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.”

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Commitment and Belief Revision

I came across this working paper by Falk and Zimmermann while browsing the tentative program of a workshop I am attending in the summer. Like reading the information projection paper by Madarasz (2012), I was shocked to see that no similar experiments had been conducted in economics or psychology! Typical of Falk’s experiments, all treatments are extremely simple. Subjects are shown a jar and asked to estimate the number of peas in it. Estimation is of course incentivized. Mainly they are interested to see if commitment changes the way/extend people process information and revise their beliefs. The commitment is generated by writing down an initial estimate (which is payoff-irrelevant) on a piece of paper. It turns out that commitment has huge effect on information processing and belief revision. In short, committed subjects are more reluctant to incorporate additional information or revise their beliefs. What interests me more are two additional treatments with subtle differences. In one, subjects wrote down their estimates but showed them to no one, not even the experimenters. Yet such private commitment is sufficient to generate the main effect! (Falk and Zimmermann are working on explaining the underlying mechanism of this commitment effect and it seems that they lean on the social root?!) In another, subjects only raised their hands once they had an estimate. In this treatment, no commitment effect was observed, which actually appears a little peculiar to me. After all, how (and why) would writing down something to oneself be any different from determining something in one’s head? As I’ve always believed, given what we have learnt along the way, understanding how individuals make decisions is all about figuring out the boundary conditions – this is a nice demonstration of the idea.

References
[1]Falk, Armin, and Florian Zimmermann. Information processing and commitment. mimeo, 2016.
[2]Madarász, Kristóf. “Information projection: Model and applications.” The Review of Economic Studies (2011): rdr044.

Renegotiation

I came across a working paper by Barron, Gibbons, Gil, and Murphy (BMMG henceforth) in which they look at the contracting between distributors and exhibitors of Spanish movie industry. The contracting seems to be half form and half relational in the sense that the distributors and exhibitors formally agree upon how to divide the revenue ex ante but renegotiation is a possibility (and usually is exercised) ex post (after the screening ends). The main part of the paper is empirical which I skipped. However they do include two simple models for illustration. The second one is a multi-unit auction model which I again skipped; the first one is more of my interest which is a relational contract model. Looking into it more closely, I realized how simplified the model it (it seems very similar to a toy model discussed in Kozsegi’s behavioral contract survey paper). In a nutshell, nothing is new there; renegotiation occurs after the resolution of uncertainty and is motivated by continuation value.

Then I grew some vague interests into renegotiation and started to think about the implications of loss aversion on that. There is a paper by Herweg (right who else; such a fan) and Schmidt published on RESud in 2014. They investigate the effects of loss aversion on renegotiation; the reference point is set to be the ex-ante contract. Unsurprisingly, loss aversion hinders renegotiation thus imposes additional welfare loss (besides the part created by the loss function of course). The paper also touches upon authority contract, which I did not exactly understand. Though it seems to me that the paper focuses on what I call ex-post flexibility, such as the likelihood of renegotiation. On the other hand, ex-ante flexibility is rarely if at all mentioned. Hart and Moore 2008 (oh how much I like this paper) suggest that loss aversion also reduces the value of ex-ante flexibility. They assume that parties feel entitled to the best possible outcome allowed by the contract and would revenge by shading if it is not achieved ex post. As a result, a more inclusive contract creates entitlements to better outcomes thus increases the chance of shading. One avenue that is potentially interesting is to look at whether these two forms of flexibility is substitute or complement and which form is optimal given different level of loss aversion.

References
[1]Barron, Daniel, et al. Relational Adaptation under Reel Authority. mimeo MIT, 2015.
[2]Herweg, Fabian, and Klaus M. Schmidt. “Loss aversion and inefficient renegotiation.” The Review of Economic Studies (2014): rdu034.
[3]Hart, Oliver, and John Moore. “Contracts as reference points.” The Quarterly Journal of Economics 123.1 (2008): 1-48.