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RELATIVELY RECURSIVE RATIOIIAL CHOICE

by

Alain A. Lewis

NIS ,TIDC TAR

JWllatficatio

Distribution/Availability Codes

Technical Report No.357 Avail and/

November 1981 Dist special

Prepared Under

Contract ONR-NO00Ih-79-C-0685, United States Office of Naval ResearchCenter for Research on Organizational Efficiency

Stanford University

INSTITUTE FOR MATHEMATICAL STUDIES IN THE SOCIAL SCIENCES .

Fourth Floor, Encina HallStanford UniversityStanford, California E TI

94 305 DIJA5LECT e

JAN4

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ABSTRACT

We have demonstrated previously (Lewis [1981]) that within the

framework of recursive functions, a distinction must be made between

representations of the paradigm of consumer choice, and realizations

of a given representation. The present paper extends our previous

framework to show, in brief fashion, that the concept of a recursive

rational choice function defined as an effectively computable represent-

ation of Richter's [1971] concept of rational choice, attains by means

of an application of Church's Thesis to the degrees of unsolvability

associated with a classification of types of subsets of the natural

numbers, a minimal bound in a measure of computational complexity

entailed by its realization in an effectively computable manner.I

4\

, • ' : I :ii I ! Ha "a

RELATIVELY RECURSIVE RATIONAL CHOICE*

by

Alain A. Lewis**

I. Introduction

The purpose of the introduction is to acquaint the reader with

the key concepts and terminology employed in the derivation of the

principal result of our previous paper, to which the present work is to

be considered as a sequel. The reader is encouraged to refer to the

earlier paper (Lewis [1981]) for a more extensive exposition of the items

that are summarized in the following discussion. The idea that is basic

to the construction and definitions is that recursive structures of number

theoretic character, by means of what is known as Church's Thesis ,

represent a very general class of effectively computable procedures, that

in another sense, typify ideal devices of artificial intelligence.

By a recursive space of alternatives is meant a pair <R(X) ,FR >

nwhere R(X) is the image of a subset, XC ]R+, that is compact and

n

convex, in a recursive metric space, M(On), which is in turn comprised

of n-tuples of R-indices of recursive real numbers; and where IF is theR

class of all recursive subsets of R(x). A recursive choice on <R(X),IF >

is a set valued function defined on the class IFR , C FR+ P(R(x)),

*This work was sponsored by Office of Naval Research Contract ONR-NO00 1 4-79-C-0685 at the Center for Research on Organizational Efficiency atStanford University.

**Department of Mathematics, College of Science, National University of

Singapore, Kent Ridge, Singapore, 0911.

tThe discussion of Church's Thesis in Chapter 1 of Rogers [1967] isan excellent introduction to this concept.

9!

such that for any AS FR , C(A) C A. We call the choice on <R(X),JFR>

RR

a recursive rational choice if (1) there exists a binary relation

>:R(x) x R(x) {1,0) termed the preference ordering; (2) there exists

a function f : R(X) - IN such that if it should be true of C 9

8E R(X) that a5 8 then f(a) > f(8); and (3) for any AE FR

C(A) = {a E A : VB E A f(a) > f() . A recursive rational choice on

<R(x),IFR> is thus rational in the sense of Richter [1971] and is

representable also in Richter's sense. The graph of a choice on <R(X),F >

is an enumerated collection of pairs of sets <F , C(JFR )> indexed

by J E IN whose domain* is {)FR , and whose co-domain is

{C(F ) } The elements of the domain of graph (C) are members ofj JIN

the class IFR , and it can be demonstrated" that if the choice on

<R(X),FR > is recursive rational, then the elements of the codomain

Sc(FR ) are also members of the class F.. A further result is that

a recursive rational choice on <R(x),FR > is also recursively representable

in the sense that the components of graph (C) are all effectively computable

by machine devices, in which case graph (C) C F R x FR. A recursive

rational choice on <R(x),FR is said to be recursively realizable, for

suitable choice of domain, if and only if graph (C) is recursively solvable

or equivalently if graph (C) is a recursive subset of IFRx FR. The

*The domain is full if 3K ED Vi J K F A FiR FR

%Cf. Lewis, [19811, Proposition V.2.

4:

I-

-3-

ment of the main theorem of Lewis ([1981] Theorem IV4) which places

a distinction between the notion of recursive representability and

recursive realizability.

Theorem 1.1: Allow <R(x), FR> to be a recursive space of

." alternatives derived from the recursive metric space of F , M(Fn),

for R(X) the recursive representation of a compact, convex subset

of IRn . Let C : F * - F be the nontrivial recursive rational+ R R

choice on <R(x), FR> and select from the class of sequences

(FR ) IN, element [F I C FR that comprises a full domain for

graph (C) C F x F . Then, per fixed selection of { IR R R~

graph (C) is recursively unsolvable and therefore is not recursively

realizable.

The method of proof of the above theorem employs the fact that

the restriction of graph (C) to its codomain was unsolvable by showing

that any predicate that was adequate for describing the codomain

belonged to a specific place in the Kleene-Mostowski Hierarchy that

4 classifies subsets of the natural numbers by the complexity of their

descriptions*. We presently exploit this feature of the proof to obtain

a lower bound on the degree of difficulty of computing the graph of

4 a recursive rational choice function by identifying the complexity of

descriptive predicates in the hierarchy with a measure of computational

!4 *A brief discussion of the Kleene-Mostowski Hierarchy is given in

Appendix I of Lewis [1981]. A fuller discussion is in Rogers 119673

Ch.l4, pp.301-334.

4

difficulty. The assertion of Church's Thesis which equates those

functions that are effectively computable by ideal computing devices

then becomes relative to the descriptive complexity of the predicates

that are used to describe the function. Functions that are effectively

computable in this sense of complexity are termed the relatively

recursive functions.

II. Relatively Recursive Rational Choice

The concept of relative recursiveness originates with the work

of the logician, Emil Post , and is concerned with the reduction of a

decision procedure for a given set of natural numbers, A, to that of

another set of natural numbers, B. Intuitively speaking, a set A of

natural numbers is reducible to a set B of natural numbers, if for a

total predicate T : IN - {1, 0), if the restriction to B, 'IBF can be

recursively realized, i.e., there exists a recursive 9 IN * such

that

f(n) = 1 when TIB (n) is true

O(n) = 0 when TIB (n) is false.

Then there exists a recursive realization for the restriction of T to

A, TVAl i.e., a recursive r(o) :i + iN ch that

*Cf. Lewii [1981], Ch.II, and Rogers [1967] p.130.

f'Degrees of Recursive Unsolvability", (Preliminary Report),4 Abstract Bulletin of A.M.S.. Vol.54. [1948], DD. 641-642.

-5-

{ r(f)(n)= 1 when TIA is true

r(=)(n) 0 when is false

Alternatively, the set A is said to be recursive relative to B, or

recursive in B.

In a subsequent article, Jointly authored with Kleene [i], Post

develops more fully the concept of relative recursiveness in terms of

degrees of unsolvability, which in turn is based on relations of

reducibility between decision procedures for subsets of the natural

numbers. The subject of degrees of unsolvability has been developed

into an extremely sophisticated branch of mathematical logic following

the article by Post and Kleene. To attempt a discussion of the substance

of the theory of degrees of unsolvability would exceed the bounds of

the present paper, and we will confine ourselves to using only those

items that bear on the relative recursiveness of rational choice,**

referring the interested reader to more comprehensive works.

A possible means of interpreting the assertion of Theorem I.1

is, that when recursively represented on subsets of the natural numbers,

a non-trivial rational choice function does not contain enough mathemat-

ical information in its graph to render the recursive solvability of

its graph. This is not to say that there may be in fact other subsets

of the natural numbers that do in fact contain enough mathematical

*The discussion in Rogers [19671, Ch.IX, pp.127-134 .

tCf. Ch. 13 of Rogers [19671.

**Gerald E. Sacks, Degrees of Unsolvability, Annals of Mathematics

No. 55, Princeton University Press, [1955], and Joseph R. Shoenfield [19711.

-6-

information. We are then led to the natural inquiry of just how much

mathematical information is required to recursively solve the graph

of a recursive rational choice function. What we wish to show in the

discussion that will follow is that by way of the notion of relative

recursiveness, it is possible to characterize which subsets of the

natural numbers contain enough information for the decision procedure

of recursively representable rational choice. Alternatively phrased,

we wish to inquire into its degrees of unsolvability.

We may first make the observation that there are three classic

notions of reducibility of decision procedures for subsets of the

natural numbers:

Definition I: A set A is generally recursive reducible to a

set B if A is recursive in B.

Definition II: A set A is Turing reducible to a set B if

there exists a Turing machine that reduces the decision procedure for

A to that of B. The reduction is performed by means of an oracle

that provides requisite information about the set B in the computation

of the set A as is required.t

Definition III: A set A is canonically reducible to a set

B if both A in IN-A are B-canonical sets in the sense of Post".

*This is due to Kleene, "Recursive Predicates and Quantifiers,"

Transactions A.M.S., Vol.53, [1945J, pp.41--73.

tCf. Rogers (1967], p.129.

**Post, [19.4.

. , . . .. .. . .r -S . . . . . . -- " : - : . . . . . .. ..

-7-

It is another significant feature of recursive function theory

that the above three notions of reducility of decision procedures are

all equivalent (cf. Post and Kleene [19541, p.379) and hence, no generality

is lost in considering sets of natural numbers that are relatively

recursive versus sets of natural numbers that are Turing reducible in

discussing results on degrees of unsolvability.

Let us next denote the fact that a set A is recursive in a set

B by the relation A < B . Then it can be shown* that for sets of-R

natural numbers the following items are true:

(i) VA[A < A]

(ii) VAVBVC[A < RB .A. B < C) => A < C1-A R -R

If we mean by A RB that both A < RB and B < BA, from items

(i) and (ii) we see that = is an equivalence relation among sets of

natural numbers fram which the following definition can be obtained.

Definition IV: The degree of unsolvability, with respect to

< R, of a set A is defined to be:

[A] ={BC N: A RB) dgA

In terms of the definition, the following facts concerning

degrees of unsolvability can be derived.

*Cf. Rogers [19671 p.78 or Schoenfield [1971] p.44 .

-8-

Proposition II.1 Allow A and B to be sets of natural

numbers. Then

(i) A R B <=> dgA = dgB

(ii) If a < b means that for some A and B a =dgA and

b=dgB, then dgA < dgB <=> A < B and dgA< dgB <=>-R

(A < RB -A. B RA)

We may view the concept of degrees of unsolvability in terms

of a sort of relativized Church's Thesis, in that if A < B maintains-R

then we should think of A as at least as easy to, or alternatively not

more difficult to compute by effective means as B is to compute by

effective means. From this, A = B would mean that A and B are

equally as difficult or equally as easy to compute by effective means.

It can also be observed that the relation < on the equivalence classes

under =R partially orders the set of degrees, an,! thus if the degree

of a set is a measure of the difficulty involved in effectively computing

that set, then the higher the degree under the order < , the more

difficulty involved in effective computation.

If it should happen that set A is recursive, i.e., E -7,0 0

then under the relation of relative recursiveness, for any set B, it

happens to be true that A < RB , and thus that dgA < dgB for any dgB.

*The W-w classification is the first classification in the Arithmetic

Hierarchy. Cf. Lewis [1981] Appendix I or Rogers [19671 Ch.14.

-. p -

-9-

Therefore, there is a smallest degree denoted as 0, and 0 is the degree

of every recursive set A, i.e., every E -7 set. On the other hand,0 0

if for some set A, it were true that dgA = 0, then the set is recursive,

i.e., E 0-w . The latter assertion follows from the fact that for any

recursive set B, A < RB Therefore we have the following

Proposition 11.2 A set of natural numbers A, is E7r 0

if, and only if dgA = 0

From the proposition, and the relationship it provides between

a recursive set of natural numbers and its degree of unsolvability we

are led, by way of the structure of the partial order on the set of

degrees, to certain deeper qualities of the Arithmetic Hierarchy by

associating degrees of unsolvability with levels and compartments in

the hierarchy of sets of natural numbers.

Let us now remark that Theorem 1.1 can be given an interpretation

at this point in terms of the theory of degrees that we have just

developed. The gist of what Theorem I.1 says is that if we view a non-

trivial recursive rational choice function as a correspondence between

classes of recursive sets the graph of the correspondence cannot be

recursive. By means of Theorem 4 Sec.5 p.24 of Sc) enfield [1971], if

G is the graph of a relation r, then dgG = dgr, and thus a further

interpretation of Theorem 1.1 is that, in viewing C as a correspondence,

since graph (C) is not recursive and the degree of a recursive set is 0,

dgC * 0 by way of Proposition 11.2 . In terms of a relativized Church's

Thesis, if we sa. that dgC * 0 then we merely say that the level of

. . . .. .- -- - - - - -

S.| . S I 5 I 'S l * , lI S * _ . . . .

-10-

difficulty incurred in an effectively computable realization of C is

not as easy as that of the recursive sets. It is desirable however, to

say more than this, to which purpose we now turn to the main result.

III. The Minimal Degree of Recursive Rational Choice

The purpose of this section is to provide a statement on the

lower bound of the degree of unsolvability of recursive rational choice

by means of associating degrees with the classification of sets provided

by the Arithmetic Hierarchy. Observe first, that an alternative means

of defining the components of the Arithmetic Hierarchy to that of using

Kleene strings*, as was done in Lewis [1981], can be obtained by

defining the En and Tn sets inductively as follows:

(i) The S-n sets are the recursive sets.0 0

(ii) A set is E if it can be defined as:

x EA <=> 3yk(,y) E B]

for B, a Tr set.

n

(iii) A set is nn+l

X EA <=> Vy[(x,y) E ]

for B, a E set.n

*A Kleene string is a quantified expression in the first order predicatecalculus of a recursive predicate on IN.

-11-

From this defining framework it is possible to prove the following

propositions by means of induction on the arity of the set*, and give

trules for when sets are E or w.n n

Proposition III.1: If A is Z (r) and B is defined byn n

X E B <> T(X) E A where T is a recursive function, then B is E n(wn).

Proposition 111.2: If A is E or Yt for some m < n,m- m

then A is E and wt.n - n

Proposition 111.3: If A is n(wn), then 11-A is n(E n).

Proposition III..: If A and B are En(w), then A U B

and ArlB are E (w).n n

The first three propositions are in fact properties of the Arithmetic

Hierarchy, and the last says that per fixed arity, the E and it setsn n

are closed under set operations. These properties are in turn useful in

enabling one to evaluate the degrees of the E and wn sets.n n

Lemma 111.5: A set of natural numbers A is E if andn+l

only if for some set B, A < ReB** and B is w n-- n

*The arity of a E or 7r set is n.n n

The proofs are found in Schoenfield [19711 pp.31-32.

**The relation A < ReB reads A is recursively enumerable in B and

weakens A < B . Cf. Schoenfield (1971] p.24.R

-12-

Proof: By definition, A < ReB is true when X E A <=> y[(X,y)] E B].

If A is E then by (ii) of the inductive definitions, A - ReB

n+l R

for Bir.pnConversely, suppose that A < B, and B is w and suppose

Re n

I indexes A in B, then X E A <[> XC E <=>B.A. X E Ta ,

where T yields elements in accordance with I with an oracle forI

B , and a is a finite sequence of elements in B. One can visualize

BTI as a Turing machine that lists the elements of A using information

about the set B. To see that A is En+l, it will suffice to show thata a

a C B and X E TI are En+ I . By way of Proposition III.4, X C TI

has a recursively enumerable graph and thus is EI. By Proposition 111.2

it is therefore Z n+. We observe next that a C B <=> Vn < ln(a)*

[Va(n) = TB(n)] where V and TB are the definig predicates of

a and B respectively. Since a is finite, by Proposition 111.4, it

will do to show that X = T B(n) is E n+I. However, the following

expression verifies this by way of Proposition III.4 directly:

x = B(n) <=> [(TB (n) = A. n EB).V. B(n) = 0A. n E B)] Q.E.D.

4 The next definition, when applied to the set of degrees, will

provide us with the means to make an assertion about the relative

* difficult in effectively computing recursive rational choice.

*1n(a) is the length of a

*

4 4

-13-

Definition V: For a set of natural numbers A , define the jump

or completion of A as:

A' = { O A(x)= I = X X E TAx X

A Awhere TA is the domain of A, for 0 a defining partial predicate

x X X*

for the set A with Gdel index X

The completion of a set A always has the property that A' < A,-Re

and that A < A' Furthermore, if A < B then A' < B'. From this-R R -R

latter fact when we consider the completion of the elements of the set

of degrees, starting with the minimal degree 0 , we can form an ascending

chain: 0 < R0 ' Of' < ' 0''O< R''' s''' < R0 '''"'' R0 1

This is significant in the light of the next lemma.

nLemma III.6: Let A be a E or a w set. Then dgA < 0,n n-

for 0n the nth completion of the recursive degree 0.

Proof: By induction, if we consider n = 0, if A is E - 0o

the result is trivial as dgA = 0 necessarily. Assume then that the

Lemma is true for wn sets, then from taking Lemma 111.5 forward, andn

by the properties of the completion operator if A were E n+l~ and B""

were rn A < B implies that dgA < dgB < On. The case for E setsn - - - n

can be obtained from Proposition 111.3 by means of the same argument. Q.E.D.

* Cf. Rogers [1967], p.132, p.135, and p.255.

tCf. Rogers op cit Theorem 1(a), p.255.

-14-

Definition VI: A set is said to be a complete Z set (or completen

wn set) if A is E (or wn), and if B is an arbitrary E set

(or w set) B < RAn -

We now demonstrate the main result of the present paper.

Theorem III.T : Allow C : F Fi to be a nontrivial recursive

rational choice on a recursive space of alternatives <R(x), :F >

R

derived from the recursive metric space of R n, M(IRn) for R(X), the

recursive representation of a compact, convex subset of Rn. Then for

any fixed choice from (F) of full domain, the degree of unsolvability

of graph (C) and therefore that of C , viewed as a correspondence on

02* sequences in (F) cannot be less than 0

Proof We begin with the observation that the following

concept of strong reducibility implies general recursive reducibility

of Df. I.

Definition VII: Allow l and 0 to be elements of U P(INJ)*j12

such that *i is n-aky, and 2 is m-dky. Then i is strongly

reducible to 9 written 9 << 9 if there are partial recursive2 1 2

n-4Ay functions fl...' fm for which 0 C Xt(X...,X) (f

*** fm(X 1""'Xn)

We next observe that the following concept of strictness implies

completeness as described in Df. VI.

I|

-15-

Definition VIII: An element of U p(jJ) , is said to be

strictly 7K (or EK) if 0 is VK (or EK), and if any other

A E U P(IN J ) is such that A << .

jEIN

Then, from the fact that strong reducility implies general recursivereducibility if a relation is strictly fK (or K ), then its graph,

in accordance with Df. VI is a complete wK (or E K) set. We have

shown in Lewis [1981], Theorem IV.4, however, that the non-recursiveness

of the graph of a recursive rational choice function per fixed choice of

full domain, occurs in fact because its codomain is strictly E2' and thus

in light of the above equivalence, the codomain of graph (C) is a restriction

of graph (C), i.e. codomain graph (C) C graph (C) in the sense of Df.X

of Lewis [19811, it follows that the degree of unsolvability for graph (c)

cannot be less than the degree of unsolvability of the codoamin. For,

allow graph (r) to be the codomain of graph (C). Then from Theorem 4

of section 5, p.24 of Schoenfield [1971], dg(r) - dg(graph (r)) and

2since dg(graph (F)) = 0 by Lemma 111.6 if graph (r) is a complete 22

set, then, from the fact that r C C, graph (C) = graph (r) U graph (C - F)

and by Proposition 111.2 and Proposition 111.4 applied in successsion,

* U P(N ) is the class of all relation on IN.jem,

tThis is Church's A-notation, and AXO(x) is read "the partialfunction <X,y> that gives y as a value when X takes an integervalue.

=~

L,"

-16-

dg(graph (C)) > 02. The assertion for C viewed as a correspondence

from F R to F follows from another application of Theorem 4 ofR R

Schoenfield, by which dg(graph (C)) = dgC and thus dgC _ 02. Q.E.D.

We have not, as of yet, verified a somewhat natural conjecture

that the degree of unsolvability of a recursive representation for

2rational choice can be no more than 0 If true, by precisely bounding

the degree of recursive rational choice in this fashion, further connections

would be possible within the realm of contemporary theoretical computer

2science, as 0 happens to be the degree of unsolvability of the inherent

ambiguity problem of computer science, and also coincides with the

decision degree of finite classes*.

*Cf. A. Ready and W. Savitch, "The Turing Degree of the Inherent

Ambiguity Problem for Context-Free Languages", Theoretical ComputerScience, Vol.1, L19761, pp.77-91, and Rogers [1967], pp.264-265.

CI

-17-

References

Kleene, Stephen Cole, and Enil L. Post [19541, "The Upper Semi-Latticeof Degrees of Recursive Unsolvability", Annals of Mathematics,Vol. 59, pp.379-402.

Lewis, Alain A., June [19811, "Recursive Rational Choice," Institute for

Mathematical Studies in the Social Sciences Technical Report No. 355,Stanford University.

Post, Eil L. [19441, "Recursively Enumerable Sets of Positive Integers

and Their Decision Problems", Bulletin A.M.S., Vol.50, pp.28 4-316 .

Richter, Marcel K. [1971], "Rational Choice", in Preferences, Utility

and Demand, edited by John Chipman, et al., Harcourt Brace,New York, pp.29-58.

Rogers, Hartley Jr. [19671, The Theory of Recursive Functions andEffective Computability, McGraw-Hill, New York.

Schoenfield, Joseph R. [1971], Degrees of Unsolvability, North Holland

Mathematics Studies, No.2, American Elseviar, New York.

I.

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4.0 .30 -O . Uj~~1E E - .k.

1 E.o 2 0 2 G* .-K a. Z0.

I ~wo ; tr:~ I&- ~ & *~-

* 0E -. I2 0EaC EN S od

jZ~~~~~ <~isr 'E~ j1 0

0. C&jL -.

5 1 - z0a -o J.t0..oo2t.

.0 S o -aZ ,0.4 & . - g & .oa.~~.- ,x .6.~.i2 ~ 0. 0 C . d . ~4

. . ..2

W 0.s 8 o .. .. a. .2 .-. 2.. .. . . .

Aa A A.t~4 I 2~~ . A 9

~ .4 C0C0 C = 0 . .4t 8 0~ 00.4 4 <<luEcc wc5~ (5 g-a~~ 2. 8 E

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5 .0 0.. 3h C:0~632r 5c .L.J~d.4.00.E~o0-2~.4JJ O.4ttd

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344. "Three Lectures In Monetary Theory," by Frank H. Hahn.

345. "Socially Optimal Investment Rules in the Presence of IncompleteMarkets and Other Second Best Distortions," by Frank Milne andDavid A. Starrett.

346. "Approximate Purification of Mixed Strategies," by Robert Aumann,Yitzhak Katznelson, Roy Radner, Robert W. Rosenthal, and BenjaminWeiss.

347. "Conditions for Transitivity of Majority Rule with AlgorithmicInterpretations," by Herve Raynaud.

348. "How Restrictive Actually are the Value Restriction Conditions,"by Herve Raynaud.

349. "Cournot Duopoly in the Style of Fulfilled Expectations Equilibrium,"by William Novshek and Hugo Sonnenschein.

350. "Law of Large Numbers for Random Sets and Allocation Processes,"

by Zvi Artstein and Sergiu Hart.

351. "Risk Perception in Psychology and Economics," by Kenneth J. Arrow.

352. "Shrunken Predictors for Autoregressive Models," by Taku Yamamoto

353. "Predation, Reputation, and Entry Deterrence," by Paul Milgrom and

John Roberts.

354. "Social and Private Production Objectives in the Sequence Economy"by David Starrett

355. "Recursive Rational Choice" by Alain Lewis

356. "Least Absolute Deviations Estimation for Censored and TruncatedRegression Models" by James Powell

357. "Relatively Recursive Rational Choice" by Alain Lewis

I.