Von Neumann-Morgenstern Expected Utility

yedlu, Fall 2024

Intro to Lotteries: an Extra Layer of Uncertainty

  • The individual must choose an action with uncertain outcomes. The individual has preference over final consequences.
  • \(X = \{x_1, ..., x_n\}\) be the finite set of consequences
  • \(p = (p_1, ..., p_n), p_i \geq 0, \Sigma_i p_i = 1\) be the lottery on \(X\)
  • \(\Delta(X) = \{p \in \mathbb{R}^n_+: \Sigma_i p_i = 1\}\) be the set of lotteries on \(X\)
  • \(\succsim \subseteq \Delta(X) \times \Delta(X)\) be the binary preference relations on the set of lotteries \(\Delta(X)\)

Mixing Lotteries

Given two lotteries \(p, q \in \Delta(X)\) and a weight \(\alpha \in [0,1]\), define the lottery

\[\alpha p + (1-\alpha) q := (\alpha p_1 + (1-\alpha) q_1, ..., \alpha p_n + (1-\alpha) q_n)\]

Special Lotteries with Certain Outcomes

Define certainty lotteries as \(\delta_i = (p_1, ..., p_n)\) with \(p_i = 1\). Specifically, \(\delta_1 = (1,...,0)\) and \(\delta_n = (0, ..., 1)\).

Expected Utility Functions

Given a utility function over consequences \(u: X \to \mathbb{R}\), we may evaluate lotteries \(p = (p_1, ..., p_n)\) by its expected utility:

\[U(p) := p_1 u(x_1) + ... + p_n u(x_n)\]

  • The \(U: \Delta(X) \to \mathbb{R}\) is said to have the expected utility form.
  • The \(u: X \to \mathbb{R}\) is called a Bernoulli payoff function.

Linear and Expected Utility

Lemma. The following two statements are logically equivalent.

  1. \(U\) is linear if for every \(p, q \in \Delta(X)\) and every \(\alpha \in [0,1]\): \[U[\alpha p + (1-\alpha)q] = \alpha U(p) + (1-\alpha) U(q)\]
  2. \(U\) is in expected utility form if there exists \(u: X \to \mathbb{R}\) such that: \[U(p) = \sum_{x \in X} u(x) p(x)\] for every \(p \in \Delta(X)\)
Proof (Click to Expand)

Parts of proof: \(2 \Rightarrow 1\). \[\begin{aligned} U[\alpha p + (1-\alpha) q] &= U(\alpha p_1 + (1-\alpha) q_1, ..., \alpha p_n + (1-\alpha) q_n) \\ &= \Sigma_{i=1}^n u(x_1) [\alpha p_1 + (1-\alpha) q_1] \\ &= \alpha \Sigma_{i=1}^n u(x_1) p_1 + (1-\alpha) \Sigma_{i=1}^n u(x_1) q_1 \\ &= \alpha U(p) + (1-\alpha) U(q) \end{aligned}\]

\(\blacksquare\)

The von Neumann-Morgenstern (vNM) Axioms

Let \(\succsim\) be a binary relation on the set of lotteries \(\Delta(X)\). - Axiom 1 (rational): \(\succsim\) is complete and transitive. - Axiom 2 (continuous): for every \(p, q, r \in \Delta(X)\) with \(p \succ q \succ r\), there exists \(\alpha, \beta \in (0,1)\) such that: \(\alpha p + (1-\alpha) r \succ q \succ \beta p + (1 - \beta) r\). - Axiom 3 (independence): for every \(p, q, r \in \Delta(X)\) and every \(\alpha \in (0,1)\), we have \(p \succ q\) if and only if:

\[\alpha p + (1-\alpha) r \succ \alpha q + (1-\alpha) r\]

The vNM Representation Theorem

The following statements are logically equivalent: 1. \(\succsim\) satisfies Axioms 1, 2, and 3; 2. There exists a linear function \(U: \Delta(X) \to \mathbb{R}\) that represents \(\succsim\).

The Uniqueness Proposition

Suppose \(U, V: \Delta(X) \to \mathbb{R}\) are linear functions. Suppose \(U\) represents \(\succsim\). Then \(V\) also represents \(\succsim\) iif there exists \(\alpha > 0\) and \(\beta \in \mathbb{R}\) such that \(V = \alpha U + \beta\).

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