The premise (you can skip to the mathematical part below):
You are driving back to the town where you were born. You haven't been home for a very long time and you are unsure if you are even on the correct way. To irony's amusement, the road suddenly splits in two and you feel completely lost as there are no directions and it is foggy and you can barely see further than 100 meters. But you know for sure that one of them is the road to your hometown, the other takes you somewhere else. On the roadside you see a petrol station so you decide to make a short pit stop and ask for directions.
As you walk into the building you see the cashier, a young man in his early 20's. You walk up to him and state your business but he says he doesn't know which road leads where because he just started working there a couple of weeks ago, but that there are two men coming here every day and drink a cup of coffee and stay there for a couple of hours without talking to anyone and then they leave. They surely know the correct road. "But be warned!" he says, "One of the men is known to be a liar, he always lies. The other one cannot say anything but the truth." The cashier points to the direction of the two men. One sits at a table and stares out the window into the seemingly endless fog as if he could make sense of it. The other man sits at the bar and reads the newspaper. You decide to walk to the man staring out of the window. You ask him only one question. When he answers, you confidently start walking back to your car and know to take the road to your right.
As you get into your car you see the young cashier rushing out of the store. He is seemingly perplexed and asks you why you didn't talk to the other man. You tell him you didn't need to, as the liar has been exposed. He then asks you what question you asked. You tell him: "What would the other say, which road leads to my hometown?".
So, I tried to reformulate this scenario into a stochastic view.
Let $r_1, r_2$ (road $1$ and $2$) be independent random variables and $\delta_*$ their respective space such that $\mathbb{E}_{\delta_*}[r_i] > 0$ and $\mathbb{E}_{\delta_*}[r_{j}] < 0$ with $i,j \in \{1,2\}$. This measure is not directly observable.
Now let $\delta_1$ and $\delta_2$ be two observable measures with the following (again, $i,j \in \{1,2\}$):
$$ \mathbb{E}_{\delta_i}[r_i] > 0 \; \text{ and } \; \mathbb{E}_{\delta_i}[r_j] < 0. $$
My interpretation of $\mathbb{E}_{\delta_i}[r_i]$ is what $\delta_i$ (man $i$) says about $r_i$ (road $i$). Now the "question" you have to ask to observe $\mathbb{E}_{\delta_*}[r_i]$ is:
\begin{eqnarray} \text{if } \mathbb{E}_{\delta_j}\left[\mathbb{E}_{\delta_i}[r_i]\right] &>& 0 \; \text{ then } \mathbb{E}_{\delta_*}[r_i] < 0\\ \text{if } \mathbb{E}_{\delta_j}\left[\mathbb{E}_{\delta_i}[r_i]\right] &<& 0 \; \text{ then } \mathbb{E}_{\delta_*}[r_i] > 0 \end{eqnarray}
Could someone verify this line of thought?
