von Neumann-Morgenstern utility theorem

Theorem

If player $i$ 's preference relation $\succsim_i$ over the set of compound lotteries $\hat{\mathcal{L}}$ is complete and transitive, and satisfies the four von Neumann-Morgenstern axioms, then this preference relation can be represented by a linear utility function.

von Neumann-Morgenstern axioms

Assume $\succsim_i$ defined over the set of compound lotteries $\hat{\mathcal{L}}$ . Player $i$ 's utility function, representing his preference relation $\succsim_i$ , is therefore a function $u_i : \hat{\mathcal{L}} \to \mathbb{R}$ satisfying

u_i(\hat{L}_1) \geq u_i(\hat{L}_2) \iff \hat{L}_1 \succsim_i \hat{L}_2, \quad \forall \hat{L}_1, \hat{L}_2 \in \hat{\mathcal{L}}

Axiom of Continuity

For every triplet of outcomes $A \succsim_i B \succsim_i C$ , there exists a number $\theta_i \in [0,1]$ such that

B \approx_i [\theta_i(A), (1-\theta_i)(C)]

(where $\approx_i$ denotes an indifference relation)

Axiom of Monotonicity

Let $\alpha, \beta$ be numbers in $[0,1]$ , and suppose that $A \succ_i B$ (strict preference). Then,

[\alpha(A) , (1-\alpha)(B)] \succsim_i [\beta(A) , (1-\beta)(B)]

if and only iff $\alpha \geq \beta$ .

Theorem

If a preference relation satisfies the Axioms of Continuity and Monotonicity, and if $A \succsim_i B \succsim_i C$ , and $A \succ_i C$ , then the value of $\theta_i$ defined in the Axiom of Continuity is unique.

Corollary

If a preference relation $\succsim_i$ over $\hat{\mathcal{L}}$ satisfies the Axioms of Continuity and Monotonicity, and if $A_K \succ_i A_1$ , then for each $k = 1,2,...,K$ there exists a unique $\theta_i^k \in [0,1]$ such that

A_k \approx_i [\theta_i^k (A_K), (1-\theta_i^k)(A_1)]

The corollary and the fact that $A_1 \approx_i [0 (A_K), 1(A_1)]$ and $A_k \approx_i [1 (A_K), 0(A_1)]$ imply that

\theta_i^1 = 0, \quad \theta_i^K = 1

Axiom of Simplification of Compound Lotteries

For each $j = 1,...,J$ , let $L_j$ be the simple lottery

L_j=[p_1^j (A_1), p_2^j (A_2),...,p_K^j (A_K)]

and let $\hat L$ be the compound lottery

\hat L = [q_1(L_1), q_2(L_1), ..., q_J(L_J)]

For each $k=1,...,K$ , define the overall probability that the outcome of $\hat L$ will be $A_k$ ,

r_k = q_1 p_k^1 + q_2 p_k^2 + ... + q_J p_k^J

Consider simple lottery

L = [r_1(A_1),r_2(A_2),...,r_K(A_K)]

Then,

\hat L \approx_i L

Axiom of Independence

Let $\hat L = [q_1(L_1), q_2(L_1), ..., q_J(L_J)]$ be a compound lottery, and let $M$ be a simple lottery. If $L_j \approx_i M$ then

\hat L \approx_i [q_1(L_1),...,q_{j-1}(L_j),q_j(M),q_{j+1}(L_{j+1}),...,q_J(L_J)]

Notes

Can extend Axioms of Simplification and Independence to compound lotteries of any order. By induction over levels of compounding, it follows that the player's preference relation over all compound lotteries (of any order) is determined by the player's preference relation over simple lotteries.
completeness: for any lotteries $L$ and $M$ , either $L \succeq M$ or $M \succeq L$ .
transitivity: if $L \succeq M$ and $M \succeq N$ , then $L \succeq N$
Archimedean property: if $L \prec M \prec N$ , then there exists probability $\varepsilon \in (0,1)$ s.t. $(1-\varepsilon)L + \varepsilon N \prec M \prec \varepsilon L + (1-\varepsilon) N$ (only one of this or continuity need be assumed)
may also state 4 axioms of VNM-rationality as: completeness, transitivity, continuity, independence

References

M. Maschler, E. Solan, and Shmuel Zamir, Game Theory, Cambridge University Press, 2013, pp. 14-17.
https://en.wikipedia.org/wiki/Von_Neumann–Morgenstern_utility_theorem#The_theorem
Neumann, John von and Morgenstern, Oskar, Theory of Games and Economic Behavior. Princeton, NJ. Princeton University Press, 1953.
https://isa-afp.org/entries/Neumann_Morgenstern_Utility.html