#### 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:

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 (C_{1},C_{2},…,C_{n}) and their corresponding probabilities (p_{1}, p_{2},…,p_{i}, with ∑p_{i}=1).

*EU(L) = U(c _{2})p_{1} + U(c_{2})p_{2} + … + 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 L_{0}L_{1} or L_{1}L_{0 }or L_{0}∿L_{1}

Reflexive L_{0}L_{0}

Transitive L_{0}L_{1} ; L_{1}L_{2 }↔ L_{0}L_{2}

2.Continuity: L_{0}L_{1} ; L_{1}L_{2}

with p∊ [0,1] so that L_{1}∿ p L_{0}+(1-p)L_{2}

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 L_{0}L_{1 }→β*L_{0}+ (1- β)*L_{2} β*L_{1} +(1- β)*L_{2}