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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
field or concept relying on these must also be rebuilt. I begin this process with the de
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.
Posted by Tyler Cowen on June 29, 2009 at 11:45 AM in Economics | Permalink
Comments
What sort of pragmatist? A Quinean neo-pragmatist? Peircean pragmaticism? Flaccid, lame, intellectually lazy pragmatism?
Posted by: Selfreferencing at Jun 29, 2009 12:55:42 PM
Three objections:
1) "therefore any argument to disprove their existence must be false."
Say what? The Axiom of Choice was not passed down from On High. It may be accepted or not accepted depending on the system you wish to use.
2) The paper tries to argue that the Axiom of Choice is necessary for the "dark theorem" regardless of the cardinality of the state space, but the Axiom of Choice (i.e., that you can always pick elements out of a collection) is always true for finite and countably infinite sets. One need not accept the full Axiom of Choice to accept that.
3) "To see this first assume that Abs[omega] < Abs[omega*]. Then we must have that [omega] and [omega*] are isomorphic by definition of a state space, which is true only if AC holds"
Looks like an always false statement to me. When two sets are "isomorphic", they must have equal cardinality.
Posted by: greenish at Jun 29, 2009 1:02:55 PM
Whoops, not always true for countably infinite sets, but always true for ordered or orderable sets (which include the integers and rationals).
Posted by: greenish at Jun 29, 2009 1:08:48 PM
Can it be true, however, that the rational expectations theory has been tested only with aggregated data on individual behavior?
And if so, is it even fair to call an individual "rational" or "irrational" because he or she deviates from some composite profile of individual behavior produced from group average?
Posted by: Michael F. Martin at Jun 29, 2009 1:22:28 PM
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.
Posted by: anon at Jun 29, 2009 1:25:13 PM
greenish: Well, if have a countably infinite set, you must have a mapping between it and the naturals. Pick one. There is your order. Ermm..
Seriously, though, isn't a two-month bender a bit long? In his first "theorem", he claims that the axiom of choice is required to construct the power set of a finite set!
Go back to graduate school. Do not collect a doctorate. Do not propose papers without the approval of a qualified supervisor.
Posted by: Right Wing-nut at Jun 29, 2009 1:29:51 PM
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.
Posted by: Right Wing-nut at Jun 29, 2009 1:37:12 PM
greenish: Well, if have a countably infinite set, you must have a mapping between it and the naturals. Pick one. There is your order. Ermm..
That's what I was thinking, but then I remembered/rediscovered Axiom of Countable Choice, and it's not provable in ZF.
Posted by: greenish at Jun 29, 2009 2:04:53 PM
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
Posted by: Christopher Ayres at Jun 29, 2009 2:05:47 PM
Oops, Tarski's Thm. says the assumption of a bijection between some set Y and Y x Y implies that AC holds.
Posted by: Christopher Ayres at Jun 29, 2009 2:11:45 PM
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.
Posted by: Barkley Rosser at Jun 29, 2009 2:19:54 PM
Dr. Rosser, for those of us that are less familiar with this area of research and are trying to assess its contributions, what exactly are the problems you have with the paper?
Posted by: anon2 at Jun 29, 2009 2:36:33 PM
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??
Anywho,
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.
Posted by: Christopher Ayres at Jun 29, 2009 3:09:13 PM
Angels can be heard snoring on the head of a pin.
;)
Posted by: Buzzcut at Jun 29, 2009 3:19:00 PM
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.
Posted by: taylor at Jun 29, 2009 3:38:20 PM
Next, I ask everyone who may think this paper is similar, what paper proved that NE is incorrect under AC, in that it chooses either
(1) the single remaining strategy when we don't incorporate knowledge into the state space (this single remaining strategy is the "Fundamental Game of 1 degree" from the abstract, sorry again I accidentally said Fundamental Group last post) after we have reduced the state space through "knowledge," or it chooses
(2) actions that are STRICTLY DOMINATED by the NE concept that both I and BBD generate?
???????
???????
???????
Posted by: Christopher Ayres at Jun 29, 2009 3:40:48 PM
Chris,
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.
Posted by: An observer at Jun 29, 2009 3:48:00 PM
An excellent point, thank you very much. I think it will make more sense shortly though on my next post...
Posted by: Christopher Ayres at Jun 29, 2009 3:52:05 PM
anon2,
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.
taylor,
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.
Posted by: Barkley Rosser at Jun 29, 2009 3:54:36 PM
I am not sure I entirely grasp the significance of this paper. Is it supposed to be an attack on the epistemological justifications for NE? If so, I thought that was already understood to be on shaky grounds. In any case it is clear that these types of justifications are not readily applicable to real economic actors.
This is why learning in games/evolutionary game theory are such hot fields right now. These fields try to understand when equilibrium behavior can be expected and when it cannot.
Posted by: Michael at Jun 29, 2009 4:59:58 PM
@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.
Posted by: anon at Jun 29, 2009 5:22:45 PM
Oops, Tarski's Thm. says the assumption of a bijection between some set Y and Y x Y implies that AC holds.
Luckily Tarski's Theorem doesn't say this because it isn't true. You have the, I assume, correct theorem in the footnote in your paper but that theorem doesn't mean what you think it means!
For example, let X be the set of nonnegative integers. Then x in X maps to ( even(x), odd(x) ) where even(x) is the decimal integer formed by concatenating the integers in the even positions and odd(x) is the integer using the integers in the odd positions working right to left. I.e., 1984 maps to ( 18, 94 ).
The inverse map (y,z) to x just interleaves the digits of y and z adding zeros as needed and also working right to left.
Well, I hope that's right...
Posted by: Foo at Jun 29, 2009 6:02:45 PM
anon,
I could fuss about details, but I largely agree with most of your last remark.
Posted by: Barkley Rosser at Jun 29, 2009 6:19:09 PM
Sorry about the response delay, I accidentally navigated away from the page!
Basically,
Although my field is of course economic theory, I am a 5th year PhD Candidate because of my work in Information Markets (a colleague of Tyler Cowen's, Robin Hanson, is of course the primary proponent of them). Suffice it to say, traditional financial economics, and in particular empirical finance, strictly maintains the assumption of common priors and "common knowledge." These people will fight my assumptions at every point (Stephen Morris has some great papers on this; see in particular "The Common Prior Assumption in Economic Theory," which gives a good idea of how difficult it is to talk about subjective probs., which turn out to be completely correct)
So, 9 days ago I looked at Aumann's paper and decided I'd try to disprove it. Well, it turns out that I had succeeded, and further I read through Geanakoplos' survey and found several problems, such as the "two-envelopes" problem, which I realized were the result of various state space issues. At this time I wrote the "Dark Theorem," completely unfamiliar with the rest of the literature, and wound up independently EVERY CONCEPT IN THIS PAPER.
I'm not trying to boast here; having come from math, which is a very "ego-intensive" field ;), I've found that such claims are usually made by those who lack self-confidence, in some kind of an attempt to gain some "relief" by make others feel the same way.
My point is that the "Dark Theorem" has allowed me to show all the concepts of this paper! In my brief literature review I've "independently" (an not a credible statement, but I make it just to urge you to see the implications of the theorem!) been able to generate 10 different econometrica papers. This is only because the theorem told me where to look, AND FOR NO OTHER REASON!
I included the quote about Tarski because it shows that even the top people in the field can make mistakes. What happened to Abel? He sent his research to Gauss, who dismissed it out of hand and without even reading it! What happened to Cantor? His ideas were wholly rejected in his lifetime, and he died in an insane asylum! What happened to Godel, whose "Incompleteness Theorem" I've alluded to in the title of my paper? He starved to death under the fear that people were trying to poison him! And what has happened to the brilliant Nash? von Neumann said it was worthless! Alain Lewis, who Tyler Cowen told me of yesterday and whose ideas, I'm told, are similar to my own? I don't know, but if this idea is true, then hopefully this will be his vindication.
Finally, I'd really like to hear of a reference to any proof that NE is
-either an assumption (without AC) or,
-weakly dominated by alternative solution concept specifically defined over AC (i.e. an alternative solution concept, such as Lexicographic NE of BBD or my concept of NSNE), for any game?
Also, Rosser, can you please explain to me how many of the "paradoxes" I list have already been solved, or my various errors that you claim? You don't have to list them all or anything, I'm just curious which ones.
Posted by: Christopher Ayres at Jun 29, 2009 6:19:23 PM
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.
Posted by: johnshade at Jun 29, 2009 7:02:23 PM
