Irrational Preferences

Irrational Preference Relations

Consider an economy in which there are $N$ types of goods, and represent each alternative faced by a consumer as an $N$-dimensional vector \[
\boldsymbol{x}=\left( x_{1},\cdots,x_{N}\right) \in\mathbb{R} ^{N}\]Here, the $n$th component $x_{n}$ of this vector $\boldsymbol{x}$ represents the consumption of good $n$. We represent the set consisting of all vectors available to the consumer as the consumption set \[
X\subset\mathbb{R} ^{N}\]A consumer facing the consumption set $X$ chooses the most desirable consumption vector by comparing the consumption vectors that are elements of $X$. We therefore formulate the consumer’s system of tastes as a binary relation $\succsim$ on $X$, and call it a preference relation. Specifically, for any two consumption vectors $\boldsymbol{x},\boldsymbol{y}\in X$, \[
\boldsymbol{x}\succsim\boldsymbol{y}\Leftrightarrow\text{the consumer prefers
}\boldsymbol{x}\text{ at least as much as }\boldsymbol{y}\]we define $\succsim$ so that this condition is satisfied.
In standard consumer theory, in order to analyze consumer choice behavior systematically, we assume that the preference relation $\succsim$ satisfies completeness and transitivity. That is, \begin{align*}
& \left( a\right) \ \forall\boldsymbol{x},\boldsymbol{y}\in X:(\boldsymbol{x}\succsim\boldsymbol{y}\vee\boldsymbol{y}\succsim
\boldsymbol{x})\\
& \left( b\right) \ \forall\boldsymbol{x},\boldsymbol{y},\boldsymbol{z}\in X:\left[ (\boldsymbol{x}\succsim\boldsymbol{y}\wedge\boldsymbol{y}\succsim\boldsymbol{z})\Rightarrow\boldsymbol{x}\succsim\boldsymbol{z}\right] \end{align*}we assume that these conditions hold. A preference relation that satisfies completeness and transitivity is called a rational preference relation. A consumer with rational preferences can compare any consumption vectors that the consumer may face and can position the results of those comparisons consistently so that no cycles arise.

Completeness and transitivity may appear to be commonsense and undemanding assumptions, but in fact situations can arise in various real-world settings in which consumer preferences are not rational. If at least one of completeness or transitivity fails to hold, the preference relation is said to be irrational. In what follows, we explain situations in which consumer preferences fail to satisfy rationality.

When There Are Too Many Alternatives

When there are very many alternatives, a consumer may be unable to form a clear comparative judgment between some alternatives, and the preference relation may fail to satisfy completeness.

Example (When There Are Too Many Alternatives)

A store sells one each of five types of shirts. Call the shirts $1,2,3,4,5$, and let the quantity of shirt $i\in\left\{ 1,2,3,4,5\right\} $ be denoted by $x_{i}\in\left\{ 0,1\right\} $. Then each consumption vector can be represented as \[
\left( x_{1},x_{2},x_{3},x_{4},x_{5}\right)
\]For example, the consumption vector $\left( 1,0,0,0,0\right) $ corresponds to the alternative “purchase only one shirt 1,” while the consumption vector $\left( 0,1,0,0,1\right) $ corresponds to the alternative “purchase one each of shirt 2 and shirt 5.” The same applies to the other cases. Since each $x_{i}$ can take the value $0$ or $1$, the consumer faces $2^{5}=32$ consumption vectors. There are $\binom{32}{2}=496$ ways to choose two from these $32$ consumption vectors, and for the preference relation to satisfy completeness, the consumer must judge which of the two presented consumption vectors is preferable for each of these $496$ pairs. Such judgments are difficult.

When Alternatives Change Continuously

When the alternatives faced by a consumer change continuously, situations can also arise in which the preference relation is not rational.

Example (When the Quantities of Goods Change Continuously)

Suppose a preference relation $\succsim$ defined on the consumption set $\mathbb{R} _{+}^{N}$ is given. Suppose the preference relation $\succsim$ defined on this set satisfies transitivity. For each good $n\in\left\{ 1,\cdots,N\right\} $, its consumption $x_{n}\in\mathbb{R} _{+}$ can take any nonnegative real value and is therefore a continuous variable. Hence, we can choose two consumption vectors that are arbitrarily close to each other, namely the consumption vectors $\boldsymbol{x}^{1},\boldsymbol{x}^{2}\in\mathbb{R} _{+}^{N}$ that are arbitrarily close to each other. Although these are strictly different consumption vectors, suppose that the consumer is unable to distinguish between them and is therefore indifferent between $\boldsymbol{x}^{1}$ and $\boldsymbol{x}^{2}$. That is, \[
\boldsymbol{x}^{1}\sim\boldsymbol{x}^{2}\]Furthermore, choose a third consumption vector that is difficult to distinguish from $\boldsymbol{x}^{2}$, namely the consumption vector $\boldsymbol{x}^{3}\in\mathbb{R} _{+}^{N}$. For the same reason as above, we have \[
\boldsymbol{x}^{2}\sim\boldsymbol{x}^{3}\]holds. If the preference relation $\succsim$ satisfies transitivity, the indifference relation $\sim$ also satisfies transitivity, and therefore \[
\boldsymbol{x}^{1}\sim\boldsymbol{x}^{3}\]holds. The same argument can be repeated indefinitely. That is, for each index
$v\in\mathbb{N} $, when we choose a consumption vector $\boldsymbol{x}^{v}\in\mathbb{R} _{+}^{N}$ that is difficult to distinguish from $\boldsymbol{x}^{v-1}\in\mathbb{R} _{+}^{N}$, as long as the preference relation $\succsim$ satisfies transitivity, \[
\boldsymbol{x}^{1}\sim\boldsymbol{x}^{v}\]must hold. Adjacent consumption vectors in the resulting sequence
$\{\boldsymbol{x}^{v}\}$ are so similar that they are difficult to distinguish, but if the index $v$ is sufficiently large, the difference between $\boldsymbol{x}^{1}$ and $\boldsymbol{x}^{v}$ becomes obvious. Thus, if $\boldsymbol{x}^{1}$ and $\boldsymbol{x}^{v}$ are presented to the consumer again, it may happen that either $\boldsymbol{x}^{1}\succ\boldsymbol{x}^{v}$ or $\boldsymbol{x}^{v}\succ\boldsymbol{x}^{1}$ holds. This contradicts $\boldsymbol{x}^{1}\sim\boldsymbol{x}^{v}$. Therefore, this consumer’s preference relation $\succsim$ does not satisfy rationality.

Example (When the Characteristics of a Good Change Continuously)

In the preceding example we considered a case in which the “quantity” of a good changes continuously, but the same problem can arise when a “characteristic” of a good changes continuously. Suppose that a consumer is trying to purchase a shirt from an online shop. The size and shape of the shirt to be purchased have already been determined, so only the color remains to be chosen. At this shop, when a customer specifies a preferred color, a shirt of that color is custom-made. In the consumption vector $\boldsymbol{x}=\left( x_{1},x_{2},\cdots,x_{N}\right) $, interpret $x_{1}$ as the consumption of a shirt of color 1, $x_{2}$ as the consumption of a shirt of color 2, and so on. Then the consumption vector corresponding to “buy one shirt of color 1” can be represented as $\boldsymbol{x}^{1}=\left( 1,0,\cdots,0\right) $, while the consumption vector corresponding to “buy one shirt of color 2” can be represented as $\boldsymbol{x}^{2}=\left( 0,1,\cdots,0\right) $. The same applies to shirts of other colors. Since there are infinitely many colors, for each index $v\in\mathbb{N} $, it is possible to construct a situation in which color $v$ and color $v+1$ are so similar that they cannot be distinguished. The same argument as above therefore applies to the resulting sequence of consumption vectors
$\{\boldsymbol{x}^{v}\}$, so a situation can arise in which the consumer’s preference relation $\succsim$ does not satisfy rationality.

Aggregation of Multiple Criteria

When there are multiple criteria for evaluating consumption vectors and the overall ranking of consumption vectors is determined by accumulating the evaluations under the individual criteria, a situation can arise in which the preference relation is not rational. .

Example (When Multiple Criteria Exist)

Suppose that when facing the consumption set
$X=\left\{ x,y,z\right\} $, under a preference relation $\succsim_{1}$ reflecting one evaluation criterion,
\[
x\succ_{1}y\succ_{1}z
\]holds, while under a preference relation $\succsim_{2}$ reflecting another evaluation criterion, \[
y\succ_{2}z\succ_{2}x
\]holds, and under a preference relation $\succsim_{3}$ reflecting another evaluation criterion, \[
z\succ_{3}x\succ_{3}y
\]holds. Suppose that all three preference relations above satisfy rationality. Define the consumer’s preference relation $\succsim$ so that the consumer “prefers a consumption vector that is ranked relatively higher under a larger number of evaluation criteria.” With this in mind, compare $x$ and $y$. Under $\succsim_{1}$ and $\succsim_{3}$, $x$ is more desirable than $y$, while under $\succsim_{2}$, $y$ is more desirable than $x$. Therefore, by the definition of $\succsim$, \begin{equation}
x\succ y\quad \cdots (1)\end{equation}holds. Comparing $y$ and $z$, under $\succsim_{1}$ and $\succsim_{2}$, $y$ is more desirable than $z$, while under $\succsim_{3}$, $z$ is more desirable than $y$. Therefore, by the definition of $\succsim$, \begin{equation}
y\succ z\quad \cdots (2)\end{equation}holds. Comparing $x$ and $z$, under $\succsim_{1}$, $x$ is more desirable than $z$, while under $\succsim_{2}$ and $\succsim_{3}$, $z$ is more desirable than $x$. Therefore, by the definition of $\succsim$, \begin{equation}
z\succ x\quad \cdots (3)\end{equation}holds. If the preference relation $\succsim$ satisfies transitivity, the strict preference relation $\succ$ also satisfies transitivity, but the simultaneous validity of $\left( 1\right) ,\left( 2\right) ,\left( 3\right) $ contradicts the transitivity of $\succ$. Therefore, this consumer’s preference relation $\succsim$ does not satisfy transitivity. Even if the preferences $\succsim_{1},\succsim_{2},\succsim_{3}$ reflecting the individual evaluation criteria each satisfy rationality, when consumption vectors are compared comprehensively using those criteria, the resulting preference relation $\succsim$ need not satisfy rationality.

Example (Condorcet Paradox)

In the preceding example, reinterpret the preferences $\succsim_{1},\succsim_{2},\succsim_{3}$ reflecting the individual evaluation criteria as the preferences of three different people, and reinterpret the consumer’s preference relation $\succsim$ as a group preference of the form \textquotedblleft the group also prefers the consumption vector that is ranked relatively higher by more people.\textquotedblright\ In other words, the preference relation $\succsim$ of this group of three people is determined by majority rule. Then, by the same argument as above, the group preference $\succsim$ can be shown not to satisfy transitivity. Thus, even if the individual preferences $\succsim_{1},\succsim_{2},\succsim_{3}$ of the members of the group each satisfy rationality, the group preference $\succsim$ determined by majority rule need not satisfy rationality. This problem is called the Condorcet paradox.

Framing Effect

Even when alternatives are substantively identical, changing the way they are presented can change consumer preferences, and as a result a situation can arise in which the preference relation is not rational. This problem is called the framing effect.

Example (Framing Effect)

Suppose a consumer is about to purchase one unit each of product $A$, with a list price of 10,000 yen, and product $B$, with a list price of 1,000 yen. Consider the following three alternatives \begin{align*}
x & :\text{purchase both }A,B\text{ at list price at a nearby store}\\
y & :\text{purchase }A\text{ at list price and }B\text{ at half price at a distant store}\\
z & :\text{purchase }A\text{ at a discount of 500 yen and }B\text{ at list price at a distant store}\end{align*}Suppose first that only $x$ and $y$ are presented to the consumer. The impact of the expression “half price” may attract the consumer enough to go to the distant store. That is, \begin{equation}
y\succ x\quad \cdots (1)\end{equation}On the other hand, if only $x$ and $z$ are presented to the consumer, the impact of a 500-yen discount on a product with a list price of 10,000 yen may not be very large, so the consumer may simply complete the purchase at the nearby store. That is, \begin{equation}
x\succ z\quad \cdots (2)\end{equation}Also, if only $y$ and $z$ are presented to the consumer, they are substantively equivalent (in the sense that the consumer buys the same products at the same store and saves 500 yen), so \begin{equation}
y\sim z\quad \cdots (3)\end{equation}may hold. Not every consumer responds in the same way, but it is also true that some consumers exhibit this sequence of responses. If the preference relation $\succsim$ satisfies transitivity, the strict preference relation $\succ$ also satisfies transitivity, but the simultaneous validity of $\left( 1\right) ,\left( 2\right) ,\left( 3\right) $ contradicts the transitivity of $\succ$. This is because $\left( 1\right) ,\left( 2\right) $ and the transitivity of $\succ$ imply $y\succ z$, which contradicts $\left( 3\right) $. Therefore, this consumer’s preference relation $\succsim$ does not satisfy transitivity. Even when alternatives are substantively identical, changing the way they are communicated or expressed can change the impression they create. The preferences of a consumer affected by the framing effect need not be rational.

Buridan’s Ass (Barrier to Choice)

Buridan’s ass is a thought experiment illustrating the difficulty of choice when multiple alternatives are equally desirable. Suppose a hungry donkey is searching for food. The donkey discovers one pile of hay in each of two different places, but the amount of hay and the distance from the donkey are exactly the same. Since there is no reason to determine which one should be chosen, the donkey cannot choose either pile of hay and is eventually said to starve to death.

Why was the donkey unable to choose? Choosing one alternative simultaneously means not choosing another, so making a choice can lead to regret or anxiety such as “Was my choice mistaken?” or “Would the other alternative have been better?” In particular, as in the story of Buridan’s ass, when the alternatives faced have similar conditions or are equally attractive, it is difficult to reach a conclusion through rational thought alone. If the anxiety associated with making a decision is greater than the pain associated with not making one, a situation can arise in which the decision itself cannot be made.

At first glance, this example may appear to describe a situation in which the consumer cannot compare two alternatives. However, if the two alternatives are equally desirable, then from the perspective of the preference relation the consumer can evaluate them as indifferent. If the two piles of hay are represented by $x,y$, respectively, \[
x\sim y
\]is an appropriate evaluation. In this case,
\begin{align*}
x & \succsim y\\
y & \succsim x
\end{align*}both hold, so completeness is not violated. .

Therefore, what Buridan’s ass illustrates is not necessarily irrationality of the preference relation. Rather, it shows that even if the preference relation is rational, the actual choice may not be uniquely determined by that relation alone. When $x\sim y$, the preference relation does not determine whether $x\succsim y$ or $y\succsim x$ should be chosen. To actually choose one, a choice rule distinct from the preference relation is required, such as choosing randomly, choosing the one noticed first, or choosing according to a fixed rule.
In this sense, having a rational preference relation is not the same as being able to choose one alternative without hesitation in every situation. The example of Buridan’s ass can be understood as illustrating the need to distinguish the rationality of preferences from actual choice behavior.

Bounded Rationality

So far, we have analyzed consumers as comparing the alternatives they face and making decisions on the basis of consistent preferences. In reality, however, the information, time, attention, memory, computational ability, and other resources available to human beings are limited. Therefore, consumers are not always able to compare all alternatives sufficiently and identify the most desirable one.

The idea that human rationality is constrained by cognitive ability, available information, and other limitations is called bounded rationality. The concept of bounded rationality was proposed by Herbert Simon.

For example, when a consumer chooses one product from a few varieties, it is relatively easy to make a decision after comparing their prices and quality. However, when there are hundreds of products and the consumer must compare a large amount of information about each, such as price, quality, durability, design, and warranty coverage, collecting and processing all the information in order to choose the optimal product requires substantial time and effort.

In such a situation, a consumer may stop searching upon finding an alternative that satisfies certain criteria, rather than comparing all alternatives and choosing the most desirable one. For example, the consumer may establish in advance a criterion such as \textquotedblleft purchase a product if it costs no more than 100,000 yen, has the necessary functions, and is made by a reliable manufacturer,\textquotedblright\ and decide to purchase once a product satisfying those conditions is found. This behavior of choosing an alternative that is sufficiently satisfactory rather than searching exhaustively for the optimal alternative is called satisficing.

Bounded rationality does not mean that consumers are irrational or make erroneous judgments. Since gathering and comparing information also requires time and cost, simplifying decision making under limited cognitive ability and time can itself have a certain rationality. Rather, the concept of bounded rationality attempts to explain actual decision making on the premise that real people do not possess unlimited information-processing capacity.

Bounded rationality must also be distinguished from the rationality of preference relations as defined in this book. Rationality of a preference relation is the formal property of satisfying completeness and transitivity. Bounded rationality, by contrast, describes the fact that real decision makers make judgments under constraints on information, cognitive ability, and so forth. Thus, even under bounded rationality, the preference relation formed by a person may itself satisfy completeness and transitivity. Conversely, when cognitive constraints prevent some alternatives from being compared sufficiently, observed judgments may fail to satisfy completeness or transitivity.

In this sense, bounded rationality is not a concept that simply rejects the assumption of rational preferences. Whereas standard consumer theory uses rational preferences as a benchmark for analysis, bounded rationality is an idea for understanding the gap between such idealized decision making and the decision making of real people.

Exercises

Problem (Framing Effect)

There are two policies $A,B$ for combating an infectious disease. Suppose the following two alternatives \begin{align*}
\text{Policy }A & :\text{200 of 600 people are saved with certainty}\\
\text{Policy }B & :\frac{1}{3}\text{ probability that everyone is saved, but a }\frac{2}{3}\text{ probability that everyone dies}\end{align*}are presented, and many people choose $A$. On the other hand, suppose the following two alternatives \begin{align*}
\text{Policy }A^{\prime} & :\text{400 of 600 people die with certainty}\\
\text{Policy }B^{\prime} & :\frac{1}{3}\text{ probability that no one dies, and a }\frac{2}{3}\text{ probability that everyone dies}\end{align*}are presented, and many people choose $B^{\prime}$. Explain why the phenomenon described above violates the assumption of rational preferences. Also, what might explain why many people make such choices?

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