#### Summary

In this LP we learn a bit more about risk, but also about uncertainty. We start by seeing again how risk is analysed using Morgenstern and von Neumann’s expected utility theory. We also learn about alternative approaches, such as the Friedman-Savage and Markowitz perspectives, but especially Daniel Kahneman’s prospect theory. We end our study of risk and uncertainty by learning how game theory can help when analysing uncertainty.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*):