Chapter 2 THEORETICAL BACKGROUND
2.4 Expected Utility Theory (EUT)
∑ (2.10)
And | is the probability density function with the mean b and covariance W. A great deal of literature has specified a range of suitable distributions of | for MMNL (refer to Revelt and Train, 1998; McFadden and Train, 2000 for details).
2.4 Expected Utility Theory (EUT)
The initial studies regarding decision making under risk were mostly carried out in a restricted context where participants were given the monetary value of lottery choices. As a result, is simply the face value of monetary outcome without any nonlinear transformation of utility. In 1738, Bernoulli originally proposed Expected Utility Theory (EUT) which assumes that individuals use subjective utility instead of monetary value to measure gamble outcomes. He concluded that subjects would not choose alternatives based on the expectation of monetary wealth, rather on the expectation of utility. Von Neumann and Morgenstern (1947) developed EUT by extending it into Game Theory. Thereafter, EUT was generally developed by Marschak (1950), Herstein and Milnor (1953) and Fishburn (1970).
As a result, EUT has operated as the normative approach to address choice under risk for more than 60 years. It should be noted that there has been an increasing interest in experimental economics which challenges the validity of EUT (De Palma et al., 2008, Kahneman, 2011).
EUT follows several assumptions of neoclassical economics, such as completeness, transitivity and reflexivity (these are also assumptions of preference ordering over prospects).
According to Hargreaves Heap (1992), there exist four extra axioms that EUT can be derived from:
Preference increasing with probability: if and and , then if and only if .
Continuity: for all prospects where , there must exist some probability p such that . Combining the assumptions of ordering
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and continuity implies that preference can be represented by the utility function which assigns a specific number to each prospect.
Strong independence: given , if , then , if this assumption holds, the utility function is determined to be additive across different states of the world.
Usual rule for combining probabilities: for prospect and , if and only if .
Based on these assumptions, the EUT utility function is expressed as:
∑ (2.11)
where is the associated probability of the outcome. Therefore, EUT provides a basic structure for decision making under risk by simply converting consequences and associated probabilities into the single scalar of utility. This normative method has been widely reported due to its intuitive appeal and mathematical capacity.
2.4.1 Attitude towards risk
According to Petty et al. (1983), the term attitude is defined as a “general, enduring, positive or negative feeling about some person, object or issue”. Rosenberg and Hovland (1960), meanwhile, stated that attitude is a “predisposition to respond to some class of stimuli with certain classes of response”. These so called stimuli correspond to risk in the domain of risky choice.
Psychologists and behavioural scientists have argued that “attitude towards risk” is an essential complementary idea in economics in so far as attitude towards risk enables a rational man in economics to behave more like a realistic man with different risk attitudes.
Moreover, it is attitude towards risk that has evoked substantive research regarding nonlinear transformations of utility and probability.
Given a gamble between two scenarios with the same expected payoff, in which one provides certain payoff while the other one is uncertain, attitudes towards risk can be categorized as risk aversion, risk neutrality and risk proneness according to the different decisions made by individuals. Specifically, if an individual prefers the scenario with the certain payoff, he is categorised as risk averse; if he prefers the scenario with the uncertain
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payoff, he is categorised as risk prone; whereas if he shows indifference between the scenarios, he is risk neutral.
The shape of the utility function can also be interpreted behaviourally with respect to attitude towards risk. Specifically, the concavity, convexity and linearity of the utility function implies risk aversion, risk proneness and risk neutrality respectively.This, therefore, means that is straightforward to illustrate the connection between risk attitudes and the curvature of the Bernoulli utility function, as shown in Figure 2.3.
Utility
sA sB sC sD Monetary value of outcomes
( )D u s
( )C ( )B
E s u s
( )A
u s ( )s
C'
C
D
B
A
( )C u s
Figure 2.3: Risk aversion and the effect of ‘certainty-equivalent’
Given two outcomes, and , the curve represents the elementary utility of each certain outcome. Let be the random variable within the interval [ ], along with the associated probabilities and for outcome and . Consequently, if we treat outcome as the probabilistic outcome of and , the utility of is the expected value of outcome is , as shown on the linear line AD. If we treat outcome as certain monetary outcome, the utility is . Notice that the point C is higher than C’, i.e., [ ] , which means that the decision maker prefers the alternative with certainty to the alternative with risk, i.e., the decision maker is risk averse. In this case, the risk attitudes can be expressed as follows:
Risk aversion if
Risk proneness if
Risk neutrality if
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Notice that risk attitudes can be explained by the effect of ‘certainty-equivalent’ as well.
Given an outcome with certainty, the utility of is . Notice that , while the monetary value of a certain outcome is less than the value of a risky outcome . Therefore, with certainty is referred to as the ‘certainty-equivalent lottery’, i.e., the uncertain lottery delivers the same utility as the certain lottery; is recognized as the ‘risk premium’, i.e., the maximum quantity of income that a decision maker prefers to pay for an allocation without risk. Consequently, risk attitudes can also be interpreted as follows:
Risk-aversion: if the utility function is concave or
Risk-neutrality: if the utility function is linear or
Risk-proneness: if the utility function is convex or .
The second derivative is generally applied to represent the shape of the utility function, in particular, a linear function has ; for a convex function, whereas for a concave function. Based on these features of utility function, Arrow (1965) and Pratt (1964) proposed a widely-used measure of aversion, termed the ‘Arrow-Pratt index of risk-aversion’.
The coefficient of absolute risk aversion is defined by
And the coefficient of relative risk aversion is defined as
where x represents the total wealth level or monetary value. For both and , if the index is greater than zero, risk aversion is implied. Arrow and Pratt also proposed the concept of constant risk aversion. That is, if both and is constant in x, the decision maker has constant absolute or relative risk aversion.
2.4.2 Limitations of EUT
Allais (1953) was the first study to provide convincing counterexamples to challenge the validity of EUT. Here the variations of the Allais paradox are described, as set out by Kahneman and Tversky (1979).
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Notice that problem 2 is obtained from problem 1 by removing the common consequence of winning 2400 with a probability 66%. From the assumption of independence of EUT, individual’s should display the preferences in problem 1 and problem 2. Nonetheless, the final data reveals that the utility for certain gain (prospect B) reduces more markedly.
Considerable violations of EUT are also apparent from other perspectives, such as inflating small probabilities, preference reversal, failure of description invariance et al. (Cox et al., 2011, Douglas, 2013, McFadden, 1999, Tversky and Kahneman, 1986). Most of these criticisms concentrate on the validity of the transitivity and independence axioms (Allais, 1979, Camerer et al., 2011, Manktelow, 2012). It seems that individuals’ biases, misconceptions and errors affect their actual decision making. However, it should be noted that almost all the attempts were simply based on laboratory experiments without empirical evidences. This is unfortunate, whilst these experimental efforts have been already directed at developing alternatives to EUT, i.e., non-EUT.