future self would prefer not to have made, despite knowing the same information. This dynamic inconsistency happens because hyperbolas distort the relative value of options with a fixed difference in delays in proportion to how far the choice-maker is from those options. Observations The phenomenon of hyperbolic discounting is implicit in Richard Herrnstein's "matching law", which states that when dividing their time or effort between two non-exclusive, ongoing sources of reward, most subjects allocate in direct proportion to the rate and size of rewards from the two sources, and in inverse proportion to their delays. That is, subjects' choices "match" these parameters. After the report of this effect in the case of delay, George Ainslie pointed out that in a single choice between a larger, later and a smaller, sooner reward, inverse proportionality to delay would be described by a plot of value by delay that had a hyperbolic shape, and that when the smaller, sooner reward is preferred, this preference can be reversed by increasing both rewards' delays by the same absolute amount. Ainslie's research showed that a substantial number of subjects reported that they would prefer $50 immediately rather than $100 in six months, but would NOT prefer $50 in 3 months rather than $100 in nine months, even though this was the same choice seen at 3 months’ greater distance. More significantly, those subjects who said they preferred $50 in 3 months to $100 in 9 months said they would NOT prefer $50 in 12 months to $100 in 18 months—again, the same pair of options at a different distance—showing that the preference-reversal effect did not depend on the excitement of getting an immediate reward. Nor does it depend on human culture; the first preference reversal findings were in rats and pigeons. Many subsequent experiments have confirmed that spontaneous preferences by both human and nonhuman subjects follow a hyperbolic curve rather than the conventional, exponential curve that would produce consistent choice over time. For instance, when offered the choice between $50 now and $100 a year from now, many people will choose the immediate $50. However, given the choice between $50 in five years or $100 in six years almost everyone will choose $100 in six years, even though that is the same choice seen at five years' greater distance. Hyperbolic discounting has also been found to relate to real-world examples of self-control. Indeed, a variety of