Incomplete and Intransitive Preferences

Contents

(1) Introduction
(2) Illustrations
(3) Proof of Existence

Selected References

(1) Introduction

Okham's razor argues that we ought to try to prove the main propositions of equilibrium theory with as few assumptions as possible. It is in this spirit that economists have tried to grind away at one or several of the six axioms of preference -- completeness, transitivity, reflexivity, continuity, convexity and non-satiation. Economists have attacked these axioms time and time again. In this section, we take up two of the more difficult ones: transitivity and completeness.

Intuitively, the problem can be illustrated as follows. It was demonstrated by Debreu (1954) that for a utility function to exist, at least four axioms were needed: completeness, reflexivity, transitivity and continuity. If completeness or transitivity were removed, then the utility function may fail to exist. The implication, for the rest of economic theory, is simple to see. Following previous notation, recall that demand is defined as:

f^h (p, e^h) = argmax U^h(x) s.t. x Î B^h(p, e^h)

i.e. demand is the most preferred bundle in the budget set. If the utility function does not exist, then this maximal element might fail to exist and thus the rest of equilibrium theory collapses.

Yet there is some intuition as to why one would try to omit both of the transitivity and completeness axioms. Just because an agent is a bit irrational and cannot compare absolutely everything, should we then argue that the agent cannot therefore make a "choice"? He should be able, even when intransitive or incomplete, to still find a most preferred bundle in his budget constraint.

Robert Aumann (1962), David Schmeidler (1969), Bezalel Peleg (1970), W. Hildenbrand, D. Schmeidler and S. Zamir (1973) had been the pioneers in trying to get rid of completeness, albeit often with the help of the nice mathematical properties of an infinite number of agents. Trout Rader (1963), Hugo Sonnenschein (1965, 1971), Wayne Shafer (1974) and others had attempted to derive a utility function without the transitivity axiom.

The great breakthrough was the work of Andreu Mas-Colell (1974). He did not require the supporting device of an infinite numbers of agents, nor did he wrestle unduly with utility function representations. Mas-Colell's effort was very clean.  Mas-Colell's contribution was so nice that many economists went on to prove it all over again --- and again and again. Seminal alternative proofs include David Gale and Andreu Mas-Colell (1975), Wayne Shafer and Hugo Sonnenschein (1975), Joseph Greenberg (1976) and Robert M. Anderson (1981). Extensions to other contexts are found in Nicholas Yannelis and N. Prabhakar (1983), M.Ali Khan and R. Vohra (1984) and Kim Border (1984).

It may be useful to motivate the effort with some simple examples of incomplete and intransitive preferences. Let an agent's preferences be denoted by >_h, so if x >_h y, then bundle x is (strictly) preferred to bundle y. Suppose that the commodity space is the set of all points on a continuous line between 0 and 1, then intransitivity is illustrated if we suppose that preferences are such that

(1) x >_h y if x > y for all x, y Î [0, 1)

(2) 0 >_h 1

In other words, by rule (1), the agent prefers numbers that are greater so, he prefers 1/2 over 1/3, 3/4 over 1/2, etc. However, he has in (2) the exceptional case that he prefers 0 to 1.This is where intransitivity comes: by (1), he prefers 1/2 to 0 and he prefers 1 to 1/2, but it does not follow that he prefers 1 over 0; rather, by (2), he prefers 0 to 1. Thus, his preferences are not transitive.

We can illustrate this example via the simple diagram in Figure 1. Let us define P(x) as the strict upper contour set of point x, i.e.

P(x) = {y Î C | y >_h x}

so P(x) denotes the set of points in the commodity set C which are strictly preferred to point x. So, in Figure 1, the strict upper contour set of 1/2 is the thick line P(1/2) (i.e. all numbers above 1/2), while that of 1/4 is the thick line P(1/4) (i.e. all numbers above 1/4). Thus, 3/4 Î P(1/2) and 3/4 Î P(1/4). However, while 1/2 Î P(1/4), 1/2 Ï P(1/2). Of course, 1/4 Ï P(1/2) and 1/4 Ï P(1/4). Notice also, following our example above, as only 0 is preferred to 1, then P(1) is the point at the bottom right of Figure 1.

We can define G_P as the graph of the preference mapping, thus:

G_P = {(x, y) Î C ´ C | y Î P(x)}

i.e. G_p illustrates the combinations of (x, y) where y lies in the upper contour set of x. Using our example, G_p will be the shaded area in Figure 1 and the point P(1). Notice that the diagonal is not included. The complement of the G_p is denoted G_p^c and is illustrated in the diagram by the unshaded area plus the dashed lines (and excluding point P(1)). Thus, (1/2, 1/2) Ï G_p, but is an element of G_p^c, while (1, 0) Î G_p but not of G_p^c.

If we assume, as we had before, that C = [0, 1], the points on a line segment between 0 and 1, then we can illustrate this as in Figure 1.

What about incompleteness? Suppose we have a similar case of preferences defined over the line segment C = [0, 1]. Suppose the rule is now that:

(1) x >_h y if x > y for all x, y Î [0, 1/2)

(2) x >_h y if x > y for all x, y Î (1/2, 1]

So, the agents always prefers the "greater" number once again. Only notice that now, the point 1/2 itself is not compared to anything: no number is preferred to 1/2 and 1/2 is preferred to no other number. Note that this is not "indifference"; it is simply that 1/2 is not being compared to anything else. Thus, in this case, preferences are "incomplete" over the line segment [0, 1].

It might be useful to introduce an example where preferences are both incomplete and intransitive. An example is illustrated in Figure 2. Once again, let C = [0, 1], the points on a line segment between 0 and 1. Now, let preferences be defined as:

(1) P(x) > 1/2 + x for all x such that 0 £ x < 1/2

(2) P(x) = Æ for x = 1/2

(3) P(x) < -1/2 + x for all x such that 1/2 < x £ 1

These preferences are illustrated in Figure 2. The shaded areas, G_p, denotes the preference graph. Thus, by (1), notice that 3/4 >_h 1/5, which is shown in Figure 2 by the point in the upper left shaded triangle. But notice also that by (3), 1/5 >_h 3/4, as shown by the point in the lower right shaded triangle. Thus, we have intransitivity as both (1) and (3) imply that 1/5 >_h 3/4 >_h 1/5. Notice also that, by (2), nothing is preferred to 1/2, thus incompleteness (the vertical line above 1/2 is not in G_p).

However, incompleteness and intransitivity does not mean it is impossible to choose the maximal element. By definition, a bundle x is "the best" if P(x) = Æ , nothing is preferred to it. In this example, obviously, 1/2 is the only maximal element, i.e. only point on the [0, 1] line segment for which this is true. Thus, even though preferences are incomplete and intransitive, we are still able to define a "choice".

Another example may help embed the concept clearly. This is shown in Figure 3, where, once again, we assume C = [0, 1]. Let preferences be defined so that:

(1) P(x) = Æ for x = 0

(2) P(x) > x for 0 < x < 1/2

(3) P(x) = Æ for x = 1/2

(4) P(x) < x for 1/2 < x < 1

(5) P(x) = Æ for x = 1.

which is illustrated in Figure 3. Notice, once again, that preferences are intransitive (e.g. 1/5 >_h 3/4 >_h 1/5) and incomplete (as P(0) = P(1/2) = P(1) = Æ ). Note that now we have three candidates for maximal element: 0, 1/2 and 1. This, incidentally, shows that the set of maximal elements need not be convex when we have incomplete and intransitive preferences.

All these examples are geared to show that incomplete and intransitive preferences need not prevent choice. Agents may still be able to find their most-preferred bundle, even while remaining "irrational" (i.e. intransitive) and "unsure" (i.e. incomplete). We now turn to proof of existence of a maximal element when there are incomplete and intransitive preferences.

Although the general effort is on the existence of equilibrium, much of the hard work boils down in this case to the ability to prove the existence of "demand" for an agent, i.e. the existence of a most-preferred bundle in the budget set. As noted, the first general proof of the existence of maximal elements in the absence of the completeness and transitivity axioms was accomplished by Andreu Mas-Colell (1974). We follow the proof of Wayne Shafer and  Hugo Sonnenschein  (1975) here.

Preferences (>_h) are circumscribed over C Ì R₊ⁿ where C is a non-empty, convex, compact set. Properly, we ought think of C as the budget set for it is bounded above. We make the following assumptions on preferences:

Mas-Colell Axioms:

(1) P(x) is convex

(2) x Ï P(x)

(3) G_p^c is closed.

where P(x) = {y Î C| y >_h x} is the upper contour set at point x, while G_p^c is the complement of the graph G_p = {(x, y) Î C | y Î P(x)}.

Axiom (1) is simply the convexity axiom; axiom (2) is the reflexivity axiom. Axiom (3), which looks a bit strange, is akin the continuity axiom. Notice that we have not imposed the axioms of transitivity or completeness. We wish simply to establish the existence of a maximal element, i.e. that there is an x Î C such that P(x) = Æ , i.e. a bundle x in C for which there is no other bundle y in C that is preferred to it.

Before we proceed with the proof, a few remarks are in order. Firstly, examine Example 1 again. Here we see that preferences are complete, but intransitive. There was no candidate for maximal element. In Examples 2 and 3, preferences were incomplete, and we had candidates for maximal elements -- which happened to be precisely the bundles that were not comparable. A cynic may point out that any bundle which is not comparable is, by definition, a maximal element. In the examples we worked out, this is true, but this should not detract from the important effort here. More precisely, in the proof we are about to set out, we are simply not imposing completeness; but that does not mean that we are imposing incompleteness. Just because we do not impose completeness does not mean that preferences cannot be complete. The theorem is general: it would also apply to cases where, like Example 1, preferences happen to be complete.

Yet Example 1 does not have a maximal element. What is being violated here? Namely, Mas-Colell Axiom (3), the closedness of complementary graph G_p^c. Because (1, 0) Î G_p, this implies that (1, 0) Ï G_p^c. Yet (1, 0) is a limit point of G_p^c, thus G_p^c is not closed. Thus it does not fulfill one of the Mas-Colell axioms -- and thus fails to have a maximal element. Conversely, note that in Examples 2 and 3, all three Mas-Colell axioms are fulfilled.

Let us now turn to the theorem:

Theorem: Let the preferences >_h of the agent be defined by a correspondence P: C ® C where C is a non-empty, convex, compact subset of Rⁿ and where (i) P(x) is convex " x Î C, (ii) x Ï P(x) for every x Î C and (iii) G_p^c is closed. Then there is an x* Î C such that x* is a maximal element, or P(x*) = Æ .

 Proof: As noted, although first proved by Mas-Colell (1974), our method follows Shafer and Sonnenschein (1975). Let us define a function h(x, y) as:

  h(x,y) = min {d[(x,y), z]½ (x,y) Î C ´ C , z Ï G_p}

Thus h(x,y) is a function h:C ´ C® R₊ where d is the distance metric operator so that h(x,y) is the minimum distance between point (x, y) and the complementary graph G_p^c. Thus, z is the point in G_p^c closest to (x,y). We can use "minimum" rather than "infimum" by the assumption that G_p^c is closed. Obviously, for any point (x, y) Î C ´ C, it will be true that either (i) y Î P(x), in which case (x, y) Ï G_p^c and thus h(x, y) > 0; or (ii) y Ï P(x), in which case (x,y) Î G_p^c and thus h(x, y) = 0. The reverse also holds: h(x, y) > 0 iff y Î P(x) and h(x,y) = 0 iff y Ï P(x). G_p^c is compact as it is a closed and bounded subset of a Euclidan space - and as G_p^c is compact, then the metric d defines a continuous function on it and so h(x, y) is continuous in C ´ C.

Recall that h(x, y) is formulated as a minimization of distance between (x,y) and z where z Î G_p^c . Since G_p^c is a closed and constant correspondence, then G_p^c is both upper semicontinuous and lower semicontinuous. Furthermore, d[(x,y), z] is a function which is jointly continuous in z and (x,y). Thus, by Berge’s Theorem, we know that h(x, y) is a continuous function over C ´ C.

 Now, let us define the following correspondence, F: C ® C where:

  F(x) = argmax {h(x, y)½ y Î C}

As h(x, y) is jointly continuous in x and y and C is a closed and constant correspondence, then, again by Berge’s Theorem, F(x) is an upper semicontinuous correspondence.  It is useful to think of F(x) as merely the set of points y Î C that maximize h(x, y) for a given x Î C. Now, recall that if x* is a maximal element, then P(x*) = Æ - thus there is no point y Î C such that h(x*, y) > 0, i.e. it must be that h(x*, y) = 0 " y Î C. Thus, if this is indeed the case, then the maximum h(x*, y) is zero and all points in C yield this, i.e. F(x*) = C.

So, we have F: C ® C as an upper semicontinuous correspondence from a non-empty, compact, convex set to another. For Kakutani’s Fixed Point theorem, we need a correspondence that is convex-valued and upper-semicontinuous. But, in our problem so far, all we have is F(x) = argmax {h(x, y) ½ y Î C} which is an upper-semicontinuous correspondence from C to C, but it is not convex-valued, so we cannot apply Kakutani’s fixed point theorem. The solution, proposed by Shafer and Sonnenschein (1975), is simply to "convexify" this correspondence by considering the convex hull of F(x):

coF(x) = co{argmax {h(x, y) ½ y Î C}}

where coF(x) denotes the convex hull of F(x). It is easy to show that if F(x) is upper semicontinuous, then coF(x) is also upper semicontinuous as the convex hull of a closed subset of Euclidean space (which F(x) is) is also a closed set - thus, the upper semicontinuity of F(x) is transferred to coF(x). Thus, coF: C ® C is an upper semicontinuous, convex-valued correspondence and C is a non-empty, convex, compact subset of Rⁿ. Thus, by Kakutani’s fixed point theorem, there is an x* Î C such that x* Î coF(x*).

Is the fixed-point x* a maximal element? If x* is a maximal element, then it must be that P(x*) = Æ . We can prove this by contradiction: Suppose not. Suppose there is z Î C such that z Î P(x*). This implies, as we argued before, that h(x*, z) > 0. But if h(x*, z) > 0, then we know that the set F(x*) will contain points y Î C such that h(x*, y) > 0. In short, as long as one element z Î C yields h(x*, z) > 0, then the maximum value of the function h(x, y) will not be zero. Thus, for all points y Î F(x*), it will be true that h(x*, y) > 0. But if this is true, then F(x*) Ì P(x*) by definition of h(.). Furthermore, if F(x*) Ì P(x*), then coF(x*) Ì P(x*). But, by Kakutani’s fixed point theorem, we saw x* Î con(F(x*)). Thus, it must be that x* Î P(x*) which is a contradiction of axiom (ii) (irreflexivity). Thus, x* is a maximal element. §

R.M. Anderson (1981) "Core Theory with Strongly Convex Preferences", Econometrica, Vol. 49 (6), p.1457-68.

R.J. Aumann (1962) "Utility Theory without the Completeness Axiom", Econometrica, Vol. 30 (3), p.445-62.

K.C. Border (1984) "A Core Existence Theorem for Games without Ordered Preferences", Econometrica, ol. 52, p.1536-42.

A.  Mas-Colell (1974) "An Equilibrium Existence Theorem without Complete or Transitive Preferences", Journal of Mathematical Economics, Vol. 1, p.237-46.

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J. Greenberg (1976) "Quasi-Equilibrium in Abstract Economies without Ordered Preferences", Journal of Mathematical Economics, Vol. 4, p.163-5.

M.A. Khan and R. Vohra (1984) "Equilibrium in Abstract Economies without Ordered Spaces and with a Measure Space of Agents", Journal of Mathematical Economics, Vol. 13 (2), p.133-42.

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H. Sonnenschein (1965) "The Relationship between Transitive Preference and the Structure of Choice Space", Econometrica, Vol. 33, p.624-34.

H. Sonnenschein (1971) "Demand Theory Without Transitive Preferences: with applications to the theory of competitive equilibrium" in J.Chipman, L.Hurwicz, M.K.Richter, and H.Sonnenschein, 1971, editors, Preferences, Utility and Demand. New York: Harcourt Brace Jovanovich.

N.C. Yannelis and N.D. Prabhakar (1983) "Existence of Maximal Elements and Equilibria in Linear Topological Spaces", Journal of Mathematical Economics, Vol. 12, p.233-45.

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