Suppose there are three kinds of commodities, X Y and Z. We ask an agent about his preferences and receive the following answers:
"I prefer Z to Y and Y to X".
"For every $n$, I prefer one unit of Z and $n-1$ units of X to $n$ units of Y".
Are these preferences consistent with rational preferences, according to the "rationality" axioms of von-Neumann and Morgenstern?
Initially, I thought that "obviously yes". von-Neumann and Morgenstern only talk about preferences on lotteries, but here, the agent's preferences are only given on sure sequences, so obviously there should be some rational preference relation on lotteries that is consistent with them.
However, one of the rationality axioms is "continuity". It says that there exists a probability $p\in(0,1)$ such that:
$$ p Z + (1-p) X \sim Y $$
The preference relation can be represented by a function $u$ such that: $u(X)=0, u(Y)=p, u(Z)=1$.
But then, if we take $n>1/p$, if the agent receives $n$ units of Y, his utility is more than 1, so the agent should prefer this over receiving one unit of Z and $n-1$ units of X...
What is wrong with these preferences?