Anticipation in Observable Behavior

45 Pages·2002·0.36 MB·English

Anticipation in Observable Behavior
(cid:3)
Botond Ko}szegi, UC Berkeley
September 2002
Abstract
There is littledoubt we are a(cid:11)ected by experiences related to anticipatingthe future. From
an economic theory perspective, one natural question is: how is this reflected in behavior? To
propose an exhaustive answer, I consider a general model of decisionmaking where rationally
formedanticipationenterstheagent’sutilityfunctioninadditiontophysicaloutcomes,andallow
forinteractionsbetween these twopayo(cid:11)components. Thepaperproves thatifadecisionmaker
who derives utilityfrom anticipationis distinguishable from one who onlycares about physical
outcomes, she has to exhibit at least one of the following phenomena. First, the agent can
exhibit informational preferences not only because{analogously to previous work (e.g. Kreps
and Porteus 1978){she cares about the insecurity of not knowing what will happen to her,
but also because she cares about disappointments. I prove that an agent who is indi(cid:11)erent to
insecurity always prefers fullover less than fullinformationif and onlyif she is disappointment
averse,butastronger conditionisneeded forher toprefer moreinformationtoless. Second, the
agentcanbetimeinconsistent because anticipatoryfeelingspassbythetimeshe hasto\invest"
in them, and this time inconsistency can be reflected in intransitivity of choices. Finally, the
agent can be prone to self-ful(cid:12)lling expectations. In developing these ideas, I also deal with
several modeling di(cid:14)culties when expectations enter utility.
Keywords: anticipation,timeinconsistency, disappointmentaversion,non-expected utility.
(cid:3)Address: UC Berkeley Dept. of Economics, 549 Evans Hall #3880, Berkeley, CA 94720. Email:
[email protected]. First version: May 2001. I thank George Akerlof, Andrew Caplin, Steven Gold-
man, Wei Li, Mark Machina, M(cid:19)at(cid:19)e Matolcsi, Markus M¨obius, Wolfgang Pesendorfer, Ben Polak, Attila P(cid:19)or, Matt
Rabin, Luca Rigotti,Tanya Rosenblat, Jacob Sagi, and ChrisShannon for interesting and helpful conversations.
1
1 Introduction
Anticipation and related feelings used to be customary ingredients in discussions of intertemporal
individual decisionmaking (Loewenstein 1992). But starting at the beginning of the 20th century,
emphasis turned more towards the analysis of mathematical models of consumption choices over
time. This revolutionstripped the theoryof virtuallyallpsychologicalaspects. Though researchers
acknowledged that psychological{aswell as cultural and social{phenomena might a(cid:11)ect parameter
values in models (Fisher 1930, for example), they considered it unnecessary to include them in the
theory itself. Samuelson stated this explicitly, claiming that the new mathematical models would
lead to the \same thing" as psychological considerations (Samuelson 1939, Samuelson 1937).
Recently, however,severalpapershaverecognized thatdecisionmakerswhocareaboutanticipa-
tion can be quite di(cid:11)erent from those who care exclusively about consumption. Unlike \standard"
1
decisionmakers, they can exhibit an intrinsic preference for information, and have a speci(cid:12)c ten-
dency toward time inconsistency (Loewenstein 1987, Caplin and Leahy 2001). The current paper
builds on and expands this work on the behavioral consequences of anticipation. I consider a rich
model of decisionmaking where anticipationenters the agent’s utility function in addition to phys-
ical outcomes, generalizing previous work in two important respects. First, the model is explicit
about the temporal placement of anticipation. Second, it allows for an interaction between antici-
pationandphysicaloutcomes,highlightingthatnotonlytheimmediatee(cid:11)ectofanticipation(called
anticipatory feelings here) is important, but also its e(cid:11)ect on preferences over other outcomes. In
the model, the individual decisionmaker is involved in a (cid:12)nite-horizon decision problem. In each
of the T periods, a physical outcome zt and an anticipation ft are realized. The agent’s choices
give her control over physical outcomes, but in general she cannot directly select from possible
expectations. Instead, ft is a rationallyformed expectation about the future, and thus is partly in
the hands of future selves. Self t maximizesthe expectation of a utilityfunction ut de(cid:12)ned over the
entire stream of physical outcomes and expectations, taking into account the physical constraints
of the problem, the behavior of future selves, and that anticipation is formed rationally. I call the
equilibrium that combines these features personal equilibrium, and show that personal equilibrium
exists under general conditions.
1 A seminal work on informational preferences is Kreps and Porteus (1978), although they do not frame the
problem in terms of anticipatory feelings. Recently, Caplin and Leahy (2001) expand on Kreps and Porteus’ work,
and Grant, Kajii,and Polak (1998, 2000) study general properties of informationalpreferences.
1
With this machinery in hand, I explore three major phenomena through which one can distin-
guishdecisionmakersinmymodelfromthoseinhabitinganexpectedutilityorKreps-Porteusworld.
Then I prove the most important result of the paper, that any di(cid:11)erence between \emotional"and
2
\unemotional" decisionmakers has to be related to at least one of these phenomena.
First,fortwodistinctreasons,thedecisionmakermightin personalequilibriumexhibit aprefer-
3
enceforinformation. Onereasonforselfttodosoisthatutcouldbenon-linearintheexpectations
ft. Intuitively,theagentmightnotbe indi(cid:11)erenttothehedonic experience thatresultsfromhaving
to live with insecurity about what will happen to her. This is the natural analogue of informa-
tional preferences in previous previous work (Kreps and Porteus 1978, Caplin and Leahy 2001,
Grant, Kajii, and Polak 1998, 2000), where they are determined by the shape of the utility func-
tion in an anticipatory component. However, in my model informational preferences can arise for
a completely di(cid:11)erent, as of yet unexplored reason. To illustrate the idea, suppose that T = 2
and that either something \good" or something \bad" will happen to the agent, with probability
one-half each. If the agent’s beliefs remain uncertain, she will be disappointed with probability
one-half and get a pleasant surprise with probability one-half. If, on the other hand, she knows
the outcome in advance, no disappointments or pleasant surprises are possible. Therefore, if she
dislikes disappointments more than she likes pleasant surprises{a situation termed disappointment
aversion{she would choose to (cid:12)nd out the outcome to forestall any blows. Section 3.3 formally
de(cid:12)nes disappointment aversion and shows that it can generate informational preferences that are
observationally distinguishable from the Kreps and Porteus (1978) model (and its extensions). It
also fully characterizes the relationship of disappointment aversion and informational preferences
and, in particular, shows how far the above intuition generalizes.
Second, time inconsistent preferences can be exhibited in the agent’s behavior in two ways. As
Loewenstein (1987) and Caplin and Leahy (2001) have pointed out, anticipatory feelings can lead
to a kind of time inconsistency, because they are irreversibly realized by the time the agent has to
\invest" in them. Since an expected utility maximizer can also be time inconsistent, the examples
2Throughoutthepaper,Iusetheterms\cold"and\unemotional"todenotedecisionmakerswhoseutilityfunction
depends only on physical outcomes. The term \emotional" refers to decisionmakers whose utility function depends
on anticipatory feelings as well.
3 Informationalpreferences can alsoariseinmodels wherethe decisionmaker caresonly about physicaloutcomes,
aslongasthepreferencesaretime-inconsistent. Thereasonisthatinformationacquisitionservesastrategicpurpose,
manipulating the actions of future selves (Carillo1997, Carilloand Mariotti 2000, B(cid:19)enabou and Tirole2002). With
anticipation, informational preferences can arise even in the absence of strategic considerations.
2
these authorsgiveare notobservationallydistinguishablefromastandard agent’s. But the passage
of feelings can indeed lead to a distinct form of behavior. For example, consider an academic who
could do three things next summer: work, relax at home, or take a holiday. Having anticipated
work, she might be too stressed and prefer to stay at home, and she might prefer a holiday over
relaxing at home. But having anticipated a vacation, a lot of its utility has passed, so she might
prefer to workinstead. Given rationalexpectations, one of these choices has to be suboptimal, and
since the current self’s choices are transitive,this is a reflection of time inconsistency. Surprisingly,
the failure to take the utility-maximizingaction can happen even if all of her temporal selves have
exactly the same utility function over the stream of feelings and physical outcomes.
But another featureof time-inconsistentanticipation-dependent preferences has not been noted
intheliterature. Quiteincontrasttocausingafailuretomaximizeajointutilityfunction,anticipa-
tioncan be used strategicallyto bringdi(cid:11)ering interestsclosertogether. If a laterself’spreferences
over physical outcomes di(cid:11)er from those of an earlier self, and depend on anticipation, the ear-
lier self will in general want to manipulate expectations, even when an expected utility maximizer
would not.
Interestingly, identifying the above e(cid:11)ects of (what intuitively seems) time inconsistency raises
a new di(cid:14)culty. As Section 3.2 argues, standard de(cid:12)nitions of time consistency fail to extend
into this framework. Thus, I o(cid:11)er a generalized de(cid:12)nition and use this de(cid:12)nition in the formal
discussion.
Third, Section 3.1 identi(cid:12)es an additional behavioral manifestation of anticipation, utility-
relevant unpredictability. Since current expectations depend on future outcomes while future pref-
erences can also depend on current expectations, the decisionmaker could be trapped in a feedback
loopthatmakesexpectationsself-ful(cid:12)lling. Forexample,ifaloverbelievesshecanwinherpassion’s
heart, she may indeed have the energy and courage to do the things he (cid:12)nds attractive, ful(cid:12)lling
her expectations. But if she believes she will not succeed, she cannot do the right things, and once
again she ful(cid:12)lls her expectations. And although the agent might not be indi(cid:11)erent between these
personal equilibria, there is no reason to expect that she would be able to coordinate on the one
with higher expected utility.
In the key section of the paper, I attempt to give the discussion on the behavioral e(cid:11)ects of
anticipatory feelings a sense of completeness by comparing my model to two natural benchmarks:
3
theexpectedutilityandKreps-Porteus(1978)models. Inthatvein,amaincontributionofthepaper
istoprovethatifagivenmodelincorporatinganticipationdoesnotfeatureatleastoneoftheabove
phenomena, it does not feature anything novel relative to expected utility maximization. That is,
if informationalpreferences, utility-relevantunpredictability, and time inconsistency are ruled out,
there is a utility function de(cid:12)ned only over physical outcomes such that in every decision problem
the agent behaves as if she was maximizing the expectation of that utility function. Similarly,
any di(cid:11)erence between my model and that of Kreps and Porteus has to do with preferences over
disappointment, unpredictability, or time inconsistency.
There is little doubt that we are a(cid:11)ected by utility from anticipation. From an economic
theory perspective, one question is how this is reflected in observable behavior. By o(cid:11)ering a way
of categorizing all behavioral di(cid:11)erences between emotional and unemotional decisionmakers, the
paper’s characterization result provides an (at least partial) answer to this question. Such a result
can also be used to think about any model incorporating anticipation: identify which of the above
conditions fails, and apply what we know about the concept to gain insight into the model.
The paper is organized as follows. Section 2 introduces notation and the concept of personal
equilibrium. My model is not an extremely novel decision-theoretic model, so Section 2 briefly
compares my setup to work in that area. Section 3 illustrates the kinds of behavior an emotional
agent mightdisplay. This sets up the proof of existence ofpersonal equilibrium (Section 4)and the
observational equivalence results (Section 5). Section 6 deals with issues arising from the fact that
preferences in the model do not have a revealed preference foundation.
2 Anticipation and Decisionmaking
2.1 Informal Discussion
This sectionformulatesamathematicalmodel of decisionmakingwhen one of the motivatingforces
behind behavior is anticipation. Although the de(cid:12)nition of the agent’s preferences and decision
problemisnotationallyheavy, thebasicideaisquiteintuitiveandamenabletoaverbaldescription.
To aid in this description, consider the following four-period model of a simple purchase decision,
illustratedinFigure1. Numbereddotsrepresentdecisionnodes,whileotherforksrepresentnature’s
moves, each occuring with probability one-half. In period 1, the agent’s choice set is degenerate,
4
but she (cid:12)nds out whether her investments made her rich (wealth WH) or poor (wealth WL). In
period 2, she can either visit an electronics shop to consider buying a large-screen TV, or she can
choosenottodoso. Ifshe visitsthestore,she(cid:12)nds outtheprice(pH orpL). In period3,depending
on the circumstances, she mighthave tomake apurchase decision. If she does, she decides whether
to buy the TV and explore all its features immediately, buy it and explore later, or pass on it.
Finally, in period 4, the agent again makes no decision, but in case she bought the TV without
researching it, she now (cid:12)nds out how fun itis (l or h). In period 4, the agent consumes her leftover
money and possibly l or h amount of the TV as shown.
Although simple, and although the only non-trivial physical outcomes occur in period 4, this
decision can generate a multitude of experiences related to anticipation. Upon learning that her
investments are doing well, self 1 might derive pleasure from anticipating owning a TV. Self 2,
in case she decides not to visit the shop, cannot look forward to enjoying the TV, and having
anticipated it in period 1 can make her feel extra bad about this. Self 3, if she has expected to buy
at a low price, might (cid:12)nd it aversive to buy the TV at the high price at node 7. In addition, if self
3 buys the TV, she may be curious to (cid:12)nd out all its features immediately, or she might want to
leave some surprises for period 4. Below, I relate allthese feelings to formal elements of my model.
There are three essential aspects of anticipation that I am trying to capture: they are realized
at a given time, they are forward-looking, and we have no direct control over them. In the above
example, self 1 might anticipate owning the TV. Crucially for behavior and realistically in this
setting, I do not assume that later selves have to comply with this expectation. Even if they do
not, at that point it is impossible to change these expectations and the fact that they a(cid:11)ected
utility. But ultimatelyanticipation has to be grounded in reality: I assume that the decisionmaker
has rational expectations. Thus, in equilibrium, the above situation cannot happen: if the agent
expects to buy the TV for sure, that is because so it will be. Finally, the agent only makes choices
overphysicaloutcomes(e.g. whether tobuy ornot),and does not directlychoose her expectations.
Self 1 makes no choice, so her anticipation is determined in equilibrium by selves 2, 3, and 4.
2.2 Rigorous Setup
This section introduces a natural model of decisionmaking in which utility is a(cid:11)ected by expec-
tations about the future. The setup of the model is an extension of the psychological expected
5
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11 28
utility model of Caplin and Leahy (2001) in two respects. First, it is explicit about the temporal
placement of feelings, an issue that turns out to be crucial but which Caplin and Leahy (2001) do
not capture fully. Second, my model allows for the interaction of expectations of outcomes and
the outcomes themselves. These features are responsible for a majority of the novel phenomena I
introduce.
The decisionmaker is involved in a T-period decision problem. In each period t 2 f1;:::;Tg, a
\physical"outcomezt 2 Zt andan\anticipation"ft 2 Ft areirreversiblyrealized. Forexample,the
agent’s enjoyment of the TV and money are physical outcomes in period 4 (z4), while her period 1
anticipatoryfeelings about the same are included in f1. For notationalconvenience, I willsuppress
arguments that are deterministic and do not depend on the agent’s choice; e.g. z1, z2, and z3 in
4
the example.
IassumethateachZtisaPolish(completeseparablemetric)space;theseoutcomesarestandard.
The crucial (and nonstandard) element of the model is the de(cid:12)nition of the expectations ft that
enter the agent’s utility function. I need to describe two things: the objects of expectations and
how expectations are formed. The (cid:12)rst of these is easy: expectations are about the future; thus,
denoting by (cid:1)(S)the set of Borel probabilitymeasures over a set S, de(cid:12)ne Ft recursively from the
back by
Ft = (cid:1)(Zt+1(cid:2)Ft+1(cid:2)Zt+2(cid:2)Ft+2(cid:2)(cid:1)(cid:1)(cid:1)(cid:2)ZT); (1)
Thus, self 3’s expectations lotteries over TV-money consumption pairs; self 2, in turn, anticipates
both consumption in period 4 and expectations in period 3. For a deterministic outcome z, let (cid:14)z
denote the probabilitymeasure over future outcomes thatplaces probabilityone on the outcome z.
Theexpectationsandphysicaloutcomesconstituteallthepayo(cid:11)sthedecisionmakercanpossibly
be worried about. In line with previous research, I assume that the agent’s preferences satisfy the
axiomsofexpected utilitytheoryovertheextended outcomespaceZ1(cid:2)F1(cid:2)Z2(cid:2)F2(cid:2)(cid:1)(cid:1)(cid:1)(cid:2)ZT (with
5
generic element(z1;f1;z2;f2;:::;zT)). Thus, self t’spreferences are representableby a continuous
4 Intheexample,theonlyrelevantphysical outcomesoccur inperiod4. Formally,thisiscaptured by makingZ1,
Z2, and Z3 allsingletons.
5 CaplinandLeahy(2001)arguethatobservedviolationsofthesubstitutionaxiomarenotduetotheaxiom’slack
ofintuitiveappeal, butariseinstead froman incompletespeci(cid:12)cation ofpreferences. Theappeal oftheaxiom comes
froma non-complementarity argument: that once a stateis realized,the decisionmaker should not care about states
that did not happen. When the agent has unspeci(cid:12)ed preferences{such as anticipatory feelings{that can depend on
unrealized states, the failure to include these preferences in the model shows up as a violation of the substitution
axiom forpreferences that are included. But once thedomain of preferences isfullyspeci(cid:12)ed, the intuitiveappeal of
7
von Neumann-Morgenstern utility function ut de(cid:12)ned over this space.
This now fully de(cid:12)nes the agent’s preferences. Throughout the paper, for any measurable func-
R
tiong andprobabilitymeasureGde(cid:12)nedong’sdomain,theshorthandEGg((cid:1))denotes g(x)dG(x).
Q
Let Yt = ts−=11(Zs(cid:2)Fs) (with generic element yt) be the set of all histories up to period t. Using
this notation, given yt, self t prefers the measure ft−1 2 Ft−1 (of outcomes starting in period t)
over ft0−1 2 Ft−1 if and only if Eft−1ut(yt;(cid:1))(cid:21) Eft0−1ut(yt;(cid:1)).
While preferences over anticipation are now given, their formation is more complicated. In
short, I assume that the agent has correct expectations. But correct expectations are determined
only in equilibrium. Therefore, anticipation is part of the speci(cid:12)cation of equilibrium, to which I
turn next.
In a speci(cid:12)c decision problem, the set of feasible physical outcomes in period t is a compact
Zt0 (cid:26) Zt. Let DT be the set of compact convex subsets of (cid:1)(ZT0). A decision problem in period
T is an element dT 2 DT. Going backwards, for each t = 2;:::;T, let Dt−1 be the set of compact
convex subsets of (cid:1)(Zt0−1(cid:2)Dt). A decision problem at time t−1 is some dt−1 2 Dt−1.6 Thus, a
decision at time t−1 induces a lottery over pairs of physical outcomes in that period and future
decision problems. When represented as a decision tree, each decision problem corresponds to a
decision node. In the TV purchase example, self four has no choice, so d4 is always trivial. But
it is not always the same singleton set. At node 17, d4 is a singleton with the gamble between
z4 = (WH −pL;h) and z40 = (WH −pL;l). At node 16, d4 has the riskless outcome z4. At node 14,
d4 hasthe risklessoutcome(WH;0). Atnode 6, self3’sproblemistochoose between node 14,node
17, and a lottery over nodes 14 and 15. Continuing this way, one can easily construct the other
selves’ decision problems. This construction allows us to capture decisions regarding information
acquisition{self 3’s last two options give the same lottery over Z4, but information revealed at
di(cid:11)erent times, allowing her to either indulge her curiosity or her taste for surprise.
0 0t 0
De(cid:12)ne expectations F and histories Y from the feasible Z ’s as above. Note that for each
t t
t 2 f1;:::;Tg, Ft0 is compact Polish in the weak-* topology, and Dt is compact Polish in the
topology generated by the Hausdor(cid:11) metric. Now, the elements of a personal equilibrium are:
the axiom comes back in full force. For expectations, given what the agent expects now and what she expected in
the past, her preferences should not depend on what else she could have expected today.
6 This speci(cid:12)cation of the agent’s decision problem is almost identical to that of Kreps and Porteus (1978); in
addition to their assumptions, I require decision problems to be convex.
8
1. A vector (cid:27) = ((cid:27)1;:::;(cid:27)T) of strategies for each self, where
(cid:27)t :Y0t(cid:2)Dt ! (cid:1)(Zt0(cid:2)Dt+1) (2)
and (cid:27)t(yt;dt)2 dt 8t;yt 2 Y0t;dt.
2. A vector (cid:30) = ((cid:30)1;:::;(cid:30)T−1) of measurable \anticipation functions" in each period, where
(cid:30)t : Y0t(cid:2)Zt0(cid:2)Dt+1 ! Ft0: (3)
The (cid:12)rst of these elements is the usual vector of strategies of the decisionmakers in the game.
Forexample, (cid:27)3 speci(cid:12)es whatself3 should dodepending on her decisionproblem (node) and what
she expected in periods 1 and 2. The new feature of personal equilibrium is the second element. It
is a map (cid:30)t that de(cid:12)nes the agent’s expectations as a function of what happened up to the end of
period t. Thus, after self 2’s choice, (cid:30)2 speci(cid:12)es, as a function of the decision node self 3 is going to
face, what self 2 expects; itmightspecify that the agentexpects to buy the TV and explore it later
1 1 1 1 0
at node 6 (which leads to expecting a lottery between (2(cid:14)z4 + 2(cid:14)z40;z4) and (2(cid:14)z4 + 2(cid:14)z40;z4), since
self 2’s anticipation includes self 3’s expectations as well). This is a qualitatively di(cid:11)erent part of
the proposed personal equilibrium{since the agent does not choose the anticipation function, she
is in general unable to select from all expectations that are possible given her decision problem.
Thus, even though expectations are realized today just as physical outcomes are, the agent has in
some sense less control over them. Self 1, making no decision, has absolutely no control over her
expectations. The chain of decisions and outcomes within a single period is illustratedin Figure 2.
self t chooses ft is
lottery over zt, dt+1 determined by
Zt(cid:2)Dt+1 realized (cid:30)t
-
t t+1
Figure 2: Sequencing of decisions and outcomes within one period
Finally,twomore pieces of notationare necessary tode(cid:12)ne the concept ofpersonal equilibrium.
Let lt(yt;γt;(cid:30)t) represent the probabilitymeasure that the lottery γt 2 (cid:1)(Zt0(cid:2)Dt+1) and the map
9