## Time consistency and stationarity

Suppose you prefer $105 today to $100 tomorrow. You also prefer $105 in 11 days to $100 in 10 days. During the next 10 days, your basic preferences don't change, so that at the end of that period (on day 10), you still prefer $105 now (on day 10) to $100 the next day. Your future self then disagrees with your earlier self about whether it's better to get $105 on day 10 or $100 on day 11.

In economics jargon, your preferences are called time inconsistent. Time inconsistency is supposed to be a failure of ideal rationality.

I'm not entirely convinced, but I can see the intuition. Let's grant (for the sake of the argument) that the various temporal stages of a diachronically ideal agent whose basic preferences don't change should all agree with one another about what they prefer, in the way your stages in the hypothetical example do not.

The standard economic model of time preference is the discounted
utility model going back to Samuelson
1937. The model entails time consistency. But it also has other,
more questionable, implications. In particular, it assumes that
rational preferences are **stationary**.

Stationarity means that if you prefer getting some good X in n units of time to getting Y in m units of time, then you also prefer getting X in n+k units of time to getting Y in m+k units of time.

(This is Fishburn and Rubinstein's (1982) definition of stationarity. Koopmans (1960) instead defines stationarity as follows: if you prefer a sequence X1,X2,X3,... to a sequence Y1,Y2,Y3,..., then you also prefer Z,X1,X2,X3,... to Z,Y1,Y2,Y3,....)

Stationarity has no descriptive or normative plausibility (as Koopmans, Fishburn and Rubinstein, and implicitly already Samuelson acknowledged).

For example, suppose you value having a glass of wine every now and then, but you don't want to have wine every day. And suppose you had no wine yesterday, and you're certain you won't have wine tomorrow and that you will have wine in 9 days. You may then reasonably prefer wine today to no wine today, even though you prefer no wine in 10 days to wine in 10 days. These preferences violate stationarity (on both the Fishburn and Rubinstein definition and the Koopmans definition).

Note that the preferences do not violate time consistency. You don't disagree with your future self about whether it's better to have wine or no wine on day 10.

So time consistency has some intuitive appeal, stationarity has none, and there are convincing examples where stationarity fails but time consistency is satisfied.

Yet a number of arguments suggest that, together with some apparently harmless background assumption, time consistency implies stationarity.

Halevy
2015 presents a result along these lines. Halevy's background
assumption is what he calls **invariance**. Invariance is a
diachronic constraint, saying that if today you prefer X in n units of
time to Y in m units of time, then in k units of time you still prefer
X in n units of time to Y in m units of time.

On the face of it, invariance simply seems to say that your basic (centred) preferences don't change. This is hardly a requirement of rationality, but since the problematic nature of time inconsistency turns on the absence of such changes, we can stipulate that we're modelling agents whose basic (centred) preferences don't change. On the basis of that stipulation, time inconsistency seems problematic and stationarity unproblematic, yet as Halevy points out, they are actually equivalent.

Here's the argument from time consistency and invariance to stationarity.

Suppose today you prefer X in n units of time to Y in m units of time. By invariance, it follows that in k units of time you still prefer X in n units of time to Y in m units of time. By time consistency, this implies that today, you prefer X in n+k units of time to Y in m+k units of time. So we've shown that if you prefer X in n units of time to Y in m units of time, then you prefer X in n+k units of time to Y in m+k units of time. That's stationarity.

What's going on? Clearly invariance is not as harmless as it appeared. In the wine example, you prefer wine today to no wine today, but in 10 days, after having had wine the day before, your preferences are reversed: you do not prefer wine today to no wine today. In contrast to genuine time inconsistency, this is not a problematic kind of reversal. It is explained by the difference in context. At both times, you prefer today being a wine day succeeded and preceded by no-wine days to today being a no-wine day succeeded and preceded by no-wine days. And at both times you prefer today being a no-wine day preceded by a wine day to today being a wine day preceded by a wine day. Still, you violate invariance, because invariance is insensitive to temporal context.

The main lesson is that we must be careful when we talk about whether an agent prefers X at t to Y at t'. In cases like the wine example, those preferences will be relative to what you believe you get before and after the relevant times. And since these beliefs can easily be different for t and for t', we get a spurious appearance of preference reversals.

Fishburn and Rubinstein do mention, on p.678, that a preference for
X at t to Y at t' "can" [??] be understood as a preference for an
entire history (0,0,0,X,0,0,0,...) to an alternative history
(0,0,0,0,0,Y,0,0,...), where X and Y are in position t and t'
respectively and 0 is some null outcome. That would take the temporal
context into account, at the cost of making the so-defined notion of
comparative preference relation (between X at t and Y at t') inapplicable to almost any realistic
scenario. But even Fishburn and Rubinstein don't insist that this is
how the preference relation *should* be understood, and in subsequent
work such as Halevy's, no such interpretation is assumed. But then
we're implicitly assuming that the preference for X at t to Y at t' is
independent of what happens earlier or later -- i.e., we're assuming
separability
of preferences across time, which is implausible, and violated in the
wine example.

(In footnote 16 of his paper, Havely says that "no separability restrictions are imposed on preferences in this paper". If I'm right, then that's false.)

Even Koopmans's concept of preference over entire time streams is not enough to adequately model the wine case, however, since the relevant streams only extend into the future. To capture the wine preferences, they must also extend into the past.

If we adjust the definition of invariance to that kind preference relation, invariance and time consistency no longer imply stationarity.