More on discontinuity at zero

[Itzhak Gilboa]

Dear all,

A new paper, "Zero Risk", dealing with discontinuous preferences over lotteries, is available at

https://itzhakgilboa.weebly.com/uploads/8/3/6/3/8363317/gmw_zero_risk.pdf

The abstract is:

When preferences are expressed in natural language, it is not uncommon to encounter discontinuities. For example, "We will never risk people's lives" seems to suggest a discontinuity at zero probability of an unacceptable outcome (death). Should such statements be taken at face value? We take an axiomatic approach, asking what modes of behavior are and are not compatible with such statements. Specifically, for decision under risk, we weaken the vMN axioms to derive two rather natural versions of discontinuity "at zero".

We have also extensively revised a previous paper, relating to discontinuity in consumer choice.  The new version, titled "Meaning and Discontinuity in Consumer Choice", is available at

https://itzhakgilboa.weebly.com/uploads/8/3/6/3/8363317/gmw_meaning_and_discontinuity.pdf

and its abstract is:

Continuity is a basic assumption in consumer theory, and it seems rather plausible when physiological processes are taken into account. But when consumption is the carrier of meaning, discontinuities may arise. Specifically, consumer preferences may behave discontinuously at zero quantities, as in the case of vegetarians who prefer not to consume any amount of animal meat. We argue that, as opposed to the example of lexicographic preferences, the discontinuity in such cases is not only in stated preferences, but can also be matched by consumption behavior. Relatedly, it can be represented by a numerical utility, and we provide an axiomatization of such a function.

Comments are, as always, welcome!

Thanks,

Stefania, Fan, and Tzachi

[Itzhak Gilboa]

Dear all,

In the context of discontinuity-at-zero issues, Stefania, Fan, and I ran across the following paper by Igor Kopylov:

    https://econpapers.repec.org/article/eeemateco/v_3a67_3ay_3a2016_3ai_3ac_3ap_3a32-37.htm

-- which we had not been aware of before.  I think that it has never been advertised here, so I'm doing it now.

Among the various (very nice) contributions of the paper, there's one that I think may be of interest to many: a simple proof of Debreu's continuous representation theorem.  I am among those who teach graduate level micro, mention Debreu's theorem but don't prove it in the general case.  A substitute is to prove the result in the case of monotone preferences.  In that case the proof is simple and intuitive, but it still feels awkward to state the general theorem without proving it.  And other proofs, such as Debreu's (or Jaffray, 1975) are a bit too complicated if you still have all of consumer theory and GE existence ahead of you in the same semester.  By contrast, Igor's proof is simple enough to do in class.  

In case you're interested, the gist of the idea is to take a countable set that "separates" the space, assign utility values to this set by induction in the most natural way (pick the midpoint of the interval of possible values) and then take sup/inf for the rest.  As stated, this doesn't sound like it's going to work, but the trick is the definition of "separates".  In most of the literature, one assumes 

     Point Separability:  There exists countable set Z such that whenever x>y there exists a z in Z such that x >= z >= y

but the condition that is used here is

     Interval Separability:  There exists countable set Z' such that whenever x>y, there are two elements z1, z2 in Z' such that x >= z1 > z2 >= y

Obviously, Interval Separability implies Point Separability.  The converse takes a few lines to prove, but it's easy enough, and has appeared in the literature.  (A countable set Z that satisfies the condition of Point Separability need not satisfy the condition of Interval Separability, but by adding at most two elements for each element in Z, one obtains a larger set Z' that does.)

And then, constructing the utility by induction as above on a countable set that satisfies Interval Separability yields a representation also on the rest of the alternatives.  Proving that, if the order is upper semicontinuous / lower semicontinuous / continuous, then so is the utility function isn't too difficult.

David Schmeidler used to say that, when a result is truly important, its proof often gets shorter over the years.  This applies to John Geanakoplos's proof of Arrow's Impossibility theorem (Economic Theory 2005, preprint 1996 -- see also Reny's 2001 Economic Letters unified proof for Arrow's and Gibbard-Satterthwaite results).  I think it also applies here.

Best,

Tzachi

[Mohammed Khan]

Dear Members of the Decision Theory List, 

It is still morning in Europe, and certainly morning here in Baltimore. I greet everyone on this (my) morning. 

I am not a regular reader of this list, leave alone post items on it, but by a chance perusal, my eyes fell on this post of some time ago by Tzachi Gilboa. It is a wonderful post, and I shall follow up the substance of it. It also reminds me  of a note in Econometrica by Mehta-Bearden on "Debreu's gap lemma" which I teach in my graduate course on "mathematical thinking and reasoning in economics" here at Hopkins. I teach it as a good illustration of the quotient topology in mathematical economics. Anyhow, like I say, I shall look up again the references Tzachi gives. 

But remaining with the list, I also read Karl Schlag's post of yesterday.  I read (this) Tzachi's message is an antidote to his lament and instruction. 

I am not really part of the decision theory community other than a welcome and kind consideration (always) from Tzachi Gilboa at his DTEA  Paris conferences -- my admiration for David Schmeidler dates to his early  GE papers and surely does not count as an entitlement to membership. 

Anyhow, to cut my verbosity short, and not intrude more than necessary on your valuable time, let me say that I sent some recent work Ani Ghosh (Cal Poly) and Metin Uyanik (Queensland)  to Tzachi and Peter Wakker for their comments day before yesterday.   It is part of our ongoing explorations of the continuity and convexity postulates in economic theory and thereby of stylized academic interest. This investigation goes back  to Schmeidler's Ectra 1971 two page note that has no references! and which Metin and I followed up in our {\it Economic Theory}  piece about five years ago  (2021) 71:411–460.

And now the punchline: My co-authors and I were heartened and very much encouraged by the prompt responses of both Tzachi and Peter, and the former thought that the work would be of interest to the members of this list, and that I should post it.  I have checked up with my co-authors, and with their agreement, do so with pleasure. 

To be sure, any criticisms and expressions of interest would be most welcome. 

Warm regards, Ali 

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