Allowing Violations of Independence

Both the common ratio and common consequence effects suggest that indifference curves are steeper in the western portion of the Marschak–Machina triangle than in the southeast portion. This led some to suppose that instead of parallel indifference curves, indifference curves fan out across the Marschak–Machina triangle. One way to salvage the axioms, or behavioral rules, that serve as the foundations for expected utility is to replace the independence axiom with the betweenness axiom.

Betweenness Axiom

If A B, and C is a compound gamble that yields A with probability p and B with probability 1 − p, then betweenness is satisfied if A C B.

The betweenness axiom allows indifference curves to have different slopes, but it still requires indifference curves to be straight lines. Thus, the indifference curves may be like those displayed in Figure 9.5. To see that indifference curves must be straight lines, consider the gambles A and B that lie on a single indifference curve. Any gamble located directly between points A and B represents a gamble that is a combination of A and B (called C in the definition). Thus, betweenness requires that the gambler be indifferent between A, B, and C, and thus C must also lie on the indifference curve.

One of the first proposed theories to employ betweenness is called weighted expected utility theory. Let p1, .. , pn be the probabilities associated with outcomes x1, , xn. Then weighted expected utility preferences suppose that people act to maximize their weighted utility, which is given by

Here, ux is a standard utility of wealth function, and vx constitutes some function used to weight the probabilities associated with particular outcomes. To derive indifference curves, consider all points p1, p2 such that

where the left side of equation 9.32 corresponds to the definition of weighted utility from equation 9.31, and the right side is just a constant. Rearranging obtains

p1 u x1 − u x3 − k v x1 − v x3 + p2 u x2 − u x3 − k v x2 − v x3

933

+ u x3 − kv x3 = 0,

reflected around the 45-degree line (see Figure 9.7). This reflection results in many of the same properties but some distinct differences. In particular, people appear to be more risk loving (more shallow curves) in the center of the triangle.

Evidence on the Shape of Probability Weights

Malcolm G. Preston and Philip Baratta were the first to experiment with using probability weights to describe deviations from expected utility theory. They conducted several experiments in which they auctioned off various simple gambles (with some probability of winning an outcome and remaining probability of winning nothing). They discovered that willingness to pay for gambles was approximately linear in the amount of the reward but highly nonlinear in the probability associated with the outcome. This led them to suppose that people were misperceiving the probabilities in very systematic ways. In particular, people appeared to overweight small probabilities and underweight large probabilities. Similar work has confirmed their early findings with respect to the general shape of these perceived probabilities. Such systematic misperceptions would explain some of the violations of expected utility such as the common ratio and common consequence effects.

One prominent theory of decision under risk supposes that people seek to maximize their perceived expected utility, where their perception is described by probability weights (discussed earlier in examples 9.3 and 9.4). Probability weights are given by a function πp, which maps probabilities onto the unit interval. Generally πp is thought to be an increasing function of the true probability, with πp > p for all p < p and πp < p for p > p. Moreover, experimental evidence suggests that π is a concave function when p < p, and a convex function when p > p. A typical probability weighting function is depicted in Figure 9.8. Generally, the fixed point (the point p such that