## Saturday, February 20, 2016

Today's José wants tomorrow's José to go on a diet, but when tomorrow arrives, the "new today" José will want the "new tomorrow" José to go on a diet, etc.

("My diet starts tomorrow" XXXL t-shirts available in the gift shop.)

As far as I know, Richard Thaler was the first economist to illustrate the inconsistency between choices in the short term and the long term with a simple pair of questions. First:

Q1: Do you prefer an apple in one year or two apples in one year and a day?

Most people choose the two apples. Then Thaler hit them with the second question:

Q2: Do you prefer an apple now, or two apples tomorrow?

And most people choose the one apple. This, trained economists and careful thinkers will say, is inconsistent. (This is one of the rare occasions when trained economists and careful thinkers will agree, so it's worth noting. :-)

Why is it inconsistent? For the same reason "my diet starts tomorrow" t-shirts are a good joke: because the decision is reversed simply by the passing of time. If instead of "in one year" and "in one year and a day" we had dates, say "on Feb 20th, 2017" and "on Feb 21st, 2017" and repeated the question every day, at some point the answer to Q1 would become "one apple," say on Feb 4, 2017.

Or maybe not. Maybe only on Feb 20th, 2017. Still, just the passing of time would reverse the choice, which is what "inconsistent over time" means.

Two common models of time preference that account for these inconsistencies are hyperbolic discounting, in which the exponential discounting used for finance (and for economics rational models) is replaced by an hyperbolic function; and a non-immediacy penalty for any delayed reward. In the second case, all future payoffs are discounted by a factor $\beta \times \delta(t)$, where $\delta(t)$ is the standard exponential discount factor and $\beta < 1$ is the non-immediacy penalty. The lower the $\beta$, the more now-oriented the decision-maker.

The reason why I've come to like the $(\beta,\delta(t))$ formulation is that it models a number of explanations that have little to do with time orientation and a lot to do with the actual circumstances of getting a reward.

For example, I give these choices to participants in one-day managerial decision-making exec-ed events:

Q3: Choose between $\$10$now or$\$20$ tomorrow. (Nearly all choose the $\$10$.) Q4: Choose between$\$10$ in a week or $\$20$in eight days. (Nearly all choose the$\$20$.)

And when we discuss the "inconsistency" participants mostly bring up the mechanics of the transaction: how exactly are they going to get the money after the event is over? (It's hypothetical, of course, in these events money comes my way; but participants play along and take the decision seriously.) If it's now, they can just get the money and walk away. So the future is discounted not just because of the opportunity cost of having the money later but rather because it's associated with more hassle and uncertainty. Of course, when both payoffs are in the future, then participants prefer the larger payoff, as both payoffs have the same hassle and uncertainty.

Given the advantages of being temporally-consistent (which includes delaying gratification for bigger rewards), these non-opportunity cost reasons for now-preference are quite important. For example, in the case of people going on diets, their experience with bad diets may make them ask "what's the point? I might as well have that  second crème brûlée and a chocolate soufflé while I'm at it…"

I think that Scott Adams was right, the best think is to stop considering goals (that is making payoff-based choices) and adopt systems that work by bypassing the choice mechanisms. For me, the Paleo diet is one of them, strength training and rowing are another. YMMV, of course.

Another possibility is to practice delaying gratification as an exercise; it will be prophylactic against temporal inconsistency. There's a problem with this, of course, sometimes it's taken too far and leads to bad choices in itself. But in general, postponing a decision for a few days or considering whether a decision would change if the timing was shifted by a couple of days is a good idea.

Living for the now is a sure way to compromise the future.

--  --  --  --

For the quants…

The notion that the choice in Q3 could be due to standard discount (that is, a matter of opportunity cost of only having the money tomorrow instead of today) becomes ludicrous when we compute the discount rate associated: annualizing a $1/2$ one-day discount factor we get a yearly rate of (drumroll please…):

$\delta(\text{1 day}) = \frac{1}{(1+r)^{1/365}}= 1/2 \quad \Rightarrow \quad r = 2^{365}-1 = 7.515 \times 10^{109}$.

Choices like those captured by Q1-Q4 have to be driven by immediacy, as any attempt to find a discount mechanism that makes sense without a discontinuity at "now" quickly run into these ridiculously high discount rates.