Myopia, amnesia, and consistent intertemporal choice

M;~th~ll~~ti~ithSocial Sciences 8 (IWU) 95- 109 North-Hollard

MYOPIA, AMNESIA, AND CONSISTENT INTERTEMPORAL CHOICE

Communicated

by F.W.

Rowh

Received 25 June 1984

Social scientists often explain choice at wcce\ri\e the future. Within

obrer\ed

streams of behavior

This paper presents four axioms for con4stent

this framework,

It is shown that an anticipatory

process if and only if binary dogenous

preference

‘ahistorical’. axioms,

choice In \uLh m~A4~.

organized.

is charr, .: erized

Amnesia.

by the condition

posed by inconsistent

amnesia:

endogenous

Each cjf thr’w rationalir!

or the ahwnce that

The paper closes with some remarks on the methodological

Myopia;

choice.

on the basis of bounded

LIMI-

process reduces to a beha\ iorally equil alent m! c)pL

choices are lexically

formation,

and the problems

Key w&x:

intertcmporal

play no role in current

strategies is argued to have some plausibility

siderations.

ot ~cv~\tr~in~~i

ot‘;tition~ C’\~L~L~III~tf!iij

myopia and amnesia are defined to be choice processes in which tuture

feasible actions and past decisions, respectively, modeling

a4 the outcome

date\. \\ith actors chou\in 3 among planned wqucni0

binar!

i~t cn-

chtjlzc\

Iv

role of the ~~w~l~~tt‘t~~~

choice processe\.

preference

formation:

binar!


1. Introduction

Social scientists often explain observed streams of behavior as the o~~c’otnr ot‘ constrained choice at successive dates. For example, economic models of ~Gng\ and investment decisions have this structure. A number of theorists have ~tuditxi r tw paradoxes which can arise when choice processes are inconsistent 01 c‘r time,’ w that plans made on an initial date are not carried out at later decision pointy. The recent literature on ‘the economics of self-control* has marshalled persuasiw mccdotal and introspective evidence that these theoretical paradoscs arise in swrpia> life.2 Nonetheless, the assumptions used in applied economics, such as the conwncy of discount rates over time, commonly ensure that choices 3rt’ intzrtttnqmdl~ consistent. This paper proposes a simple axiomatic structure for ccmsistwcy in intertmpwd choice. Within this unified structure, myopic behavior and t’n,l~~~t‘nc>uiprtft‘ertltrcc

‘See Strotz (195% Grout

Pollak

(1968,

Blackorby

et al. (1972).

Hamm~wd

413%~). c.;oIJI~.~II( I%()). .IIIJ

(1982).

‘Examples

include Winston

(1980).

Thaler

and Shefrin

(l3Sl3.

znJ

S
(13&i)

formation are characterizedin terms of binary choices between plans. The analysis sketches the role of bounded rationality colnsiderations in modeling intertemporal &oice, and demonstrates the range of choice processes which can be accommodated by the consistency axioms. The axiomatic approach also clarifies the methodological issues raised by the study of ‘inconsistent’ behavior. A constrained choice process is a function which maps sets of feasible alternatives into sets of chosen behaviors, at a sequence of points in time. The process is anticiparo/u if at each date choices are defined over planned sequences of actions extending into the future. Section 2 formalizes this concept and presents the four intcrtcmporal consisten;y axioms which are used in the remainder of the paper. A choice process is myopic with respect to some planning horizon if fhc actions which it generates prior to the horizon are independent of the actions wilich will be feasible after that date. Section 3 shows that an anticipatory choice process can be reduced to a fully myopic one if and only if choices are lexical: that is, for any two plans, the choice between plans is independent of all actions occurring after the date at which the two plans diverge. Section 4 identifies the conditions under which preferences vary endogenously, in response to past actions. A choice process is called amnesic if choices on a given date are independent of the actions taken prior to that time. It is shown that a consistent process is fully amnesic if and only if the initial choice between two plans is independent of ?he common actions occurring before the two plans diverge. Section 5 closes with some methodological remarks, and a reconsideration of the axiomatic structure,

2. Consistency anioms for constrained chaice processes A history is a >;equence of actions which have occurred prior to the current date. A plan is an infinite sequence of actions beginning at some date 1:

d=(X,J,*

1,

.

..).

I’he action occuirring at the date szz t in a &n s’ will be denoted by sY. A partial plan is a finite sequence of the form (x,, . . . . A-~). 1 shall say that the plans .I-’and Y’ diverge al Tif they can be written in the foam s’= (q,x7), y’= (q,~ ‘j, with .Y#.Y~, for some partial plan 4. The set of conceivable plans at I, given history 2, is denoted S(.Z,f]. The set of feasible plans at (z, 1) is denoted by F[z, I), or simply Fif there is no confusion about the intended history and date. F(z, I ] is always a non-empty subset of S[z, I]. Given a specific starting point (a history and date), the sets S[z, f] describe the actions conceivably available at each node in a decision tree which emanates from that starting point. An atrricipatory constrained choice process is defined by a choice jbnctiorl A which maps each possible triple (F, z, I ) (a set of feasible plans, a history, and a

date) into some non-empty subset of F. For some purposqs,

the f’ sets might ht.restricted to a certain class of subsets of S, such as the compact subsets, but here A is taken to be defined for all subsets of S. The arguments : and t allow the actor’s ‘tastes’ to change either endogenously or csogonously. Plans in the set .-I[& :. r] arc’ called choices relative to F, and the elements of F - .4 [F, z, t ] are said to bc wjttwi relative to F. A finite horizon problem can be handled by introducing the dunml\ action ‘death’, and constructing the sets of conceivable plans so that only this actim is possible after some date. 1

shall now introduce

a pair of axioms for feasible sets.

Axiom 1 (forward inclusion). If_& F(z, I], then ifs .wb,dau X’E F[(z.s.,. . . . , -L 1). .$I. Axiom 1 states that a plan which is feasible at date f has feasible \ubplan~, gi\~rl that the intervening steps of the original plan have been taken. Axiom 2 (backw3- J inclusion). Given x” E F[z, t ] and a p/art _Y’E P‘[(z,s,, . . . . A-, 1).:$I. then (A-~, . . . ,s,_ ,,_;‘) E F[z, f). This asserts that if the initial steps of a feasible plan s’ are taken, and i; b~~ic~n?~~ possible to adopt some plan ys at date s, the original constraint set F(z, f 9 murt have explicitly permitted this sequence of actions. Later constraints sets do not introduce, ex post facto, new possibilities absent from earlier feasible sets. Feasibility is taken to be a feature of the external environment in which choic-~~ are made. The objection that an actor may lack exhaustive knowledge of ai1 auilable options3 can be interpreted in one of two ways: (a) as a statement about rhc cognitive limitations of the actor being modeled, or (b) as a claim that no sharp distinction between feasible and non-feasible plans is possible, even for an c\ctt‘rnal observer. An objection of type (a) can be treated as a proposition about
can be abbreviated to A(.\;, . . . , s,

Axioms 3 (forward inclusion).

If x’ E A IF, E,t ], then its subpktn A-’E A (.\;,. . . , .q

, ).

This asserts that if a plan is chosen at time I, none of its subplans will be rejected on later dates, so long as the intervening steps of the original plan have been taken.

Axiom 4 (backward inclusion). If .v’EA [E2,t) and the plun ,vsE A(x,;, . . . , wvs_ I ). then (x,, . . . , .u,- ,*y”~EzQIF,Lfl. That is to say, if the iniaial steps of

a chosen plan are executed. and at a later date some other plan is a choice, then the full sequence must have been a choice ai: the initial date. Later choice sets cannot introduce novel subplans which were rejected

on an3arlicr date.’

Axioms, l-4 collectively define a ‘consistent’ anticipatory choice process. The axioms for choices are considerably more problematic than those for constraints, and a full discussion of their status is best deferred to Section 5. For now, it is enough to note that they summarize plausible properties of foresighted planning, and are generally satisfied in applied models of intertemporal choice.

The final axiom is a weak condition on choices at each date. Axbm

$ I[decisiveness).

X’EA(I=,IJ]. (i) ~ykA(F,z,t], (ii) If y’eA(F,z,t),

Let F be a subset of S[Z,t], with S! y’ E Fz and (1ssum?

A((x’,y’),t,t] = (x.‘,_v~). then A[(s’,y’),~, f] = (s’). then

Axiom 5 is satisfied whenever A IF, 2, t) .xises from a transitive preference ordering elements of S[z,r), provided that the feasible sets F are restricted to the which some undominated element exists. Proofs of the theorems in the next two sections involve only repeated applications of Axioms l-5, and are provided in the appendix.

over the class for

3. Myopia and lexical choice Anticipatory models of individual and ‘km behavior have been criticized on bounded rationality grounds, since they oken impute implausibly large information-processing capabilities to the actors inw,olved.-5For this reason, myopic choice qhese ar,ioms can be derived as theorems from more elementary propositions. If Axioms I and 3 are asserted for s = o+ I only, the extensions given above follow immediately from an induction argument.

Similarly, iii Axioms 2 and 4 are accepted for s = t + I ,. the more general form follows by induction and rhe use of Axioms 1 and 3. respectively. ‘This idea, recently restated by Simon (1978). is srentral to the analysis of enterprise organization developed bjyWilliamson (1975), as well as the evolutisnary view of market dynamics adopted by Nelson and Winter (1982).

is frequently proposed as an alternative basis for understanding beha\ ior. hit \k hen anticipatory choice reduces to a formally equivalent myopic process, the issue is moot, because the anticipatory model imputes no mow ‘rationality‘ to the xtor involved than would be assumed in a myopic framework. By the same token. ial this case the anticipatory formulation is redundant, and adds nothing to a myopic csplanation of observed behavior. ‘The bounded rationality critique has t‘ow if, but only if, no such reduction can be carried out. To formulate the relationship between anticipatory and myopic chckx mx c precisely, some definitions are necdtd. Suppose the fcasiblc set of plans at (E.f) is

F[z,r], so that the set of feasible partial plans with ho+on

CV[F(:,f))= (q:9=(s

T is

,,...,. v7-) for s’EF[=l,r]).

The set of partial plans with horizon T which arc ~eneratcd by .-I from I.*[:.t ] ib

BI‘[F(2,f)]=((I:(I-(.\;, ,....

for s’E‘4[F(2.f),;f)~.

.Y_r)

A partial plan generated by tl from F[z, I] will also be called ‘obser\.ablc‘ relariv to F. The choice function .+I is said to be ryyopic at (t, t) with horizon T when for all subsets F and F’ of S[z, f] such that G ‘(F) = G ‘@‘I. B ‘(I-‘) = B ‘iF#,. A is said to be fit@ myopic at (:,t)

if it is myopic for all horizons ;Tr 1.’ Myopia with horizon T implies that the actions which will be feasible after T do not affect the generation of partial plans ending with T. An>, two feasible sets \I hich permit the same ‘short -run’ actions yield the same predict ions about ‘dart -run‘ behavior. For example, myopia with horizon t implies that the action(s) obserwbk at t are independent of the actions which will be available at an? later dare. Using the axioins of Section 2, we can characterize myopia entirely in rerms of binary choice.

one of’ the +follo wi:tg coniiifiorts

@l-h’.Is ‘myopic ‘rkve’

usagt’ deparls from the littmturc and ‘naive’ as synonyms

intervening

on cwsistency

that wrrrnt

change in preferences (Pollak.

plans (Goldman.

1980). Asionw

naive and sophisticated

plans

1968).

of such change, and regards future

in intcrtznqwr.~l

CYal.. 1072:

(cf. Blackorhy

agent ignores the possibility

possibility

holds:

may

not

ln comrast

myopic, as that term is used here.

be

zh0icc. 19’bx

impknwnted

coincide. Both

\\hi& rmi

in

, a ‘sophiwcated

preferences as imposin, 0 constr;tints

3 and 4 ensure that constraints

behavkr

HaIIlmcmLii.

uw.kll!

~~rlu~‘:.

trs-if[\

1382).

.L\

Ehe iururc ciuz 1~1.m sent

rzzopnirr‘s

the

cm currently

chosen

of this type are nt\\;r binding.

SO that

naive and sophisticated

procemtts art‘ typicrtll!~

non-

ti)

,4((x’,d~~),;,~]

= (s’)

(ii)

A [(s’,_V’), z, 11= (Y’I

(iii)

A((_~‘,~QJ~

and

A[(.~“,.~“)LJl=

and

A [(X”,_Y”), Z,tl = {Y”)J

and

= (.$,_y’)

(x”L

A[(x”,.Y”)~zJ~=

(-~‘b”f 1.

The theorem asserts that the choice between two plans which diverge no later than Tdepends only on the actions taken under each plan by that date, and not on the subsequent behavior of either plan. For myopia with horizon f, this requires that A choose between any ,V and y’ having different initial actions solely on the basis of those initial actions. If A [(x’, y’), z, t] = (x’), then ever-v plan beginning with x( must be chosen over ever-v plan beginning with y,, regardless of the actions to be taken later. If A((x’, y ‘), z, t] = (x’, y’), then A cannot distinguish any pair having the same initial actions X, and yr . This property motivates the following definition. A is lexical at (z,f) when

for any subsets F and F’ of S[z, t] having the form F= ((q,xT),(qrYT))~

F’=

((q,X’T),(qJ’T))~

with x7 = S;fyT=y;, one of rhc following conditions

holds:

(i)

,~(F(z, q), (z, q), 7’1= (X T)

and

A [F’Cz,q), (z, q), Tl = (x’ ‘),

(ii)

A[F(z,q),(z,q),Tl=(y~)

and

A[F’(z,q),(z,q),Tl=(~“),

(iii)

AIFWMZ,q).

Tl= (xT,yr)

and

M”(z,qh(z,q),

Tl= (x’~Y”>-

In thic definition, F(z,q) denotes the feasible set F[(z,q), T] = (xT, y T, which prevails at date T if F(z, t) is the feasible set at date f. Similarly, F’(z, q) = (x’~, y”). Lexicality requires that on the date of divergence for x’ and y’, when A must reveal a choice between the two plans, the choice is independent of the actions to occur after that date. Alternatively, we may say that every binary choice between arbitrary x and y’ is uniquely determined by the partial plans (q,xT) and (q, yT), that is, by the actions available at the diwergence point, and the common prior history of the two plans. The following theorem makes clear the importance of this restriction on binary choice.

Theorem 2 (lexical choice). A is *fullyrnyt.>picat (2, t) if and only if it is lexical at (290. Theorem 2 gives a precise criterion for the formal reducibility of an anticipatory choice process to a fully myopic one. Rounded rationality considerations can be understood as a form of Occam’s razor: one should not impute more information processing capacity than is needed

101

‘to explain an actor’s behavior. Full myopia involves the minimum possible imputation of rationality in the context of intertemporal choice. Hence, bounded rationality suggests the principle that full myopia should be rejected only when a lcsical choice structure is inadequate to the behavioral facts at hand.

4. Amnesia and ahistorical choice Mechanisms of endogenous preference formation are represented through the second argument of A [F, t, t]. Choice behavior may also vary exogenously through the time argument. Economic analysis commonly involves stationary choice processes, where neither E nor t appear, but some models of preference endogeneity have been proposed in the theory of consumer demand.’ An assertion about preference endogeneity generally amounts to the counterfactual hypothesis that if an observed behavior stream had been different, subsequent choices would also have differed. Since only finite histories are act;lally observed, I shall consider choices relative to a fixed initial position (z, t). The process A is amnesic at T relative to (z, t) when for any subsets F[z, t] and F’[z, f I of Sk, t 19 if 4EBr-‘[F(zJ)],

4% B*-‘[F’(W)],

Fk, q), 7-1 = F’Nz,0, then A F-T,(z,d,

Tl

Tl

and

= Fj-,

= A FT, (z, 0,

Tl.

Full amnesia relative to (z, t) is defined as amnesia relative to (z, f) for all 35 t. Amnesia at T implies that choice at T depends only on the feasible set at T. 1f some feasible set FT could have arisen through two observable paths 4 and q’, the choices made at T from rqTdo not depend upon which path actually occurred. Consequently, choices at date T can be represented by a single choice function of the form A [F, T]. Requiring q E B-‘[F(z, t)] is innocuous, since 4 is of no interest unless it can be generated by A from F[z, t]. Similar remarks apply to 4’ and F’[:, t 1. As the terminology indicates, the absence of endogenous preference formation is behaviorally indistinguishable from the actor’s failure to recall past actions. From this standpoint, the bounded rationality arguments advanced on behalf of myopia might also justify a modeling strategy involving amnesia. Just as there are limits on human abilities to cope with complex future constraints, there are also limitations on memory capacity and ;he sensitivity of current behavior to past experience. As in Section 3, we proceed by characterizing amnesia in terms of binary choke.

‘See, for example, the models of habit formation in Pollak (1070, El-Safty (1976), avd Hammond (1976b).

1976). ton fVeirsackr

(13~11.

TLeorum 3. A is amnesic at T relative fo (z, t ) if and only if Jot- every distinct .~‘,_Y’ES[~IJ with (x,, . . . , XT

,)=(Y,,.--&-I),

and each partial p/an q = (q,, . , qr l

.p = (q,x’)

and

1) s&i

l

that

y”=(q,y%S[zJ],

one sf the folllowing conditions holds:

(ii)

,q{_r’,y’),~r]

= {y’)

(iii)

A[{x’,_~),~J]

= (x’,_v’)

and

A[{x”,Y”)JJ]=

and

{Y”),

A[(x”,_Y”),z,~~ = (-v”,Y”).

Amnesia at T thus requires that any choice between two plans which are identical until Tis independent of the common path taken prior to T. If some alternative path beginning at (z,f) leads to the same pairwise choice between subplans at T, the choice between subplans, and hence the initial choice at t, must be made in the same way.

&fore characterizing full amnesia, we require a further definition. A is said to be ahistorical relative to (z, t) when

The process

for any two subsets of ~[zJ] having the form

F= (((I;x*),(q,yT)f, with Xrfyr,

F’= {(q'JTMf9Y

T)>9

one of the following conditions

(i)

A[Ez,z]={x~)

and

A(F’,z,t]=(x’,),

(ii)

AV?t.t]=(y’)

and

A[F’,z,r]={y”),

(iii)

A[F,z, t] = (x’,y’)

and

holds:

A[F’,z, t) = (x”,y’,}.

A is ahistorical at (z, t) if the choice between any two plans x’ and y’ is independent

of the path taiken prior to the point at which they first diverge. Binary choice at the by the choice which will be made between xT and yT on the date of divergence. No&e that the definition does not involve plans which diverge immediately at t, since they cannot have a common history relative to (z, f ). This concept plays a role in characterizing; 2~1 amnesia parallel to that played by lexicality in Section 3. initial date t i%thus uniquely determined

Theorem 4 (ahistorical choice). A is fully amnesic relative to (z, t ) if and only if A is ahistorical at (G t ).

It is worth noting as a corollary that Axioms! I-5 by no means preclude endogenous

preference formation, since consistent examples are easily devised which violate t hc requirements of an ahistorical process. Also, A may be both fully myopic and fully amnesic at (at). In this case, the outcome of any binary choice between plans is entirely determined by the actions of the two plans at the date of their divergence. Both the actions occurring after that date and the common path taken prior to divergence are irrelevant in pairwise comparisons.

5. Consistency in intertemporal choice

The literature on intertemporal consistency cited in the references deals extensitcly with situations in which Axioms 3 and 4 are violated. It is therefore appropriate to evaluate the analytic role of these axioms. 1 shall argue that the basis for choosing between consistent and inconsistent models of behavior is fundamentally methodological, although empirical factors may play some part. An important feature of consistent choice is that if Axioms 1 and 2 are accepted, no empirical data can recute the assertion that Axioms 3 and 4 also hold. To see this, suppose that some collection of partial plans has been observed for a number of actors facing diverse constraints and that constraint sets have been defined to make each observed behavior stream feasible. One can then adopt the trivial model A [F, z, 11= F, identically for all histories, dates, and constraint sets. This model generates any behavior which is compatible with feasibility restrictions, and in particular, it generates the behaviors actually observed. It is readily checked that if Axioms 1 and 2 hold, this model also satisfies Axioms 3 and 4. Clearly, this procedure contributes nothing to our understanding of behavior. The concept of choice is not actually used, because no feasible behavior is rejected by the model, and hence thl: mode1 is not open to empirical challenge. HOWW~, the trivial mode1 does make a useful point: if one discards Axioms 3 or 4, it cannot be because no consistent choice process is compatible with the given data. Consistent intertemporal choice can only be rejected on methodological criteria, perhaps including testability, clarity, and parsimony. If the cost of preserving intertemporal consistency is that most feasible behaviors must be treated as observable, one might search for suitable modifications of Axioms 3 and 4. One might also decide to abandon these axioms if this is the price paid for conceptual clarification elsewhere. For example, inconsistencies ovr=r time might have to be accepted if choices are to be represented by a transitive preference ordering at each date. Nonetheless, Axioms 3 and 4 remain appealing in terms of simplicity and intuit i\ t’ coherence. A model which ‘chooses’ a plan at one date, and then ‘rejects’ it at a later date, obscures the notion of ‘choice’ as an explanation for observed behavior. Adopting language from the philosophy of science, we may regard the consistent? axioms as constituting elements in the core of a broad research program, rather than

as hypotheses open to direct empirical test. ’ Accordingly, the axioms are best judged by their fruitfulness in guiding the construction of models which describe behavior in a variety of particular domains.

Acknowledgemenls The support of the National Science Foundation under Grant No. SES-82 19275 is gratefully acknowledged. 1 would like to thank Thomas Juster for stimulating my interest in the problem of endogenous preference formation, and Richard Nelson tbr helpful comments on an earlier draft. Mathemalical appendix: Proofs of Theorems 1-4 Tbeorem 1 (Necessity) Supposethat there is some set of four plans s’,y’,s ‘,_v”ES[Z,t] with (S,,...) q.)=(XJ )...)_ U;‘)+(y,,**=,-)‘T)=(YI,

l *-La*

such that one of the following holds:

(0

,q(x’,y’},E,t]

(ii)

,q(x’,y’),t;t)=

(iii)

A((s’,y’),&

= (2) (y’) t] = (x’,y’)

but _)?“EA[(X”,Y”}.Z,tl, but ht

X”EA((x”,Y”}.Z,t], A4j(~‘~,y’f),z,:‘J=

Then choose F= (x’, y’) and F’= (x”, y”),

(x”)

or (_v”)=

SOthat

Gr(F)=GT(F’)={(x,,...,x~),~~~~~-~~~)}~ But if any of (i)-(iii) are true, then BT(F) # B’6EF,),contradicting the definition of myopia at (z, t) with horizon T.

(Sufficiency) Assume that the conditions stated in the uheorem hold, but that A is not myopic at (z, t) with horizon T. This wilf generate a contradiction, establishing that the stated conditions are sufficient for myopia. If A is non-myopic with horizon T, there are subsets F and F’ of S[z, t] such that G ‘(F) = G T(F’), but B’(F) #B T(F’). Then relabelling F and F’ if necessary, there is some X’E A [F, z, t J such that (x,, . . . , XT) 6 B’(F). Since G ‘(F) = G ‘(F’), there is an x”EF’such that (x,,...,xT)=(x;,...,x;), but x”$A[.F’,z,t]. ’ There is also some y” E A [F’, z, t] such that (x,, . . .., XT) #(y;, . . . , yk), since oul, . . . , XT)() BT(F’), but BT(F’) is not empty. Finally, since G T(F) = G T(F’), there is some y’ E F such that (y;, . . . , y;) = (yl, ..-, yT). ‘“see Blaug (1980) for a helpful discussion of ecounomicmethods in relation to tile philosophy of wtience, and especially chapter 2 on scientific research programs.

Summarizing these results, we have: (a) s’,ykF, with skA(I;;z,~), and (b) A-“,y” EF’, with _v”~A[F:2,1) but .~%4[F:z,f). These plans satisfy (x,, . . . , ST) = (A$ . . . , A$-‘)f (.Y,, . . , _vr) = (_q’. . . . , _v;-). Applying Axiom 5 to (a) and (b): l

This contradicts true. 2

the conditions

stated in the theorem,

which were assunwd to bc

Theorem 2 (Necessity) Suppose that A is not lesical at (z, I ). Then there are .~I,_k\.?

.r”’ E

S(z, t] of the form: s’=((J,.g-),

_&(q,yT),

stF=(q,.Pj,

_f'=(qyj.

with

such that one of the following conditions

holds:

(i)

A[(XT,yT),(:,q),t]

= (sT)

but

r~‘~E~~[{-~“,~“~,(Z1q),f].

(ii)

A[(s?,y’),(z,q),f]

= {_v’)

but

.\.‘~~rl[(s’~.y’*).(;4),f].

(iii)

A[(s7;yT},(z,q),f]

= (sT.yT}

A[(x’qy’Z),(~,q),f]

= {A-‘*}

but or (~“1.

Choose F= {A-*,y*} and F’= (s”,~“), noting that G’(F) = G ‘(F‘) = {(q,+.), (q, UT)}. For cases (i)-(iii), respectively Axioms 3 and 1 can be used to establish that :

(8

A[F,t,t]

= (s’}

(ii)

A(F,~,t]=(s’)

(iii)

A[F1z,f] = (.~‘,u’)

but

y’kA[F’,z,f],

but

x’~EA(~‘,~.~], but

A[F::,f]

= {.I”) or (_v”).

In any of these cases, B’(F) f B ‘(F’), so that .+I is not tilly nqopiz (Sufficiency) c We assume s. This generates a

at (2. f ).

that A is lesical at (2, f ), but myopia fails for ~mc contradiction, showing that iesicality is wffic:im for myopia at (2, t) of all horizons. If myopia fails at s, then by Theorem 1 there are .Y’J’,-~“J”E% [I \vith horizon

(.q, . . ..L~s)=(.u;,... ,.uJ)#(_v,,. . . . _Q=c\‘;, . . . ..Q. such that one of the following conditions

holds:

G. K. Dow / M_vopiu and amnesia

(0

,q(x’,y’),~,r]

= (x’)

but

y”~pl[(Xr’.y”),Z,f],

(ii)

,#x’,y’),z,r]

=

but

~~“~~[(~“,Y”).,~~~

(iii)

~[(x’,~‘),~,t]

= (s’,~‘)

(y’)

Iact T~S be the date at which

X'

but

A[(x”,y”),z,~]

= (-~“1 or (~“‘)a

and y’ diverge. We can rewrite the Plans in the

form:

with

But applying Axioms 3 and 4 lo (i)-(iii) gives, respectively: (i)

A[(.~‘,y”),(;,q),

T] = (_I-~)

but

j”7.~A[(-~“r,4“7‘),(~,q).

T],

(ii)

/#.&y’),(~q),

T) = (yT}

but

.u’TEA[(-~~r,‘~T).(~.q).

T],

(iii)

A[(x7,yr}&,q)*

T] = (x7.p7’)

but

A[(.~~r,_Y‘T),(z,q~,T] = {YT) 01: (u”‘)~ Since one of these conditions

holds, A cannot be lexical at (z, t).

Ll

Theorem 3 (Necessity) Assume that there exist distinct x’ and ?_‘IE Slz, t] with &VTix,,-=*

,)=ti,,-•=,YT-

I)=@,

that for some q = (q’, . . . , qT- I 1,

x”--@XT)

and y” =(q9yT) are in S[zJ],

and one of the following conditions

holds:

(0

4{x’,u’),z4=(xf:

but

y’kk[(x”,y”),z,t],

(ii)

A((x’,y’),z,t)=(y’)

but

x”~A[(x”,y”),~,t],

(iii)

A[{x’,y’),z,t]=

(x’,y’)

but

A[Tx”,y”),z, t] = (x”) or (Y”)=

Clearly, if any of (i)-(iii) hold, q #q’. Let: F[z, t ] = ((q, x ‘), (q, y ‘)) and F’[z, t I = ((q’,x ‘), (q’, y ‘)). Applying Axiom 1, we hwe

F[(z, q), T] = F’[(z,q’), T] = (x ‘9.J’*) . We have qd? ‘- ‘(F) and q’MT- ’ (F’) because A [F,z, f] and A [F’, z, f ] are nonempty. Amnesia requires A[(x’,y’),(z,qh T] =A[(xT,yT),(z,q’), T]. But if any of (i)-(G) hold, this is precluded by Axioms 3 and 4. Hence, the stated condition is necessary for amnesia at T, relative to (z, P).

(Srrfficknqv) Suppose that the stated condition holds, but amnesia fails at Trelative to (:, 1). Then there are subsets F and F’ of S[t, t] such that for some q and ~1’: WB ‘- ‘(F)

and

qk B7‘- ‘(F’),

with F((z, q), T] = F’((z, q’h T] = FP~ but

Relabelling F and F’ if necessary, choose some s 7E A [F,, (z, q), T] with x7 $ A[F,,(z,q’), T]. Since .Y’E FT, and there is some q’E F with (q,, . . . , q7. I) =q because q E B ‘- ‘(F) Axiom 2 gives (q,.t?) e F. Likewise, (q’,s ‘) i- F’. The fact that qe BT- ‘(F) al;o implies the esistence of some plan (1% 4 ]I;: 2, r ] Gt h (Q,. . . . , qr_ , ) = q. Axiom 4 then gives:

(q,-~TkA[F,z,t),

(a)

(qt,sT)$,4[F’,z,z],

but

applying Axiom 3 to the fact that xT@ .4 [Fr, (z, q’), Tl. Since choice sets are non-empty and X’B A [FT, (z, q’), T], there is some Y ‘*A- ’ with y ‘E A (FT, (z, q’), T]. Reasoning as above, (q’,,r?‘) E F’, (q,_vT, E 6 and

(q’,_vThW’,z,~l.

(b)

NOWdefine Q= ((q,xT),(q,yT)) and Q’= {(q’,x’),(q’,yT)}, where Q is a subset of F and Q’ is a subset of F’. Applying Axiom 5 to (a) and (b): (q,x’)EA[Q,z,t],

A[Q’,z,rl=((q’,~~)}.

but

But (q’,xT) and (q’,y ‘) are distinct elements of S[z, t] because s ‘SF ‘- and ((1..I-’ 1 and

(q,vT)

theorem,

are in S[;, t].

so the sufficiency

result contradicts must hold. :I! This

the conditiccn stated

in the

Theorem 4 (Necessity) Suppose that A is not ahistorical. This will be shown to implv_ that amnesia fails for some T. If A is not ahistorical, then there are x’._v’ E S[z, f] with (x,, . . . , XT_ ,)=(_v ,,...,

yr_

,)=9’*

A-~#_v~,and TX

and some q such that (q,xT), (q,vT) are in S[:, t], such that one of the folio\\ ins conditions holds:

(0

A[(x’,_~‘),~,t]=(~‘)

but

(q,yr)E.4[{(q.-~~).(q~1”)).~.t].

(ii)

A[(x’,_.F’),~J] = {,I’>

but

C9,.~Z)E~~[J(q,-~~),(q*?‘T~~~;*z].

(iii)

A((x’,y’},z,~]

= {-v’,Y’}

A(((q,sT),(q,yT)),z,~]

but

= {(9.-y’))

Or t(9~“~)).

ii. K. Dow / Myopia and amnesia

IO8

Clearly, if any of these hold, q+q’. Then aBrs’(F) I

and

Set F’= {x’, y’) and F= ((q,xr),(q,y*)).

qkBT-‘(F’),

and mE,q),

q

=mZ,q’lr

Tl=

{cxl;u’).

But if any of (i)-(iii) hold, Axioms 3 and 4 imply that

and so A is not amnesic at T. Therefore,

it cannot be fully amnesic.

(S~~fficiency)Assume that A is ahistorical, but not My amnesic. This will generatea contradiction, showing that if A is ahistorical, it is fully amnesic. Suppose amnesia fails at date s. Then using Theorem 3, there are distinct ~‘,y’~S[z,f] with

(x,,***r .ys_l)=~,,*..,Ys-I)=P', and there is somep such that (&) followSng conditions holds:

0

A[(x’,y’},z,f)

= (x’}

(ii)

A[(x’,y’},z,t]=(y’}

(iii)

A[(~‘,_Y’)J,~]

y’

but

(~,Y”)~A[{(~~,~~“),(P,Y~)},z,II,

but

(hxS)~A[{(~,xS),i~,~S)),ttl.,

= (x’, y’}

A~{(p,~~),(y,yS)),z,~]

Since x’ i md

and (p, y”) are in S[z,t], such that one of the

but = iQws))

or {(my”))-

are distinct, they diverge at some T?s.

Set

q’=(x,,...,XT-,)=~~,*~=,Yr-l),

and

Then if any of (i)-(iii) hold, p#p’, and herRa q #q’. Substituting (p,x”) = (q,xT) and (p, y”) = (q, y in (i)-(iii) above, we fill
r,

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