You mean F(t_1, t_2) where t_1 is the decision time and t_2 is the time of the event whose utility is weighed. Yes, that’s the general form, but we assume that discounting is roughly constant across time (ie depends only on t_2 - t_1).
I guess it would mesh with our instincts if discounting varied with age, but in the simpler special case where we consider only timespans that are short relative to our whole lives the theory works well; there’s room to consider how this extends to a more general theorem.
No, that’s too general; for instance, hyperbolic discounting is F(a,b) = b-a, but hyperbolic discounting is inconsistent in the relevant sense. For consistency we need F(a,b) F(b,c) = F(a,c), or equivalently F(b,c) = F(a,c) / F(a,b) = G(c)/G(b) where G(t) = F(a,t). (Note that the dependence on a has gone away.) This is equivalent to discounting things at time t by a factor G(t), which is the general form I described.
Depending only on time differences is the same thing as being invariant under time-shifting your whole life.
I started getting into this, but there’s not really much point—the important thing is that we agree that if we require that preferences be invariant under time-shifting and not reverse as the choices approach, then only exponential discounting meets these criteria (treating not discounting at all as a special case of exponential discounting)
You mean F(t_1, t_2) where t_1 is the decision time and t_2 is the time of the event whose utility is weighed. Yes, that’s the general form, but we assume that discounting is roughly constant across time (ie depends only on t_2 - t_1).
I guess it would mesh with our instincts if discounting varied with age, but in the simpler special case where we consider only timespans that are short relative to our whole lives the theory works well; there’s room to consider how this extends to a more general theorem.
No, that’s too general; for instance, hyperbolic discounting is F(a,b) = b-a, but hyperbolic discounting is inconsistent in the relevant sense. For consistency we need F(a,b) F(b,c) = F(a,c), or equivalently F(b,c) = F(a,c) / F(a,b) = G(c)/G(b) where G(t) = F(a,t). (Note that the dependence on a has gone away.) This is equivalent to discounting things at time t by a factor G(t), which is the general form I described.
Depending only on time differences is the same thing as being invariant under time-shifting your whole life.
I started getting into this, but there’s not really much point—the important thing is that we agree that if we require that preferences be invariant under time-shifting and not reverse as the choices approach, then only exponential discounting meets these criteria (treating not discounting at all as a special case of exponential discounting)
Right.