The theory of consumer choice under situations of risk and uncertainty belongs to the field of microeconomics. Risk and uncertainty are sometimes interchangeable terms but their meaning is easily misunderstood. Frank Knight in his “Risk, Uncertainty and Profit” 1921, treated this subject and posed a fundamental distinction between the two, formulating the definition that, ever since, became the most widely used.

In a situation that involves risk the outcome is unknown, but the distribution or the probability of its occurrence is known. These probabilities can either be objectively specified or subjectively, meaning that the agents will form their expectations regarding these probabilities. Sometimes it is said that risk is a known-unknown while uncertainty an unknown-unknown, since in the latter agents cannot (or will not) assign probabilities to each outcome.

Consumer choice under risk is usually analysed using the expected utility theory approach, while uncertainty is studied mainly in game theory. However, although different models have been developed for both situations, risk situations, are the ones that have received the most attention and economists have concentrated on the analysis of risk probably due to its well-defined probability distribution in comparison with the random variables that uncertainty involves.

It must be mentioned that Daniel Kahneman co-authored with Amos Tversky in 1979 the paper “Prospect Theory: An Analysis of Decision under risk”, which drastically changed the way economists analyse decisions under risk. Their analysis is known as prospect theory, which studies the results from a series of empirical data and has very different conclusions from those drawn by expected utility theory.

The theory of consumer choice under situations of risk and uncertainty belongs to the field of microeconomics. Risk and uncertainty are sometimes interchangeable terms but their meaning is easily misunderstood. Frank Knight in his “Risk, Uncertainty and Profit” 1921, treated this subject and posed a fundamental distinction between the two, formulating the definition that, ever since, became the most widely used.

In a situation that involves uncertainty the outcome is unknown and agents cannot (or will not) assign probabilities to each outcome. Sometimes it is said that uncertainty is an unknown-unknown, while risk is a known-unknown, since agents assign probabilities to each outcome.

Consumer choice under uncertainty is studied mainly in game theory, while risk is usually analysed using the expected utility theory approach.

The Saint Petersburg paradox, is a theoretical game used in economics, to represent a classical example were, by taking into account only the expected value as the only decision criterion, the decision maker will be misguided into an irrational decision. This paradox was presented and solved in Daniel Bernoulli’s “Commentarii Academiae Scientiarum Imperialis Petropolitanae” (translated as “Exposition of a new theory on the measurement of risk”), 1738, hence its name, St. Petersburg. He solved it by making the distinction between expected value and expected utility, as the latter uses weighted utility multiplied by probabilities, instead of using weighted outcomes. However, since then, alternative approaches have been used by different researches to answer this paradox.

The key of the paradox is to determine the value someone would be willing to pay in order to play a lottery game that works as follows: a fair coin is tossed, if tail appears the player is paid $2 (in case the amount paid to play is $1), if not, the coin is tossed again, until tail appears, doubling the initial gain every time the coin is tossed. For example, for toss number 3 (n=3), the payoff would be 8 (2ⁿ) and the expected value, which here equals the payoff multiplied by the probability (here, 1).

The probability that the first tail appears in the toss number n is equal to p_n=1/2ⁿ, being 2ⁿ the payoff.  Therefore, the expected value for n tosses would be: 

If we use the expected value as the decision criterion, the player should be willing to pay $∞ in order to play. However, no rational individual would accept this. For Bernoulli, the answer relied in using the maximum expected utility instead of the maximum expected value:

Furthermore, Bernoulli stated that the utility will increase with the player’s wealth (since he or she would have more money to play), and decreasing marginal utility.

Bernoulli’s solution to the Saint Petersburg, although simple and concise, was later on greatly discussed and criticised by other economists. Knut Wicksell used it in order to develop interpersonal comparisons, while Francis Y. Edgeworth criticised its logarithmic utility function. Vilfredo Pareto replaced in his analysis of the paradox wealth by consumption, and Alfred Marshall replaced it with income.

Alfred Marshall’s analysis, present in his book “Principles of Economics”, 1890, is of great interest because, simply by replacing wealth by income, it changes the main conclusions and the solution of the paradox. Marshall explains how, when assuming decreasing marginal utility, no rational individual would play the game since losses would be greater than gains. As an example, shown in the adjacent figure, let’s say the probability of winning the game is equal to the probability of losing, hence ½. If a player loses, he or she will lose the money paid to play the game (x-1); if he or she wins, the player will win exactly the amount paid (x+1). Since area A is greater than B, because of the decreasing marginal utility, no rational player would play. As Alfred Marshall puts it:

“The clerk with £100 a-year will walk to business in a much heavier rain than the clerk with £300 a-year; for the cost of a ride by tram or omnibus measures a greater benefit to the poorer man than to the richer. If the poorer man spends the money, he will suffer more from the want of it afterwards than the richer would. The benefit that is measured in the poorer man’s mind by the cost is greater than that measured by it in the richer man’s mind.”

The term expected utility was first introduced by Daniel Bernoulli who used it to solve the St. Petersburg paradox, as the expected value was not sufficient for its resolution. He introduce the term in his paper “Commentarii Academiae Scientiarum Imperialis Petropolitanae” (translated as “Exposition of a new theory on the measurement of risk”), 1738, where he solved the paradox.  However, John von Neumann and Oskar Morgenstern, in their book “Theory of Games and Economic Behavior”, 1944, considered the cornerstone of expected utility theory, provided great contributions and built a mathematical foundation for Bernoulli’s solution of the paradox. They developed a set of axioms for the preferential relations in order to guarantee that the utility function is well-behaved.

The expected utility is used to provide an answer to situations where individuals must make a decision without knowing which outcomes may result from that decision, this is, decision making under uncertainty. These individuals will choose the act that will result in the highest expected utility, being this the sum of the products of probability and utility over all possible outcomes. The decision made will also depend on the agent’s risk aversion and the utility of other agents.

The base of the expected utility theory are lotteries (L_n), each one defined by possible outcomes (C₁,C₂,…,C_n) and their corresponding probabilities (p₁, p₂,…,p_i, with ∑p_i=1).

EU(L) = U(c₂)p₁ + U(c₂)p₂ + … + U(c_n)p_n

The expected utility theory deals with the analysis of situations where individuals must make a decision without knowing which outcomes may result from that decision, this is, decision making under uncertainty. These individuals will choose the act that will result in the highest expected utility, being this the sum of the products of probability and utility over all possible outcomes. The decision made will also depend on the agent’s risk aversion and the utility of other agents.

The base of the theory are lotteries, or gambles, (L_n) each one defined by all possible outcomes or consequences (C₁,C₂,…,C_n) and their corresponding probabilities (p₁, p₂,…,p_i, with ∑p_i=1). 

EU(L) = U(c₂)p₁ + U(c₂)p₂ + … + U(c_n)p_n

The term expected utility was first introduced by Daniel Bernoulli who used it to solve the St. Petersburg paradox, as the expected value was not sufficient for its resolution.  However, John von Neumann and Oskar Morgenstern, in their book “Theory of Games and Economic Behavior”, 1944, considered the cornerstone of expected utility theory, provided great contributions and built a mathematical foundation for Bernoulli’s solution of the paradox. They developed a set of axioms for the preferential relations in order to guarantee that the utility function is well-behaved:

1.Axioms of order:

Completeness L₀L₁ or L₁L₀ or L₀∿L₁

Reflexive L₀L₀

Transitive L₀L₁ ; L₁L₂ ↔ L₀L₂

2.Continuity: L₀L₁ ; L₁L₂

with p∊ [0,1] so that L₁∿ p L₀+(1-p)L₂

3.Rationality: in order to maximize results, the highest probability will be chosen (ceteris paribus)

4.Rational Equivalence: agents will evaluate rationally the probabilities of the different results

5.Independence (sure thing principle): if L₀L₁ →β*L₀+ (1- β)*L₂ β*L₁ +(1- β)*L₂

Attitudes and behaviour towards risks have been, and still are, highly studied fields in psychology and their economic applications have been meaningful and of high importance. While some may be willing to assume risks in order to gain economic profits, others will prefer to avoid those risks, even though they may report higher expected payoffs. The first type of individuals is considered risk loving individuals, while the second ones are risk averse individuals.

The expected utility function helps us understand levels of risk aversion in a mathematical way:

Although expected utility is a term coined by Daniel Bernoulli in the 18^th century, it was John von Neumann and Oskar Morgenstern who, in their book “Theory of Games and Economic Behavior”, 1944, developed a more scientific analysis of risk aversion, nowadays known as expected utility theory. The analysis of risk aversion we are about to see is based on their work.

In order to shape an individual’s expected utility function, the different prices (payoffs) need to be ordered. After assigning random numbers to the upper and lower levels (here 0,1), intermediate values are considered asking the individual what is the probability of a given outcome so that he or she would be indifferent whether to play or to take a guaranteed prize. This has to be repeated until the function is obtained.

The expected utility function has a set of important properties:

-it increases with income;

-it is additively separable;

-it is bounded between an lower (0) and a upper limit (1);

-it is differentiable.

Now, let’s define a few terms we’ll need in order to analyse any individual’s degree of risk aversion:

-Expected value, also referred to as mathematical expectation, which is the expected payoff :

-Utility of the expected value:

-Certainty equivalent, which is the certain payoff that reports as much utility as the utility of the expected value: 

-Expected utility, being such that it is equal for the certainty equivalent and the expected value:

Risk aversion (green) may imply that an individual may refuse to play a fair game even though the game’s expected value is zero. While on the other hand, risk loving individuals (red) may choose to play the same fair game. In case of risk neutral individuals (blue), they are indifferent between playing or not. The utility function for each case can be graphically drawn (being RP the risk premium and A the Arrow-Pratt measure of absolute risk aversion):

It is sometimes important to know how averse to risk a certain individual is. To this effect there are a set of tools to measure risk in a quantitative way. The most common and frequently used measure of risk aversion are the Arrow-Pratt measures of absolute and relative risk-aversion. Named after John W. Pratt’s paper “Risk Aversion in the Small and in the Large”, 1964, and Kenneth Arrow’s “The Theory of Risk Aversion”, 1965, these are the measures:

Arrow-Pratt measure of absolute risk aversion:

Arrow-Pratt measure of relative risk aversion: 

Where x is the payoff of a given lottery and U(x) the utility derived from that payoff.