, should yield a probability p ₃ = p ₁ + p ₂. For example, suppose you are an investor considering three stocks: A, B, and C. You assess the probability that stock A will close 2 points higher today than yesterday to be 20%, and you assess the probability that stock B will close 2 points higher today than yesterday to be 15%. The stocks are two different companies in two different industries and are completely independent. Now consider the price of stock C, which you believe has a 35% probability of closing 2 points higher today. The likelihood of a 2-point increase in either stock A or B should be identical to the likelihood of a 2-point increase in stock C. The probability-weighting relationship however, does not exhibit additivity. That is for small probabilities, weights are subadditive, as we see from the extreme flatness at the lower end of the curve. It means that most decision makers consider the likely increase of either stocks A or B to be less likely than an increase in stock C.

Except for guaranteed or impossible events, weights for complementary events do not sum to 1. One implication of the subcertainty feature of the probability-weight relationship is that for all probabilities, p, with < p < 1, p(p) + p(1 − p) < 1.

According to the regressiveness principle, extreme values of some quantity do not deviate very much from the average value of that quantity. The relative flatness of the probability-weighting curve is a special type of regressiveness, suggesting that people’s decisions are not as responsive to changes in uncertainty as are the associated probabilities. Another aspect is that nonextreme high probabilities are underweighted and low ones are overweighted.

The subjective value associated with a prospect depends on the decision weights and the subjective values of potential outcomes. Prospect theory makes specific claims about the form of the relationship between various amounts of an outcome and their subjective values.¹⁸ Exhibit A1-7 illustrates the generic prospect theory value function.

Three characteristics of the value function are noteworthy. The first pertains to the decision maker’s reference point. At some focal amount of the pertinent outcome, smaller amounts are considered losses and larger amounts gains. That focal amount is the negotiator’s reference point. People are sensitive to changes in wealth.

A second feature is that the shape of the function changes markedly at the reference point. For gains, the value function is concave, exhibiting diminishing marginal value. As the starting point for increases in gains becomes larger, the significance of a constant increase lessens. A complementary phenomenon occurs in the domain of losses: Constant changes in the negative direction away from the reference point also assume diminishing significance the farther from the reference point the starting point happens to be.

Finally, the value function is noticeably steeper for losses than for gains. Stated another way, gains and losses of identical magnitude have different significance for people; losses are considered more important. We are much more disappointed about losing $75 than we are happy about making $75.