The Dark Theorem of Economics

Christopher J. Ayres sends me a paper.with the following abstract:

I begin by proving the "Dark Theorem of Economics," from which it follows that the foundations of economic theory rely on the Axiom of Choice (AC). All current solution concepts in game theory
also require the theorems implied by AC. In particular, lexicographic utility, lexicographic probability, the real line being well-ordered, and the existence of a universal space are all equivalent to AC; therefore any argument to disprove their existence must be false. Any proofs using properties that fail under AC must be redone. The concept of Nash Equilibrium becomes either a tautology (in the absence of AC) or violates rationality (in the presence of AC); we provide an example demonstrating this. Knowledge, Common Knowledge, Epistemics, Game Theory, and Macroeconomics (through the failure of Rational Expectations) must be rebuilt. Any economics Â…field or concept relying on these must also be rebuilt. I begin this process with the deÂ…finition of "Fundamental Game."

I joked with Chris that the other people who pursued this line of inquiry met with unfavorable ends.  But that doesn't mean he is wrong.  Fortunately, I am a pragmatist when it comes to the foundations of economic theory or lack thereof.  It's hard enough to define what a number is, so if you push on the foundations of micro theory, don't expect a completely comfortable journey.


What sort of pragmatist? A Quinean neo-pragmatist? Peircean pragmaticism? Flaccid, lame, intellectually lazy pragmatism?

Whoops, not always true for countably infinite sets, but always true for ordered or orderable sets (which include the integers and rationals).

Greenish makes a very good point. Overall, the paper reads like a very rough preprint which still needs a lot of work.

The topic is interesting, and a well-written review on these issues could attract a lot of interest from economists, mathematicians and people doing computable game theory/economics.

Since we comment on our own comments here, I probably better explain the point of my little joke.

We invoke the AC all the time as a semantic short cut. In my case, I used it to demonstrate that AC is not needed to handle a given collection. But AC is only needed to state the completely general case, because proof that a set is countable generally involves constructing an actual map. I say "generally", since I've been out of it long enough that I cannot recall immediately if such a proof necessarily involves such a construction. All I saw did.

For several decades, AC denier would construct their proofs using a fairly standard heuristic that would allow them to proceed as if they had accepted AC. I think that the practice has fallen by the wayside in mathematics.

First, let me note that I did make a mistake by using isomorphic when I just meant to say of the same cardinality. Sorry about that. However, everything is conceptually still correct.
I'll answer the emails in order:
greenish (1st post)
(1) The "Dark Theorem" does not assume the Axiom of Choice. It states that adding additional structure, such as knowledge, has just changed the state space unless it was a "trivial" addition, i.e. take any set and take the cross product of that set and the set {a}, where a is the only element, and the cardinality (size if we're in a finite space) remains the same. What you CANNOT do, is take a state space, say the 2x2 number of states in the prisoner's dilemma, and then allow each both agents to either know or not know whether their opponent is rational. To do this is to say that the original game, with 4 states, and the new game, with 16 states, are the same game.
(2) AC is not necessary in a finite space.
(3) Again, I'm following the logic of "traditional" game theory, which applies the possibility of both knowledge and a lack of knowledge of something an individual state, thus increasing the state space. My point: increasing the state space means you cannot compare the "new" game with the original game you were trying to solve. I show that most of the "paradoxes" in game theory are due to precisely this "meaningless" comparison. If you wish to compare two games with different numbers of states, you have just invoked AC. I note on the first page Tarski's Theorem and an interesting quote about its acceptance. The theorem says that the assumption that there is a bijection between X and X x X. This is where I misused isomorphism, sorry

I agree with greenish and anon that there are some problems with this
paper. Not a very impressive effort.

The go-to guy on this sort of stuff for some time has been Kumaraswamy
Vela Velupillai, along with some of his associates, a collection of
mathematicians, economists, and computer scientists, although some of
this dates back a ways, if not well known in the econ theory lit (duh).
Here are some decent entries in the discussion.

M. Pour-El and I. Richards, 1979, A computable ordinary differential
equation which possesses no computable solution, Annals of Mathematical
Logic, 17, 61-90.
K. Prasad, 1991, Computability and randomness of Nash equilibrium in
infinite games, Journal of Mathematical Economics, 20, 429-442.
M. Tsuji, N.C.A. da Costa, and F.A. Doria, 1998, The incompleteness of
theories of games, Journal of Philosophical Logic, 27, 553-568.
M.K. Richter and K.C. Wong, 1999, Non-computability of competitive
equilibrium, Economic Theory, 14, 1-28.
K.V. Velupillai, 2000, Computable Economics, Oxford University Press.
K.V. Velupilla, 2002, Effectivity and constructivity in economic theory,
Journal of Economic Behavior and Organization, 49, 307-325.

Oh, and I suppose I should mention my own paper, On the foundations of
mathematical economics, forthcoming in New Mathematics and Natural
Computation, which is available on my website.

Right Wing-Nut:
First, ????. If this is supposed to be a personal attack on me, I guess I'm confused? In particular since you're anonymous??

Second, you write

In his first "theorem", he claims that the axiom of choice is required to construct the power set of a finite set!

Here is what I stated:
Next, assume Ω is countably infinite. By Cantor's diagonalization argument, P(Ω) is isomorphic to the real numbers. Since Ω is isomorphic to Ω^{∗}, again we have the requirement of AC and the well-orderedness of R

I don't think you understand what has been written here. Let me explain it to you: Our goal is to show that additional structure on our existing space (such as adding knowledge of an opponents rationality) increases the cardinality of the new space. By adding "knowledge," to Ω to get some new state space Ω*, in the form of information partitions which have been generated by the power set of the natural numbers, we have just forced Ω* to be of the same cardinality of the continuum. But Ω is the cardinality of the natural numbers. So, our new state space Ω* with information partitions is NOT THE SAME as our original space Ω.

SO, any inquiries such as "why does (4,4) occur instead of (99,99)" in the game G8 in the paper, ARE COMPLETELY MEANINGLESS WITHOUT THE AXIOM OF CHOICE! (sorry to "yell," just wanted to really emphasize ;) )

This is where several "paradoxes" come from. Why does the first player just "move down" in the caterpillar game? If they assumed the other player would play irrationally....blah blah blah

When we assume knowledge is non-trivially attached to our state space, we've just changed the game. We then use that assumption of knowledge to reduce the game to something else. In every standard game, it becomes a set of exactly one element; this is what I call the Fundamental Group.

Sorry if I seem annoyed; to have taken even a cursory glance at the paper would have prevented this question (and insult, from some anonymous poster no less). I don't mind, IN FACT I REALLY WANT TO, explain my paper, but just as my opening quote says, people seem to get emotional/irrational when speaking about infinities.

So, this explanation was very long winded, and I apologize for this, but hopefully now it clears things up a bit.

When people ask these questions, they get extremely bad answers. Why? Because to make any kind of comparison between the games Ω and Ω*, when Ω* has an information partition attached to it, is is to

read this properly. Furthermore, you may wish to take a look at the examples.

i just wonder why this is such a hotly contested idea? if so much of mathematics requires ZFC, and AC has become largely accepted (?) within mathematics, then why is a discussion of its consequences (or equivalences) in economics necessary or controversial?

this brings me back to a funny story my measure theory professor told me about a prospective math phd student turning down mit because they taught ZFC.


I do not think you are doing yourself any favor with your aggressive posts. To paraphrase Carl Sagan: "Extraordinary claims require not only require extraordinary evidence but also extraordinary humility that you may be wrong".

By the way, I have no stake in the veracity of your claims.


A specific point is that there are variations on AC, and the results
here appear to depend on which version. There is also, the lack of
references to the not-so-small previous lit on this matter, with the
paper seeming to lack a bibiography.


While most mathematicians accept AC (and the law of the excluded middle and
some other things, like the continuum hypothesis, which I think Ayres should
have stayed away from), many don't, and this is a matter of ongoing
contention, with the constructivist critics fighting back hard in various
circles (the idea that a prospective grad student in math might turn down
going to MIT because they teach ZFC is not as big a joke as you might
think it is).

BTW, some of these problems can appear even in finite games. See
K. Prasad, 2009, The rationality/computability trade-off in finite games,
Journal of Economic Behavior and Organization, 69, 17-26. Ayres, you need
to do your homework a bit better on this stuff before you try out for
prime time.

@Barkley Rosser:

Actually, I think the supposed contention between classical and constructive mathematicians is somewhat of a red herring. As a rule, classical mathematicians have no problem with accepting constructive results as mathematically sound, and most of them will also readily admit that a constructive proof provides more information than a classical one.

Conversely, constructive mathematicians can make use of some classical results by reinterpreting them as negative statements (when they rely on the excluded middle or proof-by-contradiction) or by using other heuristics (as Right-Wing Nut mentioned); and much of what constructive mathematicians try to do is constructivizing existing mathematical theories.

The real interest of constructivist mathematics is as a "missing link" between formal mathematics as we all know it and the theories of computation and programming languages. Much of the theory of programming languages is more-or-less based on type theory, which is closely related to constructive mathematics. There's also a lot of interest from industry in the prospect of having viable formalisms for developing provably correct hardware and software systems.

Moreover, as mathematical proofs themselves grow more and more complicated, there is growing interest in using computers to develop precise formalizations of mathematical theories and certify them as correct. See [1] [2], as well as [3] [4] for a general interest overview. All of this has led to increased interest in the foundations of mathematics, including such "heterodox" systems as category-theoretical foundations, type theory and intuitionism.

At the end of the day, it is no surprise to see some interest in constructive foundations for mathematical economics as well.


I could fuss about details, but I largely agree with most of your last remark.

Quoting from your paper: "the non-existence of a bijection between the natural numbers and the real numbers is no longer valid".

If the non-existence of such a bijection is no longer valid, then according to your results there IS a bijection between the natural numbers and the reals, no? (You'll have a difficult time selling that one.) Or does this not mean what I think it means?

You also seem to think that the axiom of choice implies that the cardinality of a power set of a set can be equal to the cardinality of the set itself. But this is simply wrong.

Christopher Ayres,

Actually I find that I am not able to access your paper. I was looking at the one by Suber, which has no bibliography, and
my comments were on it. This leaves me a bit at loose ends as all I have read is your abstract. If what I am reading from some
of these other commentators such as Rognlie and greenish are in your paper, you have problems. For sure a power set does not
have the \ same cardinality as the set from which it is derived. It is this fact that is the key to the simplest
(non-constructive) proof of the infinity of the set of transfinite cardinals.

Interesting you should mention Alain Lewis. I did not mention him, but he was working on some of these issues back in the
1980s. People like Velupillai admire his work greatly and cite it, and Ken Arrow is reportedly a big fan of his work (he
was Arrow's student). Again, I have not seen your bibliography, but it sounds like you need to do more reading. I only
barely scratched the relevant literature on these matters. And, I would warn against getting too deeply into the continuum
hypothesis matter. Do you really want to mess around with forcing proofs? Even the Boolean alternative is not the simplest
thing to deal with, and, offhand I do not think it is all that relevant to what it appears that you want to get at.

And, regarding what appear to be your claims about game theory, well, several of the papers I listed deal with matters
that seem to be similar to what you are getting at.

I don't have time to read this in detail but what on Earth do you use "the real line being well-ordered" for? It's not well-ordered in the usual ordering. AC implies it _can_ be well-ordered, but I don't know of what use that could be in economics.

BTW, someone who was very aware of these matters from a fairly early time (1970s) was Herbert Scarf.

Just a silly point, why do you define actions in terms of payoffs and not strategies. It's just confusing to read.

I really don't see the value of this paper. Even if we were to accept Ayres' logic (a highly questionable beginning, with all the issues pointed out above, particularly the bijection between the naturals and reals), he doesn't even use his newly-invented concepts consistently.

In the "Hotelling Paradox" section, he claims that each firm has an incentive to deviate from the center position. I cannot see how, even under his new SDNE concept, he plans to defend moving away from the center and receiving a smaller share of the market. Even in his final solution, both people in the market receive precisely 1/2 of the market.

More generally, he appears to be advocating a very basic and limited-application solution concept: in symmetrical games, play that strategy for which a symmetrical choice by the other player produces the highest payoffs.

I very much want to play the one-shot Prisoner's Dilemma against Mr. Ayres.

The true "Dark Theorem of Economics": Don't send your working papers to Tyler unless you're ready to be taken to the cleaners.

Foo: Tarski's Theorem says that for any infinite set Y, there is a bijection between Y and YxY. I mistyped my statement of it on here, although in the paper this is correct. Your example, if correct, is proving that Z and Q have the same cardinality, i.e. there is a bijection between them, which is true. However, there is no bijection between R and RxR.

For an undergrad, matt, you've got a lot of advice on how to have a successful career as an academic economist.

If by a "a lot of advice" you mean "very specific advice to a single person to avoid spending time during the fifth year of his Ph.D on a subject where he lacks basic foundational knowledge," I plead guilty.

Imagine an abstract that says: "I am going to use powerful tools of the theory of continental drift to cast exciting new light on economics," and then the paper begins by saying "The theory of continental drift shows that the earth is flat. Therefore, all economic propositions that depend on the proposition that the earth is round are invalid." Then an undergrad geology major says, "Uh, you might want to rethink the basic direction of this paper, which threatens to derail your time frame for completing your Ph.D." I don't think that's particularly presumptuous. It's polite if anything.

Put up 1000 to 1 on what? That the Axiom of Choice implies that there is a bijection from the reals to the integers? How much money do you have?

Chris -

Please do yourself a favor and read Patrick Suppes' book Axiomatic Set Theory, available in reprint for cheap from Dover Books. Then see if you want to modify anything in your paper.

I am about to go out of town (again) and have a great deal to do before then,
so I do not have time to dig through and adjudicate all kinds of details here.
I did finally access the paper and have looked it over, but not in detail. I
have a few remarks.

1) My main original point holds. Much of what appears to be true, or possibly
true, has already been shown, or something close to it. The biggie is the
relationship between AxC (a preferred usage in the lit rather than "AC," btw)
and the inability to establish Nash equilibrium. I have seen your bibliography,
which does exist, but is clearly inadequate. Now that I know that you are a
grad student, I hope you take my advice to go read some of the sources I
provided earlier in this thread. You need to figure out what has been proven
already and what has not and how it relates to what you are trying to do.

2) I strongly recommend against setting up this market (hope that Robin Hanson
is reading this). I understand that you are trying to impress people with your
willingness to go out on a limb, but unless you are independently wealthy, I
would strongly suggest that this is not a good idea, even if somehow Robin thinks
it is exciting.

There are two big problems. The first is, what constitutes "winning" for
either side? Do you lose if somebody finds even one minor error in your
paper (I think that has already been done, although I did not dig through
in sufficient details to see if some of the characterizations being thrown
around here were accurate or not)? Or do you win if even one major part of
your "dark theorem" proves to be true? Also, do you win if it is true, but
has already been established by Pour-El and Richard or Prasad or somebody else?
And, finally, just how is it going to be determined "who wins"? Who are you
going to bring in to determine this? Me? Aumann? Velupillai? Prasad? Who?
(And, no, I am not going to do it, sorry.)

In any case, I think you would be spending your time more fruitfully by going
and doing some reading of the literature rather than getting into some kind of
macho fight with a bunch of people on this blog, and trying to resolve it by
some kind of betting market set up by Robin Hanson is an especially pointless
waste of time. Hey, the bottom line in the end is going to be publishing (not
to mention getting that Ph.D.), not winning or losing on some betting market.
Clearly you have gotten some publicity here, but it has hardly been all that
favorable so far, I fear. Kind of jumping the gun a bit, although maybe you
will get something useful out of the exercise.

Good luck getting it together and finishing up. Yes, this stuff has become
hot for a variety of reasons, many of them related to the rise of computable
and computational economics.

axg, if this is a hoax, I doubt that Tyler is in on it. I think that he's just a generous and thoughtful guy, and he doesn't have the time to review every paper he links for technical accuracy. (I have been the grateful beneficiary of his linking generosity before!) The premise of Chris Ayres's paper seems at least vaguely plausible, and as (apparently) a Ph.D candidate at UC Davis he has more credibility than a run-of-the-mill crank writer. This, in fact, is what makes this paper so strange, and what raises the possibility of a hoax: such a bizarre, confused paper is absolutely what I'd expect from an overexcited dabbler with no knowledge of math or micro theory, but it's unusual coming from a fifth year graduate student. My puzzlement on this point is the only reason I've stayed in the discussion this long.

As Barkley Rosser explains, this is a very bizarre proposal for an information market. The purpose of such markets is to aggregate information that is not trivially available through other means, and for a market to be functional, this information needs to be objectively verifiable. But we already have all the information we need, and it's in the most objective form possible -- mathematical proof. I can easily prove that there is a bijection between R and R^2, or that a set and its power set cannot have the same cardinality, or any of the other foundational errors that seem to be in dispute here. It seems to me that an information market in this case would consist of Chris writing the other commenters and me a check. Except, of course, that the fact that he denies these errors (or at least doesn't respond to them) suggests that he is unwilling to accept straightforward mathematical logic, at which point it's difficult to have any confidence in the basic rules of a market.

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