#### 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.#### Risk:

#### EUT and other approaches:

- Expected utility theory
**Alternative approaches**- Prospect theory

#### Uncertainty:

After *John von Neumann* and *Oskar Morgenstern* developed the *expected utility theory* in their “Theory of Games and Economic Behaviour”, 1944, various different approaches were developed. Although the *expected utility function* helps us understand the real world, it is important to remember that it is only a simplification of it. Expected utility theory does not completely reflect how agents interact in the real world. Furthermore, agents’ behaviour in the real world seems to systematically break some of the axioms. Let’s see some of the alternative approaches that where formulated after the original expected utility theory came out.

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**Friedman-Savage perspective:**

*Milton Friedman* along with Leonard J. Savage, developed in their “Utility Analysis of Choices Involving risk”, 1948, their own utility function known as the Friedman-Savage utility function. They argued that a single individual could have different *utility functions* depending on their initial wealth. The implication of an individual being, at the same time, *risk-loving and averse*, implies that its utility function has different curvatures as shown in the following figure.

**Markowitz’s perspective:**

Although the Friedman-Savage perspective somehow explained the flaws in the original expected utility theory axiomatic, the critiques and alternative theories didn’t stop there. *Harry Markowitz*, who was a student of Milton Friedman, criticized the Friedman-Savage utility function. Markowitz argued in his paper “The Utility of Wealth”, 1952, that the final concavity of their function assumes that individuals with the highest incomes would never gamble. As a prelude to *Kahneman* and Tversky’s *prospect theory*, he proposed measuring utility based on a reference level instead of in absolute values. This implied that, to individuals, small gains would provide an increasing utility, while big gains would provide a decreasing utility, as shown in the following figure.