Simple everyday examples of violations of transitivity are not difficult to come up with, but I'm having trouble thinking of some for the completeness axiom.

One possible formulation of the completeness axiom in plain English:

Given any two alternatives A and B, exactly one of the following is true: (1) I strictly prefer A to B; (2) I strictly prefer B to A; (3) I am indifferent between A and B.

One example that I was hoping could work (but didn't) came from the book/movie Sophie's Choice, where Sophie is made to choose between the following two alternatives:

A: Send her son to the gas chambers (and her daughter to the children's camp).

B: Send her daughter to the gas chambers (and her son to the children's camp).

She is initially very reluctant to express a preference for either alternative. So one possible interpretation is that she is exactly indifferent between A and B. Another is that she does not obey the completeness axiom.

However, she is then pressed further and told that if she doesn't choose, she'll simply be given alternative C: "Send both children to the gas chambers."

Faced with this alternative C that is clearly inferior to both alternatives A and B, she then does indeed choose one of the alternatives (B: Send her daughter to the gas chambers), hence strongly suggesting that she does indeed obey the completeness axiom.

The above is of course merely my interpretation of the story in Sophie's Choice. Maybe someone can give an alternative interpretation where the above turns out to be an example of the completeness axiom being violated.

(One possibility might be along these lines: Sophie's preference relation is incomplete over {A,B}. When faced only with A and B, she doesn't prefer A to B, doesn't prefer B to A, and isn't indifferent between A and B. However, when given an additional option C that is vastly inferior to A or B and which she must default to in the absence of a choice, she now prefers B to A.)

For those who seem to believe the completeness axiom is inviolable and are unaware that "within economic theory, criticism of the axioms of transitivity and completeness have quite a long history" (Putnam, 2002, p. 163), please see the following three examples of economists criticizing the axiom of completeness:

Von Neumann and Morgenstern (1953, Theory of Games and Economic Behavior, 3rd edition, p. 630):

The axiom (3:A)—or, more specifically, (3:A:a)—expresses the completeness of the ordering of all utilities, i.e. the completeness of the individual's system of preferences. It is very dubious, whether the idealization of reality which treats this postulate as a valid one, is appropriate or even convenient.

Aumann (1962, p. 446):

Of all the axioms of utility theory, the completeness axiom is perhaps the most questionable. Like others of the axioms, it is inaccurate as a description of real life; but unlike them, we find it hard to accept even from the normative viewpoint.

Anand (1987, p. 190):

Whilst completeness is one of the first assumptions used in any" formal" theory of rational choice, it is probably less acceptable as an axiom of rationality than either transitivity or independence.

  • $\begingroup$ "Maybe someone can give an alternative interpretation where the above turns out to be an example of the completeness axiom being violated." I don't think so. If someone can give an interpretation where two apples are less than one apple that means we are not talking about the same thing when we say "one" and "two". Similarly an alternative interpretation of your example would mean we are not talking about the same completeness axiom. $\endgroup$
    – Giskard
    Sep 21 '17 at 6:11
  • $\begingroup$ Thanks for improving the question with the source I provided earlier. My opinion is that the completeness axiom can only be violated when you dont know about a possible option, therefore you don't have a preference for that option. $\endgroup$ Sep 21 '17 at 12:38
  • $\begingroup$ @JoaoBotelho: No I didn't look at any source you provided. $\endgroup$
    – user18
    Sep 22 '17 at 0:26
  • $\begingroup$ @KennyLJ I don't think anyone doubts that in real life preferences are not complete. What we do doubt is that you can come up with an example that demonstrates this. $\endgroup$
    – Giskard
    Sep 23 '17 at 20:16

There is a problem in how you translate completeness into behavior. Let $R$ be any binary relation, representing preferences, on a set $X$ of alternatives and $A\subseteq X$ be a nonempty set of alternatives available. The usual assumption is that the decisionmaker chooses an alternative $a\in A$ optimally according to the relation $R$. Here are three ways to interpret this:

  1. $a R b$ for all $b\in A$.
  2. $b R a$ for no $b\in A$.
  3. $b R a$ implies $a Rb$ for all $b\in A$.

One usually uses 1. for weak (reflexive) preference relations, 2. for strict (irreflexive) preference relations, and 3. works in both cases. If we define $R'$ by $xR'y$ iff $\neg(yRx)\vee(xRy)$, then $R'$ is complete and satisfies 1. iff $R$ satisfies 3. If $R$ is irreflexive, $R'$ satisfies 1. iff $R$ satisfies 2. So completeness in itself has no behavioral implications from this perspective. It can interact with other properties. For example, transitive and complete is more restrictive than transitivity alone- completeness forces transitive to be applicable to more comparisons.

Now one can try to make a distinction between indifference and incomparability. But such a distinction necessarily uses a different framework than the usual maximization paradigm.


Indifference is different from incompleteness. A good example is indecisiveness. Eliaz and Ok have a nice discussion. Suppose you want to buy holidays for your family. You know your wife prefers Bahamas to Florida to Paris. Since you do not care particularly about the destination, you choose trying to represent their preferences. If Bahamas and Florida are available, you will choose Bahamas. If Bahamas and Paris are available, you may choose either of the two. But if Paris and Florida are available, you may also choose either of the two (absent any intensity of preferences)!.

Notice that you are not indifferent since if you strictly prefer Bahamas to Florida and you do not prefer Bahamas to Paris, you should strictly prefer Paris to Florida.


Let $X$ be the number of possible baskets of goods that one can buy from a Walmart Superstore. Even if there were only 1,000 distinct items and we could only buy at most one of each item, that'd be $2^{1000}$ possible baskets. (Note that $X \gg 2^{1000}>10^{300}\gg10^{100}>$ "any estimate of the number of particles in the universe".)

Even as a normative matter, it is debatable whether a perfectly rational being "should" have a complete preference ordering over these $X$ baskets.

But as a positive matter, many (if not all) human beings will not have a complete preference ordering over these $X$ baskets.

(Continued discussion of example.)

... Certainly not in their heads. And not on paper either.

Suppose we sat a bunch of human beings down. Ask each to go through a mere ten million of the $\left(\begin{array}{c} X\\ 2 \end{array}\right)$ possible pairwise comparisons of baskets.

Insist that for each pair of baskets, she must choose to agree with exactly one of the following statements: (i) I strictly prefer the first basket to the second; (ii) I strictly prefer the second to the first; (iii) I am exactly indifferent between the two.

Give her as much time as she needs to be absolutely sure and absolutely honest about each choice. (The flippant economist asked to do this exercise may simply and dishonestly claim that he's indifferent between all baskets, just so he doesn't violate the completeness axiom. But let's assume our subjects are all absolutely honest and carefully consider each choice.)

Then repeat the same exercise (i.e. go through all ten million pairwise comparisons again).

We will certainly discover some inconsistencies. That is, we'll certainly find for example that for some pair of baskets of goods $y$ and $z$, someone said $y \succ z$ the first time round but $z \prec y$ the second time round.

Now, the economist may say that: (1) she made a mistake; or (2) her preferences changed between the first and second runs of the exercise.

But I think there is a simpler and broader explanation that subsumes both of these explanations: This individual simply doesn't have a complete preference ordering over the $X$ Walmart baskets.

More examples from other economists/decision theorists/philosophers. The first two are similar to the above.

Anand (1987):

Surely the interpretation that a consumer has, in his or her head (or on paper) a complete ordering of all possible pairs of choice objects is unacceptable. A shopping list for seven items with a choice between two brands for each would require 91 pairwise comparisons.

On a similar note, Thrall (1954, p. 183):

From the practical point of view, if the number of judgments needed is finite but large, there is still the time difficulty. By the time the judge has reached the 1,000,000th choice, his standards of comparison are almost certainly not the same as initially.

Aumann (1962):

Or he might be willing to make rough preference statements such as, "I prefer a cup of cocoa to a 75-25 lottery of coffee and tea, but reverse my preference if the ratio is 25-75"; but he might be unwilling to fix the break-even point between coffee-tea lotteries and cocoa any more precisely. Is it "rational" to force decisions in such cases?

The last class of examples is of the Sophie's Choice variety. Not surprisingly, these examples are pursued largely by philosophers who argue (contra economists) that there is a real distinction between indifference and incomparability (or incommensurability, though naturally philosophers split hairs over whether these two terms are the same).

Putnam (1986)

I may be quite sure that if I choose the hedonistic-sensual may of life, I would prefer to have a beautiful and responsive lover to a plain and unresponsive one. Call these choices x and y, and let z be the ascetic-religious life. If I regard the two ways of life as "incomparable", then I might insist that, prior to my making my existential choice, ~xPz& ~zPx, and also ~yPz& ~zPy

See also Putnam (2004).

Raz (1986, pp. 341-2) similarly compares a teaching and a legal career.

Another perhaps distinct line of research is that initiated by Slovic and Lichtenstein (1971, etc.) and goes by the name of "Preference Reversals". These could arguably also be examples of incomplete preferences. The following example is from Tversky & Thaler (1990, JEP Anomalies article):

Imagine, if you will, that you have been asked to advise the Minister of Transportation for a small Middle Eastern country regarding the choice of a highway safety program. At the current time, about 600 people per year are killed in traffic accidents in that country. Two programs designed to reduce the number of casualties are under consideration. Program A is expected to reduce the yearly number of casualities to 570; its annual cost is estimated at \$12 million. Program B is expected to reduce the yearly number of casualities to 500; its annual cost is estimated at \$55 million. The Minister tells you to find out which program would make the electorate happier.

You hire two polling organizations. The first firm asks a group of citizens which program they like better. It finds that about two-thirds of the respondents prefer Program B which saves more lives, though at a higher cost per life saved. The other firm uses a "matching" procedure. It presents respondents with the same information about the two programs except that the cost of Program B is not specified. These citizens are asked to state the cost that would make the two programs equally attractive. The polling firm reasons that respondents' preferences for the two programs can be inferred from their responses to this question. That is, a respondent who is indifferent between the two programs at a cost of less than \$55 million should prefer A to B. On the other hand, someone who would be willing to spend over \$55 million should prefer Program B. This survey finds, however, that more than 90 percent of the respondents provided values smaller than \$55 million indicating, in effect, that they prefer Program A over Program B.

This pattern is definitely puzzling. When people are asked to choose between a pair of options, a clear majority favors B over A. When asked to price these options, however, the overwhelming majority give values implying a preference for A over B. Indeed, the implicit value of human life derived from the simple choice presented by the first firm is more than twice that derived from the matching procedure used by the other firm.

  • $\begingroup$ Nice examples! There is an important difference between your answer and the other answers: You assume that people are aware of their preferences. All other answers assume revealed preferences. That is we do not take people's word for what they would choose, we insist on observing the choice. As you yourself point out, being aware of $\left(\begin{array}{c} X\\ 2 \end{array}\right)$ preferences would be quite amazing. $\endgroup$
    – Giskard
    Sep 24 '17 at 5:59

I assume you are talking about preferences?

In case you are talking about mathematical relations in general, $\geq$ in $\mathbb{R}^2$ is not complete, as neither $(0,1) \geq (1,0)$ nor the opposite holds.

In case you are talking about preferences or decision theory, I don't think you can come up with any examples that violate completeness. The issue is that there is a set of choices which can violate transitivity, but there are none for completeness.

If a preference ordering is not complete then we are unable to forecast your decision even with perfect knowledge of your preferences. Empirically this would mean that we forecast your choosing $A$, but you choose $B$ instead. Obviously we could correct for this by forecasting $B$, so next time you would have to choose $A$ or $C$ or some other alternative. Unfortunately this could also mean that you are indifferent between the choices and then your preferences would still be complete.

There are some suggestions on how to deal with this: Perhaps if you are unable to make a choice, you do not have to make a choice. But this can be regarded as a regular alternative $D$, like all the others, and then the above reasoning applies. Perhaps you can choose more than one alternatives. Again, indifference and being unable to tell the difference would be indistinguishable.

There are probably some assumptions you could make on the preferences to solve this (e.g. all preferences are strict) but I cannot think of any that in my opinion would be realistic.

  • $\begingroup$ In my opinion, this answer is probably better suited as a comment since you're simply saying that "in your opinion, no examples are possible." $\endgroup$
    – user18
    Sep 21 '17 at 1:31
  • $\begingroup$ @KennyLJ My statement is a little stronger than that. I am saying that you would have to make additional assumptions for a counterexample to be possible. And IMO the assumptions would have to be fairly strong. $\endgroup$
    – Giskard
    Sep 21 '17 at 6:09

Based on gametheory101.com,

In essence, the only thing completeness rules out is a “decline to state” option.

If a refusal to state a preference or indifference would be the violation of the axiom, then you can use the example of a couple that refuses to answer to their family if they will marry or not. They are not saying yes, no, or that they are indifferent - they are refusing to discuss the topic.

However, you can always say that this is indifference. That's what Tassos Patokos wrote in his book Internal Game Theory (pages 33-34).

  • 1
    $\begingroup$ Downvotes without explanation? How helpful... $\endgroup$
    – luchonacho
    Sep 21 '17 at 8:48

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