Risk and uncertainty I: EUT

Summary

Everyone has to make decisions, but it is not always clear to us what outcomes can derive from these decisions. When this happens, we say we are making decisions in situations under risk or uncertainty. In this LP we learn about risk and uncertainty. We see how risk can be analysed by using expected utility instead of expected value, and how different kind of people will behave differently when facing risk.

Risk and uncertainty:

Expected utility:

St. Petersburg paradox
Expected utility
Expected utility theory

Risk aversion analysis:

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 L₁L₀or 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

Ok, all of this looks just fine. But, how do we analyse an individual’s risk aversion? Morgenstern and von Neumann also thought of that, as we’ll see next.

Risk and uncertainty I: Expected utility

Risk and uncertainty I: Risk aversion