While it presents more challenges to interpretation than the standard geometric discounting utility model, the dual-self bargaining model presented here need not be silent on questions of welfare. By choosing to view an agent as actually consisting of two individuals, there are several results that grant leverage for welfare evaluation. This gives an advantage over models in which there is a single individual whose preferences change in each period due to non-geometric time discounting. Rather than have one set of preferences for each period, here there are only two in total. There is also the observation that utilities are intrapersonal here, rather than interpersonal. Selves share a payo utility from actions, and so utility comparisons arguably carry more weight here than they would when comparing across individuals. I rst address the notion of Pareto improvements: actions that improved the utility for both selves over another action.
Lemma 8. Given two actions aandb,ais never chosen from any action set A containing both a
andb if and only if the utility vector created byb is a Pareto improvement over that created by a.
Proof: Ifbis a Pareto improvement, thenawill not be on the Pareto frontier, and so will never be
chosen. Ifais never chosen fromA={b, a}, then it must be thatbis a Pareto improvement, as the
agent would otherwise choose a strict mixing of the two actions.
This has a close relation with the notion of the unambiguous choice relation developed by Bern- heim and Rangel (2009). Essentially, ifais never chosen whenb is available, then we say thatb is
unambiguously preferred to a, written as bP∗a, in the terminology dened by their work. Thus, in
this model,bis unambiguously preferred toaif it represents a Pareto improvement overain regard
to the two selves. An immediate extension is that for a given action,a, if there exists another action, b, which is unambiguously preferred, acan denitively said to be Pareto inecient from a welfare
perspective. It also has the important implication that if an agent always makes the same decision from a binary choice (probability 1), then the choice made is a Pareto improvement over the other, and thus can be denitively said to be welfare improving.
I now turn to the more dicult question of welfare evaluation of options when neither is a Pareto improvement over the other. If we wish to make meaningful statements about welfare on such questions, it is necessary to consider some aggregation of the utilities received by the two selves. One method by which to do so is a weighted utility welfare function,W(a) =wUl(a) + (1−w)Us(a),
famously advanced by Harsanyi (1955, 1977) as a method of aggregating interpersonal social welfare. For what follows, I will conne attention to this form of welfare function.21
21A criticism of such a weighted utility function as a measurement of social welfare involves the diculty in
interpersonal utility comparisons, e.g. Sen (1977). In my model the utility comparisons are intrapersonal: the selves in fact share a payo utility function, and I argue that this lessens the strength of the critique. Second, more practically, as the selves share their payo utility function, it is desirable that any measure of welfare be invariant to
Lemma 9. Consider a choice set given byA={a, b}, suppose that bis the action preferred by the
short-term self, and that the agent is observed to choose b with probability p. Then,
p < 1 1 −w
w
γ
+ 1 ⇐⇒ wUl(a) + (1−w)Us(a)> wUl(b) + (1−w)Us(b).
Lemma 9 implies that, if welfare of an individual is evaluated by a weighting between options, observation of choice from a binary set of two actions is sucient to say which is welfare superior.22
Ifw= 0.5, so that welfare is considered as an equal weighting between the selves, then the condition
simply becomes p < 0.5, so that γ need not be known, nor which action is preferred by which
self. This is appealing for application: it would imply that in a series of random choices between two options the agent would be observed to choose the option one granting higher welfare more frequently.
We may also wish to consider welfare weightings that place greater weight on the utility of the long-term self.
Lemma 10. Consider a choice set given byA={a, b}, and suppose an agent is observed to chooseb
fromAwith probabilityp, and thatbis the action preferred by the short-term self. Further, consider
a utility weighting given bywUl(·) + (1−w)Us(·), withw≥0.5. Then,
p <0.5⇒wUl(a) + (1−w)Us(a)> wUl(b) + (1−w)Us(b).
Proof: Considerw= 0.5. From Lemma 9,p <0.5 ⇐⇒ 0.5Ul(a) + 0.5Us(a)>0.5Ul(b) + 0.5Us(b)
⇒Ul(a)−Ul(b)> Us(b)−Us(a)⇒w(Ul(a)−Ul(b))>(1−w)(Us(b)−Us(a))forw≥0.5
⇒wUl(a) + (1−w)Us(a)> wUl(b) + (1−w)Us(b)forw≥0.5.
Lemma 10 says that if a utility weighting welfare function places at least equal weight on the long-term self's utility, then observing the agent choosing the long-term self's preferred action more frequently than an alternate action implies that the more frequently chosen action grants higher welfare. This is of interest because if, for example, a bag of potato chips is resisted more often than not, then it implies that it is welfare improving to remove the bag of chips as an option. Additionally, it allows a degree of welfare evaluation to take place without taking a stance on whether an equal
ane transformations of this underlying payo function, and a weighted utility welfare function meets this criterion.
22In cases where it is not apparent, which action is preferred by which self is easily established by time delay due
weighting between selves, or a higher weighting on the long-term self, is the correct choice in a welfare function.23
Finally, consider the welfare eects of the availability of commitment devices.
Lemma 11. Consider action setAn, and a welfare functionW(a) =wUl(a) + (1−w)Us(a), with
w > 0. Consider decision 1, with tn−t1 = ∆t ≥0. Then, ∃∆ such that ∀∆t > ∆, allowing the
agent access to a zero-cost commitment action set at timet1 increases welfare.
Lemma 11 says that as long as you put positive weight on the utility of the long-term self, then a commitment option given suciently far in advance of the action being committed to is welfare improving. The implication is that if weighted utilities between selves is regarded as an acceptable method of welfare evaluation then, regardless of the weights used, commitment devices can create welfare improvements. In fact, the result will easily extend to any welfare function which is bounded, continuous, and strictly increasing in Ul. Thus, the model gives strong support to the notion that
commitment devices create welfare improvements for agents.