## Cognitive Dissonance

I just finished reading Cooper’s book on the topic and would very much like to write down what I get out of it.

Cognitive dissonance was first envisioned by Leon Festinger in the 1950s. The theory started out elegant and simple: With consistent cognitions, individuals experience uneasiness and discomfort thus are motivated to reduce the inconsistency. Here cognitions can refer to actions, beliefs, attitudes, and so on. Inconsistency can be due to inconsistent attitude and action. One of the many experimental paradigms to test the theory runs as follows: subjects are asked to evaluate a certain proposition arguing against her/his position. Lastly attitude is elicited again. For instance, subjects (usually college students) are asked about their attitudes towards increasing tuition. Expectedly most of them are against it thus are asked to write an essay/make a speech for tuition increase after which they are asked again about their attitudes on the issue. (All results will be discussed under this paradigm for exposition.) Typical findings on these experiments show that subjects change their attitudes closer to what they (are asked to) advocate. According to Festinger, subjects experience cognitive dissonance: writing an essay for tuition raise is inconsistent with their attitude against a raise. As a result of the dissonance, subjects are motivated to reduce the inconsistency. In this case (as in many other cases), changing one’s attitude is easier than changing a committed action.

The theory is appealing as it explains, elegantly, experiments and observations. However humans are hardly that simple (thus the struggle of social scientists). In the following years, many experiments and theories are raised to clarify and complete the theory. I will first discuss efforts on verifying the physiological manifestation of dissonance and then talk about modifiers found that largely rewrite the whole story.

One of the first questions raised against the theory concerns the motivated drive (Chapter 3 of the book) Festinger highlighted in his original composition. Simply put (especially intuitive to economists), psychologists wondered if people really experience tension states/arousal/emotions or anything with a physiological trait thus are motivated/driven to reduce inconsistent or is it just a metaphor/as-if argument. With the hep of misattribution and two-factor theory, experiments show that the uneasiness is experienced

• Two-factor model of Schachter and Singer (1962): an experience of emotion is based on two factors: arousal and the attachment of a cognitive label to that arousal. They conducted experiments to show that the factors are separable in the sense that how the arousal is experienced is affected by the label provided. In short, the experience of emotion is labile.
• Back to dissonance: subjects who take placebo but think they take a drug which might cause uneasiness and discomfort, do not change their attitudes as much after the essay task. Similarly, subjects who think they take a drug that might cause relaxation change their attitudes more after the task.
• These experiments show that when provided with a convenient label of their discomfort, subjects do not attribute the arousal to inconsistent cognitions thus make no effort in reducing consistency.

These are good news for the theory which solidify the original ‘guess’ of Festinger. Now the ‘bad’ news: New Look Model and Self-Standards Model (SSM henceforth) that largely kill the original idea of the theory. SSM is basically an extension of the former; some years passed before the birth of SSM but I will skip the struggle and talk directly about SSM as if it were a success in one shot (but it is really not).

• The following experimental results forced researchers to rethink about the theory: No/little attitude change is observed if (1) subjects are not presented with a choice, (2) the action does not result in any aversive and foreseeable consequence, or (3) the behavior is expected by some (social/personal) standard. For the essay task,
1. Attitude changes if subjects are asked if they would like to write such an essay (high-choice treatments) and no changes if subjects have to do so (low-choice treatments).
2. Attitude changes if the essay is to be presented to the college committee and no if it is destroyed after completion. Attitude changes if the essay might be viewed by the committee and no if it comes as a complete surprise (unforeseeable).
3. This point cannot be demonstrated with the paradigm and plays an important role in turning Festinger’s theory up-side-down. So I will discuss it in more details in the next entry.

Just as a side note: the experiments become more intriguing as the theory gets more complicated. The levels (plural really) of deception used in these experiments stunned me, an economist who sleeps with the dogma that deception is not allowed in experiments.

Okay this entry is getting too long to bear even for myself. I am gonna end here.

[1]Cooper, Joel. Cognitive dissonance: 50 years of a classic theory. Sage, 2007.
P.S. This is a very nicely written book: well-organized and fun to read. Perfect as an introduction into the theory.

## Zorn’s Lemma and Revealed Preference

I don’t think I ever understand why Axiom of Choice is always singled out; why it is special (and controversial) relative to other axioms. Gradually, I realized that probably it is not. It is just one of the axioms that found set theory. Anyways, this is not what we, as economists, should care. This entry is about Zorn’s lemma and revealed preference. I mentioned axiom of choice first just because its close relation to Zorn’s lemma. To complete the discussion of axiom of choice, let me just put the version that I personally like most here

Axiom of Choice: Cartesian product of non-empty sets is non-empty.

Now Zorn’s lemma and revealed preference. Revealed preference is one of the most fundamental result in economics. It builds a bridge between observable choice data and underlying (but invisible) preferences. There are various versions of the result but the idea is the same

Revealed Preference: Any set of choice data that satisfies some axiom of revealed preference can be rationalized by a preference relation.

In other words, however limited the data are, we can find a preference defined over the whole commodity space to make sense of the data, given that the data satisfy some conditions. Zorn’s lemma allows a compatible extension from the limited scope provided by the data to the whole commodity space.

Zorn’s Lemma: If every chain in a partial-ordered set has a upper bound, then the partial-ordered set has a maximal element.

Zorn’s lemma is essential to the following result which basically makes the revealed preference a corollary.

Szpilrajn’s Theorem on total extension of preorders: Any preorder has a compatible extension to a total order.

Below I just list the definitions of the relevant concepts assuming familiarity of common properties of binary relations

• Preorder Reflexive+Transitive
• Partial order Preorder+Antisymmetric
• Total order Partial order+Complete
• Transitive closure of a binary relation R over X is the smallest relation over X that contains R and is transitive
• Chain a totally ordered subset of a partial-ordered set
• A binary relation R* over X is a compatible extension of R if R* extends R and preserves the asymmetry of R.

Proof of Szpilrajn’s Theorem:

Let R be a preorder over X and E the set of all preorders that compatibly extend R. Clearly E is non-empty as it contains at least R itself.

Consider any non-empty chain C in E by set inclusion. We want to show that U the union of elements in C is an upper bound of C (so that we can use Zorn’s lemma). That is we need to show that all elements of C is contained in U; alternatively, U is (still) a preorder that compatibly extends R. This can be shown easily by using the properties of individual element in C (at the end of the day, one needs to show that U is reflexive, transitive, and compatibly extends R).

By Zorn’s lemma, we know that E has a maximal element S. The last step is to show that S must be total. Suppose not and there exists a pair $(x,y)$ such that neither xSy nor ySx. Consider $T=S\cup\{(x,y)\}$ and W which is the transitive closure of T. Since W includes S, if we can show that W is a compatible extension of R (reflexive is obvious), we reach a contradiction (that S cannot be the maximal element).

Again suppose W does not compatibly extend R (or S) and there exists a pair $(u,v)$ in X such that $(uSv) \wedge\ ^\neg (vSu)$ but vWu. By definition of transitive closure, we can construct a finite sequence such that $v=c_1 T c_2 T \cdots T c_{n-1} T c_n=u$. Since T and S only differ over (x,y) and if no $c_i$ is x or y, we obtain the contradiction. Otherwise, we can find $(c_i, c_{i+1})=(x,y)$. If this is the case, $y=c_{i+1} S\cdots S c_n=u S v=c_1 S \cdots c_i =x$ contradicting that S cannot rank x and y.

Thus the proof is complete and any maximal compatible extension of R is a total preorder.

The definitions and proof are from Aliprantis and Border’s book. I put it here just because I did not know why economists cared about Zorn’s lemma and this is an interesting illustration in my opinion. And I want to throw away my scratch papers.

Reference
[1]Aliprantis, Charalambos D., and C. Kim. “Border. 2006. Infinite Dimensional Analysis: A Hitchhikers Guide.”