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

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!

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.