I was reading the wikipedia entry on the liberal paradox, and I realized how absurdly sterile and worthless game theory has become. Also how poorly understood the concept of individual liberty is. No wonder so many academic theorists become proponents of totalitarianism, they seem to grasp both human interaction and liberty so poorly.

Now, let me reproduce the example given in wikipedia to show how this argument is so very flawed:

Suppose Alice and Bob have to decide whether to go to the cinema to see a chick flick, and that each has the liberty to decide whether to go themselves. If the personal preferences are based on Alice first wanting to be with Bob, then thinking it is a good film, and on Bob first wanting Alice to see it but then not wanting to go himself, then the personal preference orders might be:

• Alice wants: both to go > neither to go > Alice to go > Bob to go
• Bob wants: Alice to go > both to go > neither to go > Bob to go

There are two Pareto efficient solutions: either Alice goes alone or they both go. Clearly Bob will not go on his own: he would not set off alone, but if he did then Alice would follow, and Alice's personal liberty means the joint preference must have both to go > Bob to go. However, since Alice also has personal liberty if Bob does not go, the joint preference must have neither to go > Alice to go. But Bob has personal liberty too, so the joint preference must have Alice to go > both to go and neither to go > Bob to go. Combining these gives

• Joint preference: neither to go > Alice to go > both to go > Bob to go

and in particular neither to go > both to go. So the result of these individual preferences and personal liberty is that neither go to see the film.

But this is Pareto inefficient given that Alice and Bob each think both to go > neither to go.

Now, this sounds plausible, and it uses nice formal logical notation to argue the point, but it misses on two points.

First, there is an assumption that no communication occurs between the two. Assuming that Bob is simply informed of Alice's preferences, he can then alter his hierarchy based upon his knowledge of Alice's response. In fact, judging from the initial premise, Bob's hierarchy is wrong anyway, it is illogical. If he wants to spend time with Alice, even if he truly detests the movie, how would "Alice goes alone" be his first choice? That would be his first preference only if he wanted to both avoid Alice and the movie. Thus not only does the example assume absolute isolation in decision making, but sets up an absurd initial premise.

The second problem is that the assumption is that liberty means that one must at all times have absolute liberty, EVEN FROM ONES OWN PREFERENCES. That is absurd. The assumption is that if Bob decides to go, then Alice's going will invalidate her freedom. It is not so. Alice's choice to go is based on her own preferences, not on some compulsion. SO I don't see why saying Bob's going makes it inevitable Alice will go and thus is "less free". Alice has said she will choose to go, so it is still her decision.

If this is the way formal logic, economics, and game theory are now taught at the university level, I am very glad I passed up my the doctoral program into which I was accepted. I would much rather stick with simple human logic.

Of course, this is a silly contrived theory, much like those for which Turing is so famous. The ability to create a paradox does not prove anything about the real world. Turing's famous "Everyone shaves themselves or are shaved by the barber, but who shaves the barber?" is just as meaningless as me saying "All assertions are false." Of course both are paradoxes, but that is because they are contrived statements which cannot be applied to the real world.

If I say I only eat liquid ice, does it invalidate the existence of ice? Or prove I don't eat? Then how does this silly contrived logic invalidate the statement that libertarian solutions produce optimal solutions? The theoretical basis of that assertion has been proven to my satisfaction, this contrived paradox does not even stand the test. For it to prove anything we have not only to allow an absurdly inconsistent set of desires, but also redefine liberty in a strangely counterintuitive way. I just do not buy into it.

Of course, I have only based this on wikipedia's description, so I may be fighting a strawman. I had not heard of this silly theory before, so I will have to do more research. If there are better argument, I will come back tot his. If not, then I will let it go with this post.

POSTSCRIPT

By the way, the Turing stopping "problem" only exists because of the absurd machine he created which used OUTPUT to test a program. If you use the source code, then there is no problem. I even wrote code to do it. It checked every line, and evaluated the branching and the possible range of values for each variable and told me where it was impossible, possible, likely or certain a loop would result in a perpetual loop. In other words, Turing only created a "paradox" by making his logic so peculiar. And yet many in the computer industry think that perfect debugging machines cannot exist because of his logical sleight of hand.

Turing was clever, but some of his paradoxes were more flash than truth.

ADDENDUM  

I reread the example about a dozen times, as I was sure I had to be misunderstanding something. They could not seriously mean that if Bob went, then Alice would follow and that somehow invalidated Alice's liberty, but, yet, that appears to be exactly what they mean.

So, they think if I say "If you hit me, then I will fight back" means I am surrendering autonomy. That is crazy.

I feel like I did when I finally actually read Turing and saw the problem in several of his silly paradoxes. Can professional mathematicians, economists, philosophers and logicians really be this far divorced from reality that they can't see that words and even numbers can create situations which may be paradoxical, but only because they in no way match something seen in reality?

Allow me to give a personal example. In 8th grade geometry I came up with a pair of proofs that prove all numbers are equal. Here are the pair of proofs:

First, an indirect proof that infinity minus any number is still infinite.

suppose infinity minus any number is not infinite
then infinity minus finite number a equals finite number b
but then finite number a plus finite number b must equal infinity
but two finite numbers equal a finite number
thus infinity minus any number is still infinite.

Then the direct proof:

a>b
inf.-a=inf
inf-b=inf
inf-a=inf-b
-a=-b
a=b

I will omit the proof that if a>b and a=b, then all number are equal to one another, as it should be easy enough to follow my reasoning from this point.Suffice it to say that for any a and b, where a>b, if a=b then all numbers are equal to one another.

And thus I proved all numbers are equal. Even as an eighth grader I recognized it was sleight of hand, that "infinity" is not really a number and to treat it as such was to twist mathematics.

But it seems to me that these logicians are doing just that, creating hypotheses that do not match reality and then using them to "prove" things.

Turing's debugging machine, the definition of liberty in the theory above, those sort of concepts are akin to my misuse of infinity, theoretical constructs with no matching referent in reality which allow you to produce results which do not mirror the real world.

Well, I still have not read the original theory, I will reserve judgment until I do. But from what I have seen, I just can't believe that this passes for serious thought on economics.

UPDATE 06/30/2008

I said that Bob's hierarchy makes no sense, yet I did not explain why. The reason is that his hierarchy doesn't fit the expressed wishes as stated.

The one question I have is if "neither goes" means they do something else together, or they each do something independently. It makes quite a difference which way we decide to read that option. I think that "neither goes" implies jointly doing something other than the movie, but I am not certain. However, there is some evidence I am right in this assumption. So, let us start looking at the hierarchies of desires.

Alice's hierarchy, assuming she wants to see Bob and the movie, and assuming "neither goes" means they do something else together is: both go, neither go, Alice goes, Bob goes, which is what they stated, so it reinforces my belief that "neither goes" means they do something else together. But if that is the case, then Bob clearly should have had an alternate hierarchy.

So, let us see, if we assume "neither goes" means they do something else together, then it would seem there should be two possible hierarchies, depending upon which desire is stronger. However, regardless of which desire is stronger, avoiding the movie or seeing Alice, Bob's hierarchy is neither goes, both go, Alice goes, and Bob goes. In neither case would we get the provided hierarchy Alice goes, both go, neither goes, Bob goes. Even if we assume that "neither goes" means they do something separately, we do not get that hierarchy. In that case we get the sequence both go, then either Alice goes, neither goes, or neither goes, Alice goes (as those two are equal), and finally, Bob goes, which also fails to match the listed hierarchy of preferences.

The only way that hierarchy makes sense is if Bob wants to avoid Alice and see the movie, but wants to avoid Alice more. If Alice goes to the movie, he can avoid her for sure, but if he goes to the movie, Alice might show up.

Actually, that is the biggest flaw in the whole theory, the four possibilities. There really aren't four. If the two want to be together, then should Alice or Bob decide to go, the other will go as well, so the only real choices are both go and neither go. Unless they are artificially forced to reveal their decisions simultaneously and never change their minds, the other two options (Bob or Alice goes) will turn into both go, assuming Bob wants to see Alice more than he wants to avoid the movie.

Even assuming Bob really hates the movie, more than he wants to see Alice, there are still only three possibilities: neither, Alice or both, because if Bob goes, then Alice will as well. So there is no possibility of Bob going alone as a final outcome, which makes the entire exercise absurd.

As I said, still tracking down original sources and hoping they have better examples, as this one is just absurd. Hopefully it is just the example chosen and not the entire theory. Then again, my experience with those using game theory in the past has not filled me with confidence that the formal theory is any better than the example.

Correction (08/08/2008)

 I rewrote large parts of the 06/30 update, as the original version had so strange wording and organization. I could understand what I intended to say, but it appeared I started writing, stopped at some point, and forgot what I was doing when I returned. So I have cleaned it up to make it much more clear. I also added a few bits of explanation to make it more clear. (I also fixed a few spelling errors and one formatting error, and corrected a mistake in my 8th grade proof in the section above.)

For those curious about the bad wording, the second through fourth paragraphs originally read:

Let us look at Bob's hierarchy of desires His desires are to see Alice, then to avoid seeing the movie. That is based on the assumption that neither goes means that he and Alice do something else, if it means he does nto see Alice, then the choices are: Both go, Alice goes and neither goes being equal, then Bob goes.

Alice's hierarchy, assuming she wants to see Bob and the movie, and assuming "neither goes" means they do something else together is
: both go, neither go, Alice goes, Bob goes, which si what they stated, so it reinforces my belief that "neither goes" means they do something else together. But if that is the case, then Bob clearly should have had an alternate hierarchy.

So, let us see, if we assume "neither goes" means they do something else together, then there are two possible hierarchies. First, as we state earlier, if Bob wants to see Alice more than avoid the movie, the hierarchy is neither, both, Alice, Bob. If he wants to avoid the movie more than see Alice then it is still neither, both, Alice, Bob. In neither case would we get the provided hierarchy Alice, both, neither, Bob.

I think the new version is clearly superior.