<p>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.</p>
<p>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 (<span class="math-container">$v$</span> is an affine positive transformation of <span class="math-container">$u$</span> if there exist numbers <span class="math-container">$\alpha&gt;0$</span> and <span class="math-container">$\beta$</span> such that <span class="math-container">$v(a)=\alpha u(a)+\beta$</span> for all <span class="math-container">$a\in A$</span>). 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.</p>

CC BY-SA 4.0
2024-02-24T11:51:39.823Z
2024-02-24T11:51:39.823Z
Michael Greinecker
<p>Let <span class="math-container">$i$</span> be an <em>agent</em>, and let <span class="math-container">$A=\{x,y,z\}$</span> be a <em>set of three alternatives</em>. Then, suppose that player <span class="math-container">$i$</span>’s <em>linear order</em> (i.e., complete, transitive, antisymmetric and reflexive binary relation) on <span class="math-container">$A$</span>, denoted <span class="math-container">$P_i$</span>, is given by <span class="math-container">$aP_ibP_ic$</span> (i.e., player <span class="math-container">$i$</span> strictly prefers <span class="math-container">$a$</span> over <span class="math-container">$b$</span>, and <span class="math-container">$b$</span> over <span class="math-container">$c$</span>). Also, let
<span class="math-container">\begin{gather}
\Lambda=\left\{\lambda\in[0,1]^A\mid\sum_{a\in A}\lambda(a)=1\right\}
\end{gather}</span>
be the <em>set of all lotteries over the alternative set <span class="math-container">$A$</span></em>.</p>
<p>Further, let <span class="math-container">$u_i:A\to\mathbb{R}_+$</span> be any non-negative <em>utility function</em> representing player <span class="math-container">$i$</span>’s strict preference: namely,
<span class="math-container">\begin{gather}
u_i(a)&gt;u_i(b)&gt;u_i(c)
\end{gather}</span>
and let <span class="math-container">$v_i:\Lambda\to\mathbb{R}_+$</span> be player <span class="math-container">$i$</span>’s <em>expected utility function</em>: namely, for all lotteries <span class="math-container">$\lambda\in\Lambda$</span>,
<span class="math-container">\begin{gather}
v_i(\lambda)=\sum_{a\in A}\lambda(a)u_i(a)
\end{gather}</span></p>
<p>Now, consider the lotteries <span class="math-container">$\lambda,\lambda’$</span>, where <span class="math-container">$\lambda(a)=(1/3)$</span> for all alternatives <span class="math-container">$a\in A$</span>, and <span class="math-container">$\lambda’(b)=1$</span> (and, thus, <span class="math-container">$\lambda’(a)=\lambda’(c)=0$</span>). Then, it is possible to construct two utility functions <span class="math-container">$u_i,u’_i:A\to\mathbb{R}_+$</span> satisfying
<span class="math-container">\begin{gather}
v_i(\lambda)=\sum_{a\in A}\lambda(a)u_i(a)&gt;\sum_{a\in A}\lambda’(a)u_i(a)=v_i(\lambda’)\\
v_i’(\lambda)=\sum_{a\in A}\lambda(a)u_i’(a)&lt;\sum_{a\in A}\lambda’(a)u_i’(a)=v_i’(\lambda’)
\end{gather}</span></p>
<p>To see so, let <span class="math-container">$u_i(a)=11$</span>, <span class="math-container">$u_i(b)=3$</span> and <span class="math-container">$u_i(c)=1$</span>; and let <span class="math-container">$u_i’(a)=8$</span>, <span class="math-container">$u_i’(b)=7$</span> and <span class="math-container">$u_i’(c)=0$</span>. Then,
<span class="math-container">\begin{gather}
v_i(\lambda)=(1/3)11+(1/3)3+(1/3)1=(15/3)=5&gt;3=(1/1)3=v_i(\lambda’)\\
v_i’(\lambda)=(1/3)8+(1/3)7+(1/3)0=(15/3)=5&lt;7=(1/1)7=v_i’(\lambda’)\\
\end{gather}</span></p>
<p>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.</p>
<p>Hence: <strong>how can I ensure that two utility functions representing the same preference over alternatives generate expected utility functions representing the same order over lotteries?</strong></p>
<p>Any help will be much appreciated.</p>

<microeconomics><utility><preferences><theory><expected-utility>
How to force two utility functions representing the same preference to generate expected utility functions representing the same order on lotteries?
EoDmnFOr3q
# How to force two utility functions representing the same preference to generate expected utility functions representing the same order on lotteries?

(Asked by: username_0 on 2024-02-24)

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.

**Tags:** microeconomics, utility, preferences, theory, expected-utility


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## 1 answers

### Answer by username_1 on 2024-02-24. Score: 2 | Accepted answer

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.


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