Pollak, Robert A. “Additive von Neumann-Morgenstern utility functions.” Econometrica 35.3/4 (1967): 485-494.

Axiomatically characterize the class of (log-)additive utility functions with axioms on preferences.

Definitions and notations

Utility function forms

A vNM utility functionis ordinally additive if there exist $T$ functions $v^t(x_t)$ and a twice differentiable function $F$, $F’>0$, such that $F[V(X)]=\sum\limits_{t=1}^{T} v^t(x_t)$.

• A log-additive utility function is ordinally additive.

Alternatives: lottery tickets

Let $X_a,Y_a$ be $T$-dimensional vector representing consumption paths:$X_a=(x_{a1},…,x_{aT}), Y_a=(y_{a1},…,y_{aT})$, and let $\gamma_a\in[0,1]$ be a number. A simple lottery ticket $L_a=(\gamma_a,X_a,Y_a)$ is an alternative that, once chosen, yields $X_a$ with probability $\gamma_a$ and $Y_a$ with probability $1-\gamma_a$. \(V(L_{a}) = \gamma_{a} V(X_{a}) + (1-\gamma_{a}) V(Y_a)\)

• Two simple lottery tickets $L_a,L_b$ are a pair of k-standard lottery tickets $(L_a,L_b)$ if (1) $\gamma_a=\gamma_b=\frac{1}{2}$, and (2) have a common value on the first consumption path $x_{ak}=x_{bk}=x_k$.
• Two simple lottery tickets $L_a,L_b$ are a pair of k-normal lottery tickets $\langle L_a,L_b\rangle$ if (1) $\gamma_a=\gamma_b=\frac{1}{2}$, and (2) have a common value on both consumption paths $x_{ak}=x_{bk}=y_{ak}=y_{bk}=z_k$.

Characterization theorems

Strong additivity axiom: an individual’s preference between two k-standard lottery tickets in a given pair is independent of the level of $x_k$ for all pairs of k-standard lotteries, and all choice of $k$.

• If $V(L_a(x_k))>V(L_b(x_k))$ for some $x_k$, then $V(L_a(x_k’))>V(L_b(x_k’))$ for all $x_k’$.

Thm 1. An individual’s preferences satisfy the strong additive axiom if and only if his von Neumann-Morgenstern utility function is additive.

• proof sketch: construct two consumption paths, and differentiate the utility functions with repect to $k$.This gives an equation showing that $\partial V(X) / \partial x_k$ depends only on $x_k$. The necessity part is trivial.

Weak additivity axiom: an individual’s preference between two k-normal lottery tickets in a given pair is independent of the level of $z_k$ for all pairs of k-normal lotteries, and all choice of $k$.

• If $V(L_a(z_k))>V(L_b(z_k))$ for some $z_k$, then $V(L_a(z_k’))>V(L_b(z_k’))$ for all $z_k’$.
• Weak additivity axiom is weaker in the sense that, it restricts preference on k-normal lotteries, which is a subset of k-standard lotteries.

Thm 2. An individual’s preferences satisfy the weak additive axiom if and only if his von Neumann-Morgenstern utility function is additive or log-additive.

• proof sketch: to show the suficient condition, a lemma points out (by construction) that weak additivity implies ordinally additive property. By differentiating two indifferent consumption paths with respect to $x_k$, we can show that $H’(V)=G’‘(S)/G’(S)=c$ is a constant. This shows that $G$ must be a linear or exponential transformation.

Implications for sequential decision making

Consider the preference satisfying weak additivity axiom: \(V(x_1,...,x_T)=G(\sum\limits_{t=1}^{T}v^t(x_t)),\) where $G$ is a linear or exponential transformation.

• A single choice is made between different streams of income
• Q: why should the decision maker considers reward in each period independently.