Transitivity and consistency

315

Economics Letters 33 (1990) 315-317 North-Holland

TRANSITIVITY

AND CONSISTENCY

*

Nick BAIGENT Tulane University, New Orleans, L.A 701 IS, USA Received 23 October 1989 Accepted 18 December 1989

In the presence of the Weak Chemoff property, the expansion consistency property known as /3 characterises transitivity of the base relation. It is shown that p may be replaced by a new property. The new property is a substitution property. Since. it IS not an expansion consistency property, it is argued that it is some form of equivalent treatment rather than expansion consistency that is the essential requirement for transitivity of the base relation. The content of this paper is taken from the section on collective rationality in Baigent (1989) which contains more results similar to the one offered here.

1, Introduction Choice consistency has at least two distinct aspects. One is concerned with whether or not choices are rationalisable by a preference. The other is concerned with pair-wise choices over triples of alternatives and takes the form of well-known properties such as transitivity and quasi transitivity. For some purposes it is the latter, and not the former, which is important. In Arrow’s famous impossibility theorem for example, the transitivity of choices is what precipitates the result, independently of whether the choices are rationalisable by a preference. 1 Therefore, it is important to know exactly what sort of consistency is associated with transitive and quasitransitive choices. The existing literature appears clear on this question. ’ Contraction consistency is required for the weakest such property, triple acyclicity. Anything beyond that, such as quasi transitivity or transitivity, requires some form of expansion consistency. 3 The result presented in this paper shows that this is not, in fact, the case. In a new characterisation of quasi transitive and transitive choices, the usual expansion consistency condition will be replaced by a substitution condition. This suggests that it is not expansion consistency but some sort of ‘equal treatment’ property that is associated with quasi transitive or transitive choices. Since the choices in this paper need not necessarily be rationalisable by a preference, the quasi transitivity and transitivity of choices can take different forms. Here, attention is focussed on pair-wise choices rather than choices from larger sets of alternatives. That is, the result is stated in terms of the base relation rather than the revealed preference relation. However, an exactly analagous result has been proved in terms of the revealed preference relations. 4 * I am very grateful to Kotaro Suzumura for helpful conversations. ’ Sen (1977, 1986). : For especially good surveys of these results see Sen (1986) and Suzumura (1983). The analysis of properties of choice functions in terms of expansion and contraction consistency conditions was introduced in Sen (1977). 4 Baigent (1989). Ol65-1765/90/$03.50

0 1990 - Elsevier Science Publishers B.V. (North-Holland)

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316

The exposition begins by describing the formal framework. This is followed by the axioms and the result. A short discussion concludes the paper.

2. The analysis Let X denote a finite set of mutually exclusive alternatives where 1X 12 3, and let II(X) denote the set of all non empty subsets of X. A choice function is a function C : II(X) -+ II(X) such that, for any A E II(X), C(A) c A. For any choice function C on X, the base relation of C is the binary relation R, on X defined as follows: For all x, y E X, xR,y if, and only if, x E C({ x, y}). The asymmetric and symmetric component of R, will be written P, and I,, respectively. They denote strict preference and indifference in the usual way. A binary relation R on X is transitive if, and only if, for any x, y, z E X, xRy and yRz implies xRz; it is quasi transitive if its asymmetric component is transitive and it is triple acyclic if, xPy and yPz implies xRz, where P is the asymmetric component of R. The first property is a weak version of what has been called the Chernoff property and also property 0~. It may be expressed as follows: Property C({x,

WC

(Weak

Chernoff ).

For all x, y, z E X, x E C({ x, y, z })nC({

x, y }) implies x E

z)).

This property is a weak version of a standard contraction consistency condition and is very well known. In particular, it is known that the base relation of any choice function that it satisfies is triple acyclic. The following expansion consistency property is also well known: Property p. For any x, y E X: C({ x, y}) = (x, y) implies x, y EA, either x, y E C(A) or x, y E (A - C(A)).

that, for all A E n(X)

such that

It is well known that a choice function satisfying WC and j3 must have a base relation that is transitive. In the same way, given the WC property, the following expansion consistency property is known to characterise quasi transitive choice: Property 6.

For all x, y E X and all A, B E II(X):

x, y~Band

BcAimplies{y}#C(A).

In the characterisations of transitivity and quasi transitivity, these expansion consistency properties may be replaced by the following substitution properties: Property YE

S (Substitution).

For all x, y, z E X: if {x, y } = C({ x, y }) then, x E C({ x, z }) implies

C({x, z>).

Property WS (Weak Substitution). C({ x, y }> implies y E C({ y, z I>-

For

all

x, y, ZEX:

if

{x, y} = C({x,

y)>

then,

{X> =

The substitution property requires that alternatives that are equivalent in pair-wise choices may be substituted for each other in pair-wise choices against third alternatives. Clearly, WS is strictly weaker than S. In comparing Properties p and S, it is important to emphasise both the similarity and the difference. They are different in that they focus on quite different things, namely, expansion consistency and substitution. On the other hand, they are similar in that both involve some kind of

N. Baigent / Transitivity

and consistency

317

equivalent treatment. Property fi involves what Arrow and Hurwicz (1977) called optimal equivalence and it requires that if two alternatives are treated equivalently in pair-wise choice then they should always both be chosen or both rejected whenever both are available. Property S on the other hand, requires that if two alternatives are treated equivalently in pair-wise choice, then the choice of either one against a third alternative is invariant with respect to substitution by the other. Thus, in the statement of Property S above, x and y are treated equivalently in choices of either against z. For an extended discussion of equivalent treatment see Baigent (1989). The result may now be stated as follows: Theorem 1. A choice function has a transitive base relation if, and only if, it has properties it has a quasi transitive base relation if, and only if, it has properties WC and WS.

WC and S;

Proof. Only the sufficiency of WC and S (WS) for transitivity (quasi transitivity) of the base relation will be proved, necessity being transparent. To prove sufficiency, assume that xR,y and yR,z, and consider the following possibilities which are mutually exclusive and exhaustive: (i) xP,y and yZcZ; (ii) xZ,y and yP,z; (iii) xZ,y and yZ=z; (iv) xP,y and yP,z. In case (i), if xP,x then yR,s would follow from Property S. Since this contradicts xP,y, it must be the case that xP,z. In case (ii), xR,z follows from Property S. If xZcz, then zR,y would follow from Property S. Since this contradicts yP,z, it must be the case that xP,z. In case (iii), xR,z and zR,x both follow from Property S. Therefore, it must be the case that xZ~Z. In case (iv), xR,z follows from the fact that the triple acyclicity of R, is implied by Property WC. If xZ~Z, then zR,y would follow from WS. Since this contradicts yR,z, it must be the case that xP,z. While this case is just one of the possibilities given transitivity, it would be the only possibility given quasi transitivity. Therefore, Theorem 1 is proved •I

3. Conclusion It has been shown that, in the presence of the Weak Chernoff property, the Substitution property ensures that pair-wise choices are transitive and that a weakend Substitution property ensures that pair-wise choices are quasi transitive. Neither the Weak Chemoff nor the Substitution properties are expansion consistency properties. Since it is easy to construct examples that show the logical independence of Properties Z3 and S, it is clear that expansion consistency is not an essential requirement for transitivity. What then is the essential requirement for going beyond triple acyclicity in choices over pairs of alternatives? All that seems possible to say, is that some form of equivalent treatment is required by all results on this issue. However, as pointed out above, the type of equal treatment required in this paper differs from that required by other results.

References Arrow, K.J. and L. Hurwicz, 1977, An optimality criterion for decision-making under ignorance, in: K.J. Arrow and L. Hurwicz, eds., Studies in resource allocation processes (Cambridge University Press, Cambridge). Baigent, N., 1989, A survey of Arrowian social theory, Mimeo. Sen, A.K., 1977, Social choice theory: A re-examination, Econometrica 45, 53-89. Sen, A.K., 1986, Social choice theory, in: K.J. Arrow and M. Intriligator, eds, Handbook of mathematical economics, Vol. 3 (North-Holland, Amsterdam). Suzumura, K., 1983, Rational choice, collective decisions and social welfare (Cambridge University Press, Cambridge).