<p>Say you have preference over <span class="math-container">$N$</span> <em>deterministic</em> outcomes, say <span class="math-container">$N$</span> bundles of goods, with <span class="math-container">$1\succ 2\succ\cdots\succ N$</span>. The usual way to represent this preference is by assigning utility numbers <span class="math-container">$u_i\in\mathbb R$</span> to each bundle <span class="math-container">$i$</span>, such that <span class="math-container">$u_1&gt;u_2&gt;\cdots&gt;u_N$</span>. </p>

<p>In contrast, you can also interpret the "outcomes" as lottery tickets that pay either a high payoff (say <span class="math-container">$\$1$</span>) or a low payoff (e.g. <span class="math-container">$\$0$</span>) with varying probabilities. So each lottery <span class="math-container">$L_i=(p_i,1-p_i)$</span> is a distribution over the two payoffs, where <span class="math-container">$p_i$</span> is the probability that you get <span class="math-container">$\$1$</span> and <span class="math-container">$1-p_i$</span> is the probability of getting <span class="math-container">$\$0$</span>. Thus, by choosing <span class="math-container">$p_i$</span>'s that satisfy <span class="math-container">$p_1&gt;p_2&gt;\cdots&gt;p_N$</span>, you have a preference representation over the <span class="math-container">$N$</span> lotteries in the expected utility framework. </p>

CC BY-SA 4.0
2019-07-29T17:53:42.850Z
2019-07-29T17:53:42.850Z
Herr K.
<p>I have read the following statement and I am having difficulty understanding the second part:</p>

<blockquote>
  <p>Any set of preference orderings over a finite set of outcomes can be represented either by deterministic payoffs, one for each ordinal rank, or by only two payoffs, if the latter are stochastic. </p>
</blockquote>

<p>I get how it can be represented by deterministic payoffs but I do not see how it can be represented by two payoffs if they are stochastic. </p>

<p>In hoping that someone here gets it, thanks in advance! </p>

<microeconomics><preferences>
Representing preference orderings over a finite set of outcomes by two payoffs
Ali
# Representing preference orderings over a finite set of outcomes by two payoffs

(Asked by: username_0 on 2019-07-27)

I have read the following statement and I am having difficulty understanding the second part:

> Any set of preference orderings over a finite set of outcomes can be represented either by deterministic payoffs, one for each ordinal rank, or by only two payoffs, if the latter are stochastic.

I get how it can be represented by deterministic payoffs but I do not see how it can be represented by two payoffs if they are stochastic.

In hoping that someone here gets it, thanks in advance!

**Tags:** microeconomics, preferences

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## 1 answers

### Answer by username_1 on 2019-07-29. Score: 2 | Accepted answer

Say you have preference over $N$ *deterministic* outcomes, say $N$ bundles of goods, with $1\succ 2\succ\cdots\succ N$. The usual way to represent this preference is by assigning utility numbers $u\_i\in\mathbb R$ to each bundle $i$, such that $u\_1>u\_2>\cdots>u\_N$.

In contrast, you can also interpret the "outcomes" as lottery tickets that pay either a high payoff (say $\$1$) or a low payoff (e.g. $\$0$) with varying probabilities. So each lottery $L\_i=(p\_i,1-p\_i)$ is a distribution over the two payoffs, where $p\_i$ is the probability that you get $\$1$ and $1-p\_i$ is the probability of getting $\$0$. Thus, by choosing $p\_i$'s that satisfy $p\_1>p\_2>\cdots>p\_N$, you have a preference representation over the $N$ lotteries in the expected utility framework.


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