# Existence Theorems are Reductios

by on September 8, 2003

John Quiggin gives a modest defence of existence theorems in economics, one of the three real vices of economists according to Deirdre McCloskey.

bq. Existence theorems, for McCloskey are the archetypal example of ‘blackboard economics’, mathematical games yielding purely qualitative results that can be overturned with modest changes in assumptions. They were the high point of mathematical economics in the 50s and 60s … There are a wide variety of ‘impossibility theorems’ demonstrating the non-existence of index numbers with various properties [an area of research interest for John]. Familiarity with such theorems can save a lot of pointless effort, and they are therefore worth looking for. But an impossibility theorem is just the negative form of an existence theorem (or, if you prefer, an existence theorem proves the impossibility of the corresponding impossibility theorem).

bq. This is a rather prosaic defence, that certainly does not justify the high status accorded to the kind of theory exemplified by existence theorems. But the argument can be pushed a bit further by considering the most famous impossibility theorem, that of Arrow who showed (roughly speaking) that no voting system having a set of seemingly desirable properties could work for all possible sets of voter preferences. This impossibility theorem precluded a lot of potential effort in designing ideal voting systems. [Emphasis added.]

This is a nice parallel. Actually, it’s so nice that it may prove more than John intended. (I absolve him of responsibility for what follows.)

Arrow’s Impossibility Theorem illustrates the point neatly. We begin with the assumptions. Roughly speaking, Arrow made a list of four criteria that reasonable people might think any method of aggregating people’s preferences ought to have. These conditions are (1) That the full range of everyone’s preferences be considered, (2) That if everyone prefers x over y then the group decision should as well, (3) That the position of x relative to y in the group prefernece depends only on the position of x relative to y in each individual’s preference, and (4) That there isn’t a dictator — i.e., someone who gets to have their preference enforced over everyone else’s. We then move forward to the impossibility result: there is in fact no method satisfying all four criteria. Any voting system would necessarily violate at least one of them. The proof is striking because the initial assumptions are so plausible, even weak, but they cannot all be satisfied together and so we find that the desirable result is impossible. And so we give up our quest for what we now realise is a chimera — the idea of a perfect method of aggregating individual preferences.

As John says, existence theorems are the negative form of impossibility theorems. The classic existence theorems in economics — such as those for general equilibrium, also due to Kenneth Arrow, along with Gerard Debreu — illustrate the point neatly. We begin with the result. Roughly speaking, Arrow and Debreu wanted to show that supply and demand could be in balance in all markets at once. We then move backward to the assumptions necessary to make possible such a result. These include (1) All individuals are perfectly rational, (2) All trades take place simultaneously and instantaneously, (3) There is perfect information about all markets for all products in all conditions both now and at any point in the future, (4) Money does not exist. With these (and other) assumptions in place, the existence of a general equilibrium can be proved. The proof is striking because the initial assumptions are so implausible, even absurd, but they must all be satisfied together in order for the desirable result to be possible. And so we give up our quest for what we now recognise is a chimera — the idea that our world could ever contain economies capable of general equilibrium.

Whoops. I suppose many economists wouldn’t take that last step along the road. For their own reasons, most economists do not treat existence theorems of this sort in the obvious way — i.e., as a kind of useful reductio ad absurdum, or at least ad ridiculum. I sometimes wonder why economists so rarely adopt this interpretation. (I hear the phrase “F-twist” on the breeze.) The most plausible interpretation of the results is, “Well, we got the desired result … But look at the assumptions we had to make to do it. Absurd. The result can therefore never obtain. QED.”

Update: John Quiggin responds. Maybe I was wrong about the realism of the Arrow-Debreu temporality assumptions, because his comments seem to have appeared pretty much instantaneously.

{ 5 comments }

1

Juan 09.08.03 at 12:44 am

Just a small point: that a result follows from certain assumptions doesn’t imply that those assumptions are necessary to generate the result. That may, of course, be true, but it’s being true (if it is) is not part of the proof of the result in question. In general, reductios work when the absurdity is a necessary condition of the result to be reduced, not when it is a sufficient condition. I might just be missing something.

2

Kieran Healy 09.08.03 at 1:06 am

that a result follows from certain assumptions doesnâ€™t imply that those assumptions are necessary to generate the result.

Yes, that’s right, I fudged over that in the post. (I plead “It’s a blog, not the AER.”) The promise of many such theorems is that we’ll find a way to prove a more general case later. Often this never happens.

3

Michael Kremer 09.08.03 at 1:19 am

But now (after reply to Juan) your terminology (“existence theorems”) is confusing.

Amplifying on Juan’s point — the result that IF such and such, then there is a general equilibrum, is an “existence theorem”. The result that the existence of a general equilibrium REQUIRES such and such is NOT an “existence theorem.” It’s really of the form “if there is a general equilibrium, then such and such” and that assumes existence as the antecedent of the if-then, rather than proving it as the consequent. But this means it’s really already an impossibility theorem in disguise.

For, notice that:

Conditions C are necessary for a K to exist

is equivalent to

If a K exists, then conditions C hold

which is equivalent to

If conditions C fail to hold, then no K exists

which is an IMPOSSIBILITY theorem.

I think that what you’re really noticing is that the search for an existence theorem can lead to both necessary AND sufficient conditions for the existence of something, in other words, a claim of the form:

a K exists if and only if conditions C obtain

which is really both an existence theorem (if conditions C, then a K exists) and an impossibility theorem (if conditions C fail, then no K exists) at once.

This, however, isn’t a defence of existence theorems, but of the search for them.

4

zizka 09.08.03 at 3:45 am

In mathematics I have read people talking about certain solutions or proofs being “possible in principle”, in the sense that they are not impossible like squaring the circle, but would merely require a computer half the size of the known universe to solve. Mathemeticians also speak of “very large numbers” which are still not infinite and thus can be handled just the same as ordinary numbers: for example, if you take the estimated number of electrons in the universe, square it, and keep squaring the result 100 times or so.

I agree with the skeptical conclusion that proving that something is formally possible isn’t proving much in a science which is meant to have any real- world (world of historical time, world of historical experience) applications.

5

Walt Pohl 09.08.03 at 4:16 am

I disagree with most of the comments here. The existence of general equilibria is not a priori obvious under _any_ conditions. If general equilibria didn’t exist even for perfect markets, then that would be a significant fact.

I didn’t know about the “no money” condition, though. How does the proof depend on that?

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