At present, when POTUS ceremoniously signs an important bill, he typically does so with a multitude of pens, rendering each stroke of his signature with a different pen. These pens are later handed out to those present who had important connections with the new law. I suppose the theory is that the recipients will one day be able to point with pride to their pens as having been used by POTUS in making thus-and-such bill into law.
A QUESTION FOR MEMBERS OF THE AMALGAMATED MATHEMATICIANS' UNION: Suppose that POTUS were to select one pen out of a boxful, use it for the entire signature, then drop it back in the box so that it could not be distinguished from its litter-mates, and finally given out as in previous practice. Should the recipients under the new practice attach the same significance to their pens as before?
This should be categorized as a soft-question.strong text
Depends on the individuals' preferences. How much more does one value a pen used by the President to write his entire signature than one used to sign only one $n^{\textrm{th}}$ of his signature? How much does knowing a pen was used by the President with certainty, as opposed to with a $1/n$ chance, increase its value to you?
Let's think about this from a VNM utility framework. Denote the fraction of signature signed with a given pen by $p \in [0, 1]$. If the agent places a lot of value in owning a pen used to write the entire signature, we might expect her preferences to be convex; as an example, say her utility over $p$ is described by the utility function $u(p) = p^2$. Then, if her goal is to maximize expected utility, she will prefer the scenario in which one pen is used for the entire signature: in this case, $p = 1$ with probability $1/n$ and $p = 0$ with probability $(n-1)/n$, so that her expected utility is $\mathbb{E}[u] = 1^2/n + 0^2(n-1)/n = 1/n$, while in the other case, where $p = 1/n$ with probability $1$, her expected utility is $\mathbb{E}[u] = (1/n)^2 = 1/n^2 < 1/n$. As an alternative to this, we could also imagine that her utility function has a discontinuity at $p = 1$ in the form of some spike; in this case, as long as the spike is sufficiently large, maximizing expected utility would still lead to the conclusion that she prefers the chance of receiving a pen used for the entire signature.
If, on the other hand, she places more value in knowing with certainty that her pen was used for part of the signature, then it might make sense to say she has concave preferences over $p$, i.e. $u(p) = \sqrt{p}$ or $u(p) = \ln(p)$. Then the same expected utility analysis will yield the result that she prefers the pen used for a fraction of the signature with certainty.
Staying within VNM utility theory, the first agent would be described as risk seeking, and the second as risk averse.
Of course, there are many other ways to think about this--many of which make much more sense than my way--that do not involve examining how much "utility" a person derives from owning a pen used by the President to write a given fraction of his signature.