Representing preference orderings over a finite set of outcomes by two payoffs

Question

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!

Answer 1

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.