Wait, but isn’t the exponential curve self-similar in that way, not the hyperbolic curve? I notice that I am confused. (Edit to clarify: I’m the only one who said hyperbolic, this is entirely my own confusion.)
Justification: waiting x seconds at time a should result in the same discount ratio as waiting xseconds at time b. If f(x) is the discounting function, this is equivalent to saying that f(a+x)f(a)=f(b+x)f(b) . If we let f(x)=e−x, then this holds: e−(a+x)e−a=e−x=e−(b+x)e−b. But if f(x)=1x , then aa+x≠bb+x unless a=b. (To see why, just cross-multiply.)
It turns out that I noticed a real thing. “Although exponential discounting has been widely used in economics, a large body of evidence suggests that it does not explain people’s choices. People choose as if they discount future rewards at a greater rate when the delay occurs sooner in time.”
Hyperbolic discounting is, in fact, irrational as you describe, in the sense that an otherwise rational agent will self-modify away from it. “People [...] seem to show inconsistencies in their choices over time.” (By the way, thanks for making the key mathematical idea of discounting clear.)
(That last quote is also amusing: dry understatement.)
Wait, but isn’t the exponential curve self-similar in that way, not the hyperbolic curve? I notice that I am confused. (Edit to clarify: I’m the only one who said hyperbolic, this is entirely my own confusion.)
Justification: waiting x seconds at time a should result in the same discount ratio as waiting xseconds at time b. If f(x) is the discounting function, this is equivalent to saying that f(a+x)f(a)=f(b+x)f(b) . If we let f(x)=e−x, then this holds: e−(a+x)e−a=e−x=e−(b+x)e−b. But if f(x)=1x , then aa+x≠bb+x unless a=b. (To see why, just cross-multiply.)
It turns out that I noticed a real thing. “Although exponential discounting has been widely used in economics, a large body of evidence suggests that it does not explain people’s choices. People choose as if they discount future rewards at a greater rate when the delay occurs sooner in time.”
Hyperbolic discounting is, in fact, irrational as you describe, in the sense that an otherwise rational agent will self-modify away from it. “People [...] seem to show inconsistencies in their choices over time.” (By the way, thanks for making the key mathematical idea of discounting clear.)
(That last quote is also amusing: dry understatement.)