How to force two utility functions representing the same preference to generate expected utility functions representing the same order on lotteries?

Question

Let $i$ be an agent, and let $A=\{x,y,z\}$ be a set of three alternatives. Then, suppose that player $i$’s linear order (i.e., complete, transitive, antisymmetric and reflexive binary relation) on $A$, denoted $P_i$, is given by $aP_ibP_ic$ (i.e., player $i$ strictly prefers $a$ over $b$, and $b$ over $c$). Also, let \begin{gather} \Lambda=\left\{\lambda\in[0,1]^A\mid\sum_{a\in A}\lambda(a)=1\right\} \end{gather} be the set of all lotteries over the alternative set $A$.

Further, let $u_i:A\to\mathbb{R}_+$ be any non-negative utility function representing player $i$’s strict preference: namely, \begin{gather} u_i(a)>u_i(b)>u_i(c) \end{gather} and let $v_i:\Lambda\to\mathbb{R}_+$ be player $i$’s expected utility function: namely, for all lotteries $\lambda\in\Lambda$, \begin{gather} v_i(\lambda)=\sum_{a\in A}\lambda(a)u_i(a) \end{gather}

Now, consider the lotteries $\lambda,\lambda’$, where $\lambda(a)=(1/3)$ for all alternatives $a\in A$, and $\lambda’(b)=1$ (and, thus, $\lambda’(a)=\lambda’(c)=0$). Then, it is possible to construct two utility functions $u_i,u’_i:A\to\mathbb{R}_+$ satisfying \begin{gather} v_i(\lambda)=\sum_{a\in A}\lambda(a)u_i(a)>\sum_{a\in A}\lambda’(a)u_i(a)=v_i(\lambda’)\\ v_i’(\lambda)=\sum_{a\in A}\lambda(a)u_i’(a)<\sum_{a\in A}\lambda’(a)u_i’(a)=v_i’(\lambda’) \end{gather}

To see so, let $u_i(a)=11$, $u_i(b)=3$ and $u_i(c)=1$; and let $u_i’(a)=8$, $u_i’(b)=7$ and $u_i’(c)=0$. Then, \begin{gather} v_i(\lambda)=(1/3)11+(1/3)3+(1/3)1=(15/3)=5>3=(1/1)3=v_i(\lambda’)\\ v_i’(\lambda)=(1/3)8+(1/3)7+(1/3)0=(15/3)=5<7=(1/1)7=v_i’(\lambda’)\\ \end{gather}

Thus, two utility functions representing the same preference over alternatives may lead to two different expected utility functions representing different orders over lotteries. I am in the process of writing a proof for which this fact poses a challenge. I am thus wondering which is the most common or natural assumption to ensure that any two utility functions representing the same preference over alternatives generate two expected utility functions representing identical orders over lotteries.

Hence: how can I ensure that two utility functions representing the same preference over alternatives generate expected utility functions representing the same order over lotteries?

Any help will be much appreciated.

Answer 1

That is hopeless. The preference order over certain outcomes determines the preferences over every compatible expected utility representation if and only if there are at most two indifference classes. In particular, if preferences are linear, there can be at most two alternatives.

It is a standard result that two expected utility functions represent the same preferences over lotteries if and only if they are positive affine transformations of each other ($v$ is an affine positive transformation of $u$ if there exist numbers $\alpha>0$ and $\beta$ such that $v(a)=\alpha u(a)+\beta$ for all $a\in A$). If there are at most two utility levels, then all expected utility functions that induce the same order on certain outcomes must be positive affine transformations of each other. Otherwise, that never holds.