Decision theoretic foundations for axioms of rational preference

SVEN OVE HANSSON

DECISION THEORETIC FOUNDATIONS FOR AXIOMS OF RATIONAL PREFERENCE

ABSTRACT. Rationality postulates for preferences are developed from two basic decision theoretic principles, namely: (1) the logic of preference is determined by paradigmatic cases in which preferences are choice-guiding, and (2) excessive comparison costs should be avoided. It is shown how the logical requirements on preferences depend on the structure of comparison costs. The preference postulates necessary for choice guidance in a single decision problem are much weaker than completeness and transitivity. Stronger postulates, such as completeness and transitivity, can be derived under the further assumption that the original preference relation should also be capable of guiding choice after any restriction of the original set of alternatives.

i. INTRODUCTION

Rationality postulates for preferences such as transitivity and completeness have a central role in moral philosophy, economics, and decision theory. In spite o f this, arguments in favour of such postulates are hard to come by, and direct appeal to intuition has become a common resort.

The purpose of this essay is to show how some rationality postulates in preference theory can be argued for with decision theoretic arguments. A pivotal argument will refer to the needs of a decision-maker who foresees that some of the presently available altematives may be lost in the future.

Throughout the essay, P will denote strict preference ('better than'), I will denote indifference ('equal in value to'), and R will denote weak preference ('at least as good as'). The three relations are connected by the two logical relationships AIB ~ ARB & BPu4 and APB ++ ARB & --n(B.P~4).

Preferences are assumed to refer to a set #t of altematives. In Section 2, two basic decision theoretic principles for preferences

are introduced. In Section 3, these principles are applied to preferences that refer to a fixed set of altematives, and in Section 4 changes of the set o f akernatives are taken into account. In Section 5, the argument is summarized and the decision theoretic approach is defended.

Synthese 109: 401--412, 1997. I~) 1997 Kluwer Academic Publishers. Printed in the Netherlands.

402 SVEN OVE HANSSON

2. DECISION THEORETIC PRINCIPLES FOR PREFERENCES

The following principle is commonly assumed to hold for rational preferences. 1

Preference~choice identity: Preference is hypothetical choice.

According this principle, I prefer A to B if and only if, were I to choose between A and 8, I would choose A. 2 Preference/choice identity has immediate consequences for the logic of preferences:

Choice-guiding principle: The logical properties of preferences should be compatible with using preferences as guides for choices.

Although preference/choice identity is mostly taken for granted, it is not in general a plausible principle. We often enter tain- and express-preferences in matters in which we have no choice. It is in many cases unnatural, and in some cases outright wrong, to interpret such preferences as direct expressions of hypothetical choice.

As an example, when comparing the prizes of a lottery, you may well prefer the possible event of winning • to that of winning B, although if you were allowed to choose between A and B, you would choose B. The act of choosing A may have negative characteristics (such as shame at choosing something useless) that the event of winning A does not have. Since you cannot, by definition, choose to win, preferences such as these cannot be choice-guiding. They do not reveal dispositions to choose or to act. 3

We must conclude that preference/choice identity is untenable. This also puts the choice-guiding principle at risk. The latter can, however, be saved if we accept a weaker and much more plausible premiss:

Preference/choice paradigm Our intuitions about the structure of preferences are determined by paradigmatic cases in which preferences are used to guide choices.

In many - but not all - cases our preferences serve to guide our choices and decisions. This usage is so common and so dominant that it has the role of a paradigm. The structural requirements on preferences derive from the paradigmatic cases. By analogy or habit of mind we expect the non-paradigmatic cases to satisfy requirements whose rationale is, strictly speaking, only present in the paradigmatic cases. For this reason,

DECISION THEORETIC FOUNDATIONS FOR AXIOMS OF RATIONAL PREFERENCE 403

the choice-guiding principle seems to hold for all preferences, although not all preferences are choice-guiding.

Let us consider another lottery example. Suppose that there are three prizes, A, B, and c. You prefer the event of winning A to that of winning B, that of winning B to that of winning c, and that of winning c to that of winning A. Preferences with this logical structure cannot be used to guide choices. In this case, choice-guiding is precluded by the nature of the alternatives, but nevertheless the logical structure is conceived as unsatisfactory since you have not 'made up your mind' sufficiently on how you compare the three alternatives. The reason for this, I propose, is that we have a strong tendency to apply requirements that derive from the paradigmatic cases to non-paradigmatic cases as well.

The distinction between the principles of preference/choice identity and preference/choice paradigm may seem unimportant to readers accustomed to regard the former as uncontroversial. In what follows I will use only the choice-guiding principle, that follows from either of them. Since valid criticism can be raised against preference/choice identity, the argument has been strengthened by replacing it by the more plausible principle of preference/choice paradigm.

We were not born with a full set of preferences, sufficient for the vicissitudes of life. Most of our preferences are the outcome of postnatal mental processes. Many of these require mental efforts: the acquisition of preferences may cost time and effort. A rational subject should avoid comparison costs that are not matched by the expected advantages of having more determinate or more well-founded preferences. 4 This is what the following principle demands:

Comparison cost avoidance: A rational subject avoids excessive comparison costs.

An immediate consequence of comparison cost avoidance is that preferences cannot in general be taken to satisfy completeness (ARB V BRA). This conclusion is at variance with tradition in formal studies of preference, 5 but it conforms with actual patterns of preference. As an example, in the choice between three brands of tinned soup, A, B, and c, I prefer A to both B and c. As long as A is available I do not need to make up my mind whether I prefer B to c, prefer c to B, or consider B and c to be of equal value. Similarly, a voter in a multi-party election can do without ranking the parties or candidates that she or he does not vote for. 6

In what follows, the two principles of choice-guiding and comparison cost avoidance will be used to derive rationality postulates for preferences.


3. RATIONALITY POSTULATES FOR A FIXED ALTERNATIVE SET

For a preference relation to be choice-guiding, it must supply at least one alternative that is eligible, that can reasonably be chosen. The minimal formal criterion for eligibility is that the chosen alternative is no worse than any other alternative:

Weak eligibility: There is at least one alternative ~t such that for all B, -~(BPA).

Preference patterns that do not satisfy weak eligibility are not effective choice-guides. Suppose, for instance, that a subject has the three alternatives A, B, and c to choose between. If she strictly prefers A to B, B to c, and c to ~t, then the choice supported by her preferences will depend on at what stage she halts her deliberations. As was noted with respect to the lottery example, preferences with this structure are not useful to guide choices.

An alternative A such that for all B, -~(BPA), wilt be called a weakly eligible alternative. (Even if the subject's choice is guided by the preference relation, a weakly eligible alternative need not be a chosen alternative. She may pick one out of several eligible alternatives .7)

Let A be a weakly eligible alternative, and let B be an alternative that is not (weakly) eligible. Furthermore, suppose that A and B are ' comparable', that either ,4P8, AIB, or BPA holds. 8 Then, by the definition of weak eligibility, BPA does not hold. It would also be strange for A and B to be equal in value, for AIB to hold. If preferences are choice-guiding, then two alternatives should not be considered to be of equal value if one of them is eligible and the other is not. We may therefore conclude, as a consequence of the choice-guiding principle, that if A but not B is weakly eligible, then a and B are not equal in value. In an equivalent formulation:

Top-transitivity of weak eligibility: ffMB, and -~(CPA) for all c, then -~(cPa) for all c.

An alternative may be weakly eligible in spite of the fact that it has not been compared to all other alternatives. This corresponds to actuN patterns of choice-guiding preferences. The supermarket has about ten brands of tomato ketchup. I do not know that any of these is better than the one that I buy, but quite a few of them I have never tasted. In this case, comparison costs are so high (compared to the supposed gains from making a better choice) that I am satisfied with only weak eligibility.

However, there are other situations in which the subject wants to have compared the chosen altemative to all other alternatives, in spite of the comparison costs involved. In particular, this holds when the alternatives


are so few and the expected differences between them so big, that the expected comparison costs are low in relation to the expected gains from a more well-informed decision. In such cases at least one weakly eligible alternative should be comparable to all other alternatives, or equivalently:

Strong eligibility: There is at least one alternative A such that for all B, A/~.

An alternative A such that for all B, ARB will be called a strongly eligible altemative.9

Suppose that A is strongly eligible and that B is not. Choice-guiding preferences should not then put A and ~ on an equal footing. Hence it should not be the case that AIB. In an equivalent formulation:

Top-transitivity of strong eligibility: IfMB, and ARc for all c, then BRc for all c.

Both weak and strong eligibility allow for the existence of more than one (weakly, respectively strongly) eligible alternative. In general, this is just as it should be. Choice-guiding preferences may leave us with several alternatives, each of which may be chosen. However, situations are imag- inable in which the preference relation should provide exactly one eligible alternative. (A possible example could be one in which it is important to be capable of a quick decision, so that hesitation between several optimal alternatives must be avoided.) Then the eligibility conditions should be strengthened as follows:

Weak exclusive eligibility: There is exactly one alternative A such that for all B, --,(sPA).

Strong exclusive eligibility: There is exactly one alternative A such that for all B, ARB.

4. RESTRICTABILITY

The sets of alternatives that our preferences refer to are not immutable. To the contrary, new alternatives can become available, and old ones can be lost. Such changes may force us to renewed deliberations, l°

The reopening of an issue can be uneconomical in terms of comparison costs. In other words, comparison costs may be so structured that the costs of reconsidering an issue is high, whereas the marginal costs of making


one's deliberations sufficient for possible future changes of the alternative set are low. In such cases, the principle of comparison cost avoidance urges the subject to perform her original deliberations thoroughly enough to minimize the risk of having to reopen the issue. There are two principal means to achieve this, namely (1) to include potential new alternatives in the original set of alternatives, and (2) to make sufficient comparisons to ensure that (weak or strong) eligibility holds not only for the original set of alternatives but also if one or several of the original alternatives are lost.

If no alternative is considered to be exempt from possibly being lost in the future, it may be a cost-minimizing strategy to pursue one's deliberations until (weak or strong) eligibility holds for all non-empty subsets of the alternative set. A rationality criterion will be said to hold restrictably for a set of alternatives if and only if it holds for all non-empty subsets of that set of alternatives.11

As will be seen from the following theorem, if the eligibility properties from section 3 are required to hold restrictably, then we obtain rationality criteria of the more well-known types, such as completeness, acyclicity, and various types of transitivity.

THEOREM. Let R be a preference relation over some finite set ~4 of alternatives.

1. It satisfies restrictable weak eligibility if and only if it satisfies acyclicity. ~2

2. It satisfies restrictable strong eligibility if and only if it satisfies completeness and acyclicity.

3. It satisfies restrictable top-transitive weak eligibility if and only if it satisfies acyclicity and PI-transitivity.

4. It satisfies restrictable top-transitive strong eligibility if and only if it satisfies completeness and transitivity.

5. It satisfies restrictable weak exclusive eligibility if and only if it satisfies completeness, acyclicity, and asymmetry.

6. It satisfies restrictable strong exclusive eligibility if and only if it satisfies completeness, acyclicity, and asymmetry. 13

Acyclicity is the property A1PA2 & A2PA3 & . . . An-IPAn -+ --'(AnPA1). PI-transitivity is APB & MC --+ APc. Asymmetry is ARB --+ ~BRA. The proof of the theorem is elementary and based on well-known principles. It can be found in the Appendix.


TABLE I

Possible structures of comparison costs

Net loss from Net gain from

covering all possible covering all possible

future losses of future losses of

alternatives alternatives

Net loss from

considering all

alternatives

Top-transitive weak Restrictable top-

eligibility transitive weak

eligibility

(= acyclicity

+ PI-transitivity)

Top-transitive strong Restrictable top-

eligibility transitive strong

eligibility

(= completeness

+ transitivity)

Net gain from

considering all

alternatives

5. CONCLUSION

Top-transitive weak eligibility is a fairly weak requirement on a choice- guiding preference relation. Depending on the structure of the comparison costs, additional logical requirements may be warranted, in particular (1) top-transitive strong eligibility, and (2) restrictability. This leaves us with four major possibilities that are summarized in Table I.

It is the major claim of this paper that the demands of rationality on the logical structure of preferences depend on the decision theoretic context. In cases when preferences are intended to be choice-guiding, rationality postulates should be chosen according to decision theoretic criteria of the types summarized in the table. In the non-paradigmatic cases when preferences are not choice-guiding, only weak postulates that are common to all paradigmatic cases are applicable. Hence, it can be required that non- choice-guiding preferences should satisfy top-transitive weak eligibility, but not necessarily any stronger set of postulates.

Only under special conditions are completeness and transitivity required by rationality. They are so when (1) preferences are used to guide choices, (2) a net gain is expected from considering all the alternatives, and (3) a net gain is expected from covering all possible future losses of alternatives.

It might be protested that this use of decision theoretic arguments is inadmissible since it puts the cart before the horse. Traditionally, preference

408 SVENOVE HANSSON

theory is seen as one of the ground-pillars on which decision theory is buik. To use decision theoretic arguments at the foundations of preference theory would therefore disturb the hierarchical structure of disciplines.

In my view, this hierarchical structure is chimerical. In all arguments for or against logical postulates for preferences, we discuss which of them are better than the others, and our problem is a decision prob lem- we have to decide what logic to use. In analysis or regimentation of basic concepts, this type of (semi-)circularity seems unavoidable. When developing a regimented object-language version of a fundamental concept, we cannot avoid non-regimented uses of essentially the same concept in the meta-language. This is a pervasive feature of conceptual analysis. In this essay, it may have been more conspicuous than in many other such undertakings, but its conspicuousness does not necessarily make it more vicious.

I would like to thank an anonymous referee for valuable comments on an earlier version of this paper.

APPENDIX: PROOF OF THE THEOREM

Part 1: For one direction, suppose that acyclicity does not hold. Then there are A 1, • • • A,~ E .4 such that .41PA2,..., An- 1 PAn and AnPA 1. Weak eligibility is not satisfied for the subset {`41, • • •, An} of.4.

For the other direction, suppose for reductio that acyclicity but not restrictable weak eligibility is satisfied. We are going to show that .4 is infinite, contrary to the assumptions. Since restrictable weak eligibility is violated, there is some subset B o f .4 for which weak eligibility does not hold. Let 4̀1 E B. Since weak eligibility is not satisfied, there is some A2 E/3 such that `42P`41. Similarly, there is some A3 such that A3PA2, etc . . . . If any two elements on the list &, `42, A3 • .. are identical, then acyclicity is violated. Thus, B is infinite, and consequently so is .4, contrary to the conditions.

Part 2: For one direction, suppose that R satisfies restrictable strong eligibility. To see that it satisfies completeness let .4, B E .4. Since strong eligibility holds restrictably for .4, strong eligibility holds for the subset {`4, B} of.4, so that either ̀ 4RB or BRA. Since restrictable strong eligibility implies restrictable weak eligibility, acyclicity follows from Part 1.

For the other direction, suppose for reductio that R is complete and acyclic but violates restrictable strong eligibility. There is then some subset /3 of.4, for which strong eligibility does not hold. Let`41 C/3. There is then some ̀ 42 E/3 such that -~(̀ 41 R`42). By completeness,`42 P`41. Similarly, there is some A3 E/3 such that -~(`42RA3) and consequently A3P`42, etc. Suppose


that any two elements of the list A 1, A2, A3,... are identical. Then acyclicity is violated. Thus B is infinite, and so is -4, contrary to the conditions.

Part 3: First suppose that R satisfies restrictable top-transitive weak eligibility. Acyclicity follows from Part 1. For PI-transitivity, let A, e, and c be three elements of -4 such that APe and Mc. Suppose that -%4Pc). Then -,(xPc) for all x E {A, e, c}, and by top-transitive weak eligibility for that set e l c yields -~(xPB) for all x E {A, e, c}, contrary to xPe. We may conclude that APc.

For the other direction, suppose that acyclicity and PI-transitivity are satisfied. It follows from Part 1 that R satisfies restrictable weak eligibility. For restrictable top-transitivity of weak eligibility, let B be a subset of .4 and A and e two elements of/3 such that Ale and that for all x E/3, -~(xPe). For reductio, suppose that for some c, cPA. Then it follows by Me and PI-transitivity that cPe, contrary to the conditions. We may conclude that ~(XPA) holds for all x E/3, so that top-transitivity of weak eligibility holds for R in B. Since this applies to all subsets/3 of.4, top-transitivity of weak eligibility holds restrictably in .4.

Part 4: First suppose that restrictable top-transitive strong eligibility is satisfied. Completeness follows by Part 2. For transitivity, let ARe and ere . Since top-transitivity of strong eligibility holds restrictably for .4, top-transitivity of strong eligibility holds for {A, e, c}. There are three c ase s:

Case g AIe: By completeness, e r e , so that e/Lv for all x E {A, e, C}. It follows by top-transitivity of strong eligibility, as applied to {A, e, c}, that ARx for a l lxE {A, e, C}, SO thatARc.

Case ii, APe and ePc: Suppose that CPA. Then strong eligibility does not hold for {A, e, C}, contrary to the conditions. It follows that ~(cPA) and by completeness that ARC.

Case Jig APB and MC: Suppose that cRA. By completeness cRc, so that cRx for all x E {A, e, c}. By top-transitivity and Mc, e/Lv, for all x E {A, e, c}, so that eRA, contrary to APe. By this contradiction, -~(cRA). By completeness, ARc.

For the other direction, suppose that completeness and transitivity are satisfied. Transitivity implies acyclicity, so that restrictable strong eligibility follows from Part 2. For restrictable top-transitivity of strong eligibility, let B be a subset of .4 with A, g E/3 and such that Ale and that ARc for all x E/3. Then for all x, eRA and ARc yield eRc, so that top-transitivity of strong eligibility holds in/3. Since this applies to all subsets/3 of -4, top-transitivity of strong eligibility holds restrictably in A.

Part 5: For one direction, suppose that R satisfies restrictable weak exclusive eligibility. It follows by Part 1 that acyclicity is satisfied.

410 SVENOVE HANSSON

To see that asymmetry holds, suppose to the contrary that this is not the case. Then there are elements ,4 and B of ,A such that ARB & BRA. Then ~(APB) & ~(BPA), contrary to the weak exclusive eligibility of the subset {A, s} of A.

To see that completeness holds, suppose to the contrary that there are elements A and B of A such that -,(ARe) & -,(eRA). This too implies -~(APe) & -,(ePA), contrary to the weak exclusive eligibility of {A, e}.

For the other direction of the proof, suppose that completeness, acyclicity and asymmetry are satisfied. It follows from Part 1 that restrictable weak eligibility holds. Suppose that restrictable weak exclusive eligibility does not hold. Then there is a set/3 C ~4 with two elements A1, A2 E /3, such that -~xP& and -~xPA2 both hold for all x E/3. It follows that -~A2P& and -,A1PA2. By completeness, -~A2P.q yields A1RA2 and -~A1PA2 yields A2-RAI. By asymmetry, A1RA2 --+ -~A2RA1. It follows by this contradiction that weak exclusive eligibility holds.

Part 6: For one direction, suppose that R satisfies restrictable strong exclusive eligibility. It follows from Part 2 that completeness and acyclicity are satisfied, For asymmetry, suppose to the contrary that it does not hold. Then there are A and e in .A such that ARe and eRA. By completeness, ARA and e_Re. This contradicts the exclusive strong eligibility of {A, e}. It can be concluded that asymmetry holds.

For the other direction of the proof, suppose that completeness, acyclicity, and asymmetry are satisfied. It follows from Part 2 that restrictable strong eligibility holds. Suppose that restrictable exclusive eligibility does not hold. Then there is a set/3 C_ ,,4 with two elements A 1, A2 E/3, such that for all x E/3, A1/Lg and A2/Lr. It follows that A l RA2 and A2RA 1, contrary to asymmetry. This contradiction concludes the proof.

NOTES

1 For discussions of the relation between choice and preference, see Sen (1973), Broome (1978), Reynolds and Paris (1979), Hansson (1988), and Hansson (1992). z Conversely, it is common to identify choice with revealed preference. 3 For arguments against the converse thesis that choice is revealed preference, see Hansson (1988) and (1992). 4 Cf. Halldin (1986). Halldin does not distinguish between decision-costs and comparison- costs. 5 Most studies of preference logic have assumed that weak preference is complete. (The major exception is W. E. Armstrong. See, e.g., Armstrong 1950.) This is also a standard assumption in applications of preference logic to economics and to social decision theory. In economics it may reflect a presumption that everything can be "measured with the mea- suring rod of money" (Broome 1978, p. 332). 6 These examples also show that comparability need not be transitive, i.e., B may be corn-

DECISION THEORETIC FOUNDATIONS FOR AXIOMS OF RATIONAL PREFERENCE 4 1 1

parable to both A and C without A being comparable to C. (Cf. Davidson et al. 1955, p. 146.) 7 Ullmann-Margalit and Morgenbesser (1977). 8 "Comparability" is the established term for this property (equivalently ARB V BRA). The misleading nature of this term is particularly evident from its negation. "Incomparability" of the two alternatives A and B does not mean that it is impossible to compare A and B, only that a comparison of them with a determinate outcome has not been performed. ("Uncompared" would have been a somewhat better term.) 9 Herzberger (1973, p. 197) used the terms "liberal maximalization" and "stringent maximalization" for essentially the same concepts that I call weak and strong eligibility. 10 Some aspects of changes in preference are discussed in Hansson (1993) and Hansson (1995). 11 Restrictable strong eligibility may also be expressed as the existence of a well-defined choice function G based on R, such that for all B C_ A:

(I) G(B) = {x C B[ For all Y E B, xRY}, and

(2) if/3 ¢ (~, then C(13) ¢ 0.

12 Another formulation of this part of the theorem is given as Theorem 1 in Jarnison and Lau (1973). 13 This part of the theorem essentially coincides with Theorem 4 of Sen (1971), although the interpretation is different. (Sen takes a choice function as primitive, and his "preference relation" is the ranking that can be derived from that function.)

REFERENCES

Armstrong, W. E.: 1950, 'A Note on the Theory of Consumer's Behaviour', Oxford Eco- nomic Papers 2, 119-22.

Broome, J.: 1978, 'Choice and Value in Economics', Oxford Economic Papers 30, 313-33. D avidson, D. et al.: 1955, 'Outline of a Formal Theory of Value I', Philosophy of Science

22, 140-60. Halldin, C.: 1986, 'Preference and the Cost of Preferential Choice', Theory and Decision

21, 35--63. Hansson, S. O.: 1988, 'Rights and the Liberal Paradoxes', Social Choice and Welfare 5,

287-302. Hansson, S. O.: 1992, 'A Procedural Model of Voting', Theory and Decision 32, 269-301. Hansson, S. O.: 1993, 'Money-Pumps, Self-Torturers, and the Demons of Real Life',

Australasian Journal of Philosophy 71,476-85. Hansson, S. O.: 1995, 'Changes in Preference', Theory and Decision 38, 1-28. Herzberger, H. G.: 1973, 'Ordinal Preference and Rational Choice', Econometrica 412,

187-237. Jamison, D. T. and J. L. Lau: 1973, 'Semiorders and the Theory of Choice', Econometrica

41, 901-12. Reynolds, J. F. and D. C. Paris: 1979, 'The Concept of 'Choice' and Arrow's Theorem',

Ethics 89, 354-71.

412 SVEN OVEHANSSON

Sen, A.: 1971, 'Choice Functions and Revealed Preference', Review of Economic Studies 38, 307-17.

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Ullmann-Margalit, E. and S. Morgenbesser: 1977, 'Picking and Choosing', SocialResearch 44, 757-85.

Uppsala University Department of Philosophy Villav~igen 5 S-752 36 Uppsala Sweden E-mail: [email protected].

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Decision theoretic foundations for axioms of rational preference