If you treat this as a Bayesian problem, your context and experiences could lead you to specify a prior which has a finite mean, in which case for some values you will have a posterior probability of > 2/3 of it being the larger value.
E.g. if the maximum volume of currency that could fit in an envelope is $10k, it would be possible to have a uniform prior between 0-10k, at which point I would stick if my envelope had $5.01k.
A more extreme prior could be one capped at the largest ever academic research grant (I bet it's not more than $100m).
It could also be that you have a prior that has infinite mean, in which case it is not surprising that you will always want to switch if you draw a specific value.
If you have risk aversion, this further increases the range that you would like to stick, since switching could decrease your expected utility, even if it increases your expected return. There is a separate paradox in decision theory that we should be approximately risk-neural for 'small' amounts.
E.g. if the maximum volume of currency that could fit in an envelope is $10k, it would be possible to have a uniform prior between 0-10k, at which point I would stick if my envelope had $5.01k.
A more extreme prior could be one capped at the largest ever academic research grant (I bet it's not more than $100m).
It could also be that you have a prior that has infinite mean, in which case it is not surprising that you will always want to switch if you draw a specific value.
If you have risk aversion, this further increases the range that you would like to stick, since switching could decrease your expected utility, even if it increases your expected return. There is a separate paradox in decision theory that we should be approximately risk-neural for 'small' amounts.