Abstract
This chapter presents the foundations of decision making problems in a risky environment. By introducing suitable axioms on the preference relation, the existence of an expected utility function representation is proved. We discuss the notions of risk aversion, risk premium and certainty equivalent and characterize stochastic dominance criteria for comparing random variables. The chapter ends with a discussion of mean-variance preferences and their relation with stochastic dominance and expected utility.
Behaviour is substantively rational when it is appropriate to the achievement of given goals within the limits imposed by given conditions and constraints. […] behaviour is procedurally rational when it is the outcome of appropriate deliberation. is the outcome of appropriate deliberation. Simon (1976)
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Notes
- 1.
Θ is a collection of subsets of Ω that includes the empty set and satisfies the following conditions:
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if E ∈ Θ, then its complement E c also belongs to Θ;
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if \(\{E_{n}\}_{n\in \mathbb{N}} \subseteq \varTheta\) then \(\bigcup _{n=1}^{\infty }E_{n} \in \varTheta\).
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- 2.
In the case where Ω is a finite set, the probability measure \(\mathbb{P}:\varTheta \rightarrow [0, 1]\) is assumed to satisfy the following axioms:
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1.
for every E 1, E 2 ∈ Θ such that E 1 ∩ E 2 = ∅ we have \(\mathbb{P}(E_{1} \cup E_{2}) = \mathbb{P}(E_{1}) + \mathbb{P}(E_{2})\);
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2.
\(\mathbb{P}(\varOmega ) = 1\).
These axioms lead to some fundamental properties of the probability measure \(\mathbb{P}\):
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for every E ∈ Θ, it holds that \(\mathbb{P}(E) \leq 1\) and \(\mathbb{P}(E) + \mathbb{P}(E^{c}) = 1\);
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for every E 1, E 2 ∈ Θ with E 1 ⊂ E 2, it holds that \(\mathbb{P}(E_{1}) \leq \mathbb{P}(E_{2})\) and \(\mathbb{P}(E_{2}\setminus E_{1}) = \mathbb{P}(E_{2}) - \mathbb{P}(E_{1})\);
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for every E 1, E 2, …, E n ∈ Θ, \(n \in \mathbb{N}\), it holds that
$$\displaystyle{\mathbb{P}(E_{1} \cup E_{2} \cup \ldots \cup E_{n}) \leq \mathbb{P}(E_{1}) + \mathbb{P}(E_{2}) +\ldots +\mathbb{P}(E_{n}),}$$with equality holding for disjoint events ( finite additivity).
In the more general case where Ω is not a finite set, then finite additivity has to be replaced by σ-additivity: for any countable collection of disjoint events \(\{E_{k}\}_{k\in \mathbb{N}} \subseteq \varTheta\) it holds that
$$\displaystyle{\mathbb{P}\bigg(\,\bigcup _{k=1}^{\infty }E_{ k}\bigg) =\sum _{ k=1}^{\infty }\mathbb{P}(E_{ k}).}$$We refer to Chung [448] for a classical account of probability theory.
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1.
- 3.
Let \(\tilde{x}\) be a random variable with finite expected value \(\mathbb{E}[\tilde{x}]\). Jensen’s inequality establishes that, for any concave function \(g: \mathbb{R} \rightarrow \mathbb{R}\), it holds that \(\mathbb{E}\left [g(\tilde{x})\right ] \leq g(\mathbb{E}[\tilde{x}])\).
- 4.
The notation ≡d stands for equality in law (or in distribution).
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Barucci, E., Fontana, C. (2017). Choices Under Risk. In: Financial Markets Theory. Springer Finance(). Springer, London. https://doi.org/10.1007/978-1-4471-7322-9_2
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