x = payoff from winning the gamble

y = cost of losing the gamble

Where the payoff equals the cost, x = y, and the EMV is equal to zero.

If the utility from the individuals’ present wealth is U(W*)^{}, then using basic mathematics and geometric representation it can be shown that the utility for a risk averse individual who accepts a gamble (with EMV equal to zero) can be represented as below:

UA(W*) = (p)U(W* + x) + (1 - p)U(W* - x)

= (1/2)U(W* + x) + (1/2)U(W* - x)

where UA(W*) is the individual’s current wealth plus the outcome of the actuarially fair bet.

Since the individual is risk averse, the utility for his current wealth is greater than the utility of his current wealth plus the outcome of the actuarially fair bet (with EMV equal to zero), as shown below.

U(W*) > UA(W*)

Graphically the risk averse individual’s utility can be shown as follows:

Utility of Wealth from a Fair Bet

It can be seen from the diagram that the a risk averse individual – with a corresponding concave utility function (to exhibit his diminishing marginal utility of wealth since he is risk averse) – will not accept an actuarially fair bet since he obtains higher utility by ‘sticking’ with his current wealth (with utility U(W*)).

Using the same diagram the willingness of the risk averse individual to pay some amount to avoid a gamble can be shown. It can be seen that wealth Wi provides the same utility as U(W*). Since Wi is less than W* (on the graph) this individual will be willing to pay the difference between W* and Wi to avoid such a gamble. The quantity of wealth that this individual is foregoing to avoid risk is their risk premium. Therefore, as long as the cost of buying insurance is less than the risk premium, the individual will find it preferable to purchase insurance rather than be ‘exposed’ to risk. Thus by purchasing insurance, this risk averse individual has avoided actuarially fair gamble A (which is a fair bet) at a cost.

(ii)

The von Neumann-Morgenstern model is a utility index that can be used for analysing economic behaviour under conditions of uncertainty. In order to examine any problems that may beset this model, the axioms need to be highlighted. The bases for this model are five axioms that enable the theorem to gauge individual choice using expected utility. By examining the assumptions any problems with this model can be scrutinised.

Comparability and completeness is the first axiom. It is assumed that two choices can be compared, with the options that one alternative is preferable, indifferent or an inferior choice to the other option. The second axiom is transitivity (and consistency). Under this condition is A is preferred to B, and B is preferred to C, then in order to remain consistent, A must be preferable to C. Strong independence assumes that consumers will notice is uncertainty is introduced into a situation (will consistency when there is certainty), this is the third axiom. Forth is the assumption that preferences are measurable (since they are constructed around probabilities). Finally, the fifth axiom is that preferences can be ranked. These assumptions allow a cardinal utility index^{} to be created, although this does not allow an absolute scale to be founded, it is a relative measure. As it will be shown the axiom of strong independence is the main source of problems for this model.

The problems that can be outlined are those of violation of linearity and preference reversal. Systematic violation of linearity (or of the independence axiom) is one of the major problems that has beset the von Neumann Morgenstern model (despite that fashion that this model dispatches the St Petersbury Paradox^{}). The famous Allais Paradox, and associated fanning out, is such example of this difficulty. This problem involves obtaining preferred options regarding two pairs of gambles; diagrammatically it can be represented as below:

Expected utility indifference curves and the Allais Paradox.

p1 = probability outcome x1 occurs

p3 = probability outcome x3 occurs

Using the expected utility hypothesis a preference for a1 (between options a1 and a2 in a gamble) would suggest also choosing a4, which has similar characteristics to the attainable probabilities involved in the initial gamble (when choosing between a3 and a4 in a second gamble). However, research by Slovic and Tversky (1974)^{} suggests that the modal has been a1 and a3. This implies that indifference curves are not parallel but infact fan out, against the independence axiom (this is shown below):

Fanning out and the Allais Paradox

Despite the problem that this might create for the von Neumann Morgenstern model, it has been argued that any evidence of individuals’ choices violating the independent axiom would correct themselves once the nature of the violation was revealed. But studies by Slovic and Tversky (1974)^{} emphasise that there is no predominant swing toward the expected utility choices.

Further evidence of fanning out can be seen in the common consequence effect (a special case of a general empirical pattern). Under common consequence better off individuals would be if outcome T occurred, then the more risk averse they will be over what they will receive in the event of the alternate outcome H. This effect becomes apparent when a choice between two gambles has to be made. The independence axiom states that b1 and b3 or b2 and b4 will be chosen (where the choices are b1 versus b2 in gamble one, and b3 versus b4 in gamble two), since there is a preferred factor in either b1 and b3 or in b2 and b4. But research has found that there is a tendency for individuals to choose b1 and b4 (Chew and Waller, 1986^{}). The violation of the independence axiom is against the von Neumann Morgenstern model.

A second class of systematic violation is the common ratio effect (named so because the analysis concerns the value of probability X being divided by probability Y). Again two gambles are involved with c1 versus c2 and c3 versus c4. These options can be shown in the following illustration:

Indifference curves that fan out and the common ratio effect

If the individual’s indifference curves are steep then he will choose c1 and c4, or c2 and c3 if his indifference curves are flat, since these combinations are similar in type. But studies have shown a tendency to deviate from these predictions towards c1 and c4^{}, again showing fanning out.

Further to the previous violations of linearity is the preference reversal phenomenon. This concept outlines a situation where an individual has a choice between a bet where has a higher chance of winning (p-bet), and one where he can win a greater amount (£-bet). The von Neumann Morgenstern model implies that the bet that will be chosen will be the one assigned the higher certainty equivalent. But Liechtenstein and Slovic^{} report that a systematic tendency for subjects to choose the £-bet despite assigning a higher value to p-bet.

All of these inconsistencies with the assumptions – upon which the model is based – cause basic problems in using the von Neumann Morgenstern model. Even though the majority of these difficulties can be dealt with to some degree it is not within the scope of this paper to tackle such solutions, (which involves examining researchers’ experimenting with non-expected utility models of preferences for generalised non-linear functional forms) this could be used as a field of further research.

Bibliography

Nicholson Microeconomic Theory 1998

Machina Choice under Uncertainty: Problems Solved and Unsolved 1987

Journal of Economic Perspectives

Gravelle and Rees Microeconomics 1992

Varian Intermediate Microeconomics 1999

From the von Neumann-Morgenstern index, which will be discussed later.

Which does not involve cardinal utility.

Where the expected payoff from a particular gamble is infinite.

Referred to in Machina, Choice under Uncertainty, Journal of Economic Perspectives, p128.

Referred to in Machina, Choice under Uncertainty, Journal of Economic Perspectives, p129.

Referred to in Machina, p129.

With studies by MacCrimmon and Larsson (1979), Machina, p132.