CRITIQUES OF EXPECTED UTILITY THEORY

3.4 CRITIQUES OF EXPECTED UTILITY THEORY

Theory and practice do not always concur when we use expected utility theory. There are many reasons for such a statement. Are decision-makers irrational? Are they careless? Are they uninformed or clueless? Do they lack the proper incentives to reach a rational decision. Of course, the axioms of rationality that underlie expected utility theory may be violated. Empirical and psychological research has sought to test the real premises of decision-makers under uncertainty. To assess potential violations, we consider a number of cases. Consider first the example below called the St Petersburg Paradox that has motivated the development of the utility approach.

3.4.1 Bernoulli, Buffon, Cramer and Feller

Daniel Bernoulli in the early 1700s suggested a problem whose solution was not considered acceptable in practice, albeit it seemed to be appropriate from a theoretical viewpoint. This is called the St Petersburg Paradox. The paradox is framed in a tossing game stating how much one would be willing to pay for a game where a fair coin is thrown until it falls ‘heads’. If it occurs at the r th throw, the player receives 2 r dollars from the bank. Thus, the gain doubles at each throw. In an expected sense, the probability of obtaining ‘heads’ at the kth throw is 1/2 k , since the pay-out is also equal to 2 k , the expected value of the game is:

(1/2 k )2 k =1+1+···=∞

k=1

Thus, the fair amount to pay to play this game is infinite, which clearly does not reflect the decision makers’ behaviour. Bernoulli thus suggested a logarithmic utility function whose expected utility:

Eu (x) = p (i )u(2 i −1

The expected utility of the game equals Eu(x) = log(2). Mathematicians such as Buffon, Cramer, Feller and others have attempted to

provide a solution that would seem to be appropriate. Buffon and Cramer suggest that the game be limited (in the sense that the bank has a limited amount of money and, therefore, it can only pay a limited amount). Say that the bank has only a million dollars. In this case, we will have the following amounts,

(1/2 k )2 k + 10 6 (1/2 k ) = 1 + 1 + · · · 1 + 1.19 ≈ 21

k=1

k=20

52 EXPECTED UTILITY

Therefore, the fair amount to play this game is 21 dollars only. Any larger amount would be favourable to the bank.

Gabriel Cramer, on the other hand, suggested a square root utility function. Then for the St Petersburg game, we have:

Eu (x) = p (i )u(i ) =

And therefore, for Cramer, the value of the game is 1/(2 − 2). Feller suggests another approach however, seeking a mechanism for the gains and payments to

be equivalent in the long run. In other words, a lottery will be fair if: Accumulated gains

= R → 1 as n → ∞

Accumulated fees

N n or P

R −1 n → ∞ as n → ∞ Feller noted that the game is fair if R n = n log 2 (n). Thus if the accumulated

entrance fee to the game is proportional to the number of games, it will not be a fair game.

3.4.2 Allais Paradox

The strongest attack on expected utility theory can be found in Allais’ Paradox, which doubts the strong independence assumption needed for consistent choice in expected utility. Allais proved that the assumption of linearity in probabilities applied in calculating the expected utility is often doubtful in practice. Explicitly, the independence axiom also called the ‘sure-thing’ principle asserts that two alternatives that have a common outcome under a particular state of nature should imply that ordering should be independent of the value of their common outcome. This is not always the case and counter examples abound, in particular due to Allais.

For example, let us confront people with two lotteries. First, we have to pick one of the two gambles given by ( p 1 , p 2 ) below. The first gamble consists of $100 000 for sure (probability 1) while the other is $5 000 000 with probability

0.1, 1 million with probability 0.89 and nothing with probability 0.01 as stated below.

A second set of gambles ( p 3 , p 4 ) consists of:

and p

EXPECTED UTILITY AND FINANCE

53 Confronted with choosing between p 1 and p 2 , people chose p 1 while confronting

p 1 ,p 3 and p 4 , people preferred p 3 which is in contradiction with the strong independence axiom of utility theory. In other words, if gamble 1 is selected over gamble 2, while in presenting people with gambles 1, 3 and 4 results in their selecting 3 over 1, there is necessarily a contradiction, since if we were to compare

gambles 3 and 2, clearly, 3 is not as good as 2. This contradiction means, therefore, that application of expected utility theory does not always represent investors’ and decision-maker’s psychology. Since then, a large number of studies have been done, seeking to bridge a gap between investors’ psychology and the concepts of utility theory. Some essential references include Kahnemann and Tverski (1979), Machina on anticipated utility (1982, 1987) and Quiggin (1985) as well as many other others. In these approaches, the expected utility framework is ‘extended’ by stating that an uncertain prospect can be measured by an ‘expected utility’ u (.) interpreted either as the choice of a utility function (as was the case in traditional expected utility) or by a preferred probability distribution (P) (or a function g (P) assumed over the probability distribution). Then, the probabilities used to calculate the expected utility would be ‘subjective’ estimates, or beliefs, about the probabilities of returns, imbedding ‘something else’ above and beyond the objective assessment of uncertain prospects. Thus, the objective index used to value the relative desirability of uncertain prospects is also a function of the model used for probabilities P(.). This is in contrast to the utility function, expressing a behaviour imbedded in the choice of the function u(.) only, which stands solely for the investor’s psychology, as we saw above. For example, what if we were to determine probabilities P ∗ (.) such that the price of random prospects ˜ R could be uniquely defined by the following expected value?

R ˜ dP ∗ (˜ R )

In this case, once such a well-defined transformation of these probabilities is reached, all uncertain prospects may be valued uniquely, thereby simplifying greatly the problem of financial valuation of risk assets such as stocks, default bonds and the like. This approach, defined in terms of economic exchange mar- ket mechanisms (albeit subject to specific assumptions regarding markets and individual behaviours), underlies the modern theories of finance and ‘risk-neutral pricing’. This is also an essential topic of our study in subsequent chapters.