I've come across this little parable purporting to show why exponential discounting is superior to hyperbolic discounting1:
The greater bowing [of the hyperbolic discount curve] means that if a hyperbolic discounter engaged in trade with someone who used an exponential curve, she'd soon be relieved of her money. Ms. Exponential could buy Ms. Hyperbolic's winter coat cheaply every spring, for instance, because the distance to the next winter would depress Ms. H's valuation of it more than Ms. E's. Ms. E could then sell the coat back to Ms. H every fall when the approach of winter sent Ms. H's valuation of it into a high spike.
The figure that the excerpt refers to looks somewhat like the one shown below, the most notable difference being that I've added the legend to indicate which curve is which2, together with the analytical form of the actual discount functions used3.
But it seems to me that the argument, as presented above, is spurious. It is clear that whose valuation would be more depressed, depends on the time. Therefore, the exact same argument with the roles of Ms. E and Ms. H reversed, would work for any timepoint between the point at which the curves intersect and the vertical axis.
In fact, for certain choices of coefficients for the hyperbolic and exponential curves, the exponential curve is more depressed than the hyperbolic one for all time points. For example:
It turns out that the green exponential curve above intersects the hyperbolic curve at only one value of $t$, namely $t = 0$ (i.e. at the time indicated by the vertical axis). For all $t < 0$, the green exponential curve is strictly below the hyperbolic one.
This means that, if Ms. E's exponential discounting curve were the green one, then Ms. H would be able to quickly immiserate her by applying the strategy described in the excerpt, and this would be true irrespective of the length of the time interval between the buying and the selling back of the winter coat.
In summary, the excerpt's argument for the superiority of exponential discounting over hyperbolic discounting does not hold water, in my opinion.
Now, I realize that the excerpt is not being particularly rigorous, and that there may be a more convincing way to demonstrate the superiority of exponential discounting over hyperbolic discounting. If so, what is it? In particular, I want to know the following:
How can someone who uses exponential discounting take financial advantage unilaterally of someone who uses hyperbolic discounting?
(By unilaterally I mean that the strategy is available only to someone who uses exponential discounting vis-à-vis somoneone who uses hyperbolic discounting, and not viceversa.)
1 The reference I have for this passage is to Breakdown of will (2001) by George Ainslie (pp. 30-31). I don't have the book, though.
2 I have added the labels "hyperbolic" and "exponential", according to my interpretation of what the author means by "greater bowing". I'm not a native speaker of English, so please correct me if this interpretation is backwards.
3 Note that all these functions have $(-\infty, 0]$ as their domains. This choice was required in order to match the appearance of the original curves. Also, I should stress that the functional forms I used for all these curves are my own, chosen so as to approximate the appearance of the original curves. The text of the excerpt does not give the functional form of the depicted curves.