## Computable Entrepreneurship

*2 March 2009 at 1:24 pm* *Dick Langlois* *
10 comments *

| Dick Langlois |

I just returned from New York, where I was a discussant at a session on entrepreneurship. (Peter was supposed to have been part of the session — too bad he couldn’t make it.) I discussed a presentation by my old friend Roger Koppl. I have written before about Roger’s work on forensic science administration. This presentation, which drew on a couple of Roger’s recent papers (see here and here), was called “Who Needs Entrepreneurs?” Here is the abstract:

The mathematics of “computable economics” proves that entrepreneurship policy is unlikely to succeed if it presumes policy makers can replace the unplanned results of the entrepreneurial market process with ex ante judgments about which enterprises are best. It is mathematically impossible for policy makers or their assignees to make the required computations of opportunity costs. Some business professors dream of finding a grand algorithm that will allow them to guide entrepreneurial decisions and to judge in advance which decisions are good and which bad. The logic of computable economics, however, reveals this dream to be a form of magical thinking.

This is fascinating stuff that should be of considerable interest to O&M readers.

Entry filed under: - Langlois -, Austrian Economics, Entrepreneurship, Management Theory.

1.Peter Klein | 2 March 2009 at 1:44 pmI was sorry to have missed it. I heard that Kirzner was also unable to make his scheduled panel appearance. Whose absence do you think caused a bigger stir? :-)

2.spostrel | 2 March 2009 at 6:39 pmI clicked through to the JEBO book review linked in the post and stumbled on a very unconvincing (i.e., of dubious relevance) argument about the alleged non-computability of Nash equilibrium in a game of matching pennies. This is not encouraging further exploration.

3.Roger Koppl | 2 March 2009 at 8:05 pmSpostrel,

Look again. I said a best reply strategy is not computable. That turns out to be a different thing. See my article with Barkley Rosser in Metroeconomica 2002. See also Prasad, who says, “Even for games with computable equilibrium points, best responses of the players may not be computable.” http://garnet.acns.fsu.edu/∼kprasad/toff.pdf

4.Prévision, planification et computabilité « Rationalité Limitée | 4 March 2009 at 2:36 am[…] de l’application de cette approche en économie avant de tomber, via le blog “Organizations and Markets“, sur certains travaux de l’économiste Roger Koppl. L’argument de la (non) […]

5.spostrel | 4 March 2009 at 9:58 pmI read the example and the argument carefully. Unfortunately, a best response is defined to be (in every game theory class and book I’ve ever seen) the optimal strategy to choose taking the other player’s choice as a parameter. NOT trying to guess what the other player will do, but saying IF he does x, then my best response is y, for each possible x in his strategy space.

Of course, if you redefine terms to rule out the Nash program a priori, then you can make sensational-sounding statements that are not relevant upon close examination. Which is what I think has happened here.

Also, hanging one’s theoretical analysis on the Constructivist philosophy of mathematics greatly limits its appeal. Only a small fraction of the people who’ve heard of Constructivism actually subscribe to it (it’s not exactly mainstream mathematics), and of course the majority of applied math users have never heard of it at all. Although I did write “God created the integers. All the rest was the work of man” on a college calculus exam when I got stuck on a problem with transcendental numbers. The TAs didn’t buy it.

6.Roger Koppl | 5 March 2009 at 9:09 amSpostrel,

I think you’re still a bit off target.

Please allow me to start with the easier point, “the Constructivist philosophy of mathematics.” The philosophy of math is simply not at issue here. Let’s say the axiom of choice is somehow objectively true and that non-constructive proofs in mathematics are entirely legitimate and must be accepted. So what? Freedom to use fixed-point proofs does not enable to me compute the equilibrium whose existence is established by such a proof. Rejecting the constructivist philosophy of math does not enable anyone to compute what cannot be computed.

Before moving to the more difficult point about best replies, let me update my information on K. Prasad’s article: it is now forthcoming at JEBO and can be downloaded from Sciencedirect.

Not to best replies. It is true that we have not been in the habit of distinguishing the contestant’s computation of a best reply strategy from the observer’s computation of a Nash equilibrium. Until rather recently it looked as if there was simply nothing more say about matching pennies once you identified its unique Nash equilibrium. It all seemed so simple. It was rather elementary undergraduate stuff. All that changed with this paper:

Lewis, A.A. (1992). Some aspects of constructive mathematics that are relevant to the foundations of neoclassical mathematical economics and the theory of games. Mathematical social sciences, 24, 209–235.

After Lewis came

Tsuji, M., da Costa, N.C.A. & Doria, F.A. (1998). The incompleteness of theories of games. Journal of philosophical logic, 27, 553–564.

and other works by da Costa and Doria.

These recent results show that computability issue crop up in contexts we had thought of as “finite,” because our vague descriptions allow an infinite variety of finite games to fit the description. The issue is a little different with matching pennies. But here, too, we have computability problems cropping up in a context we thought immune to them. And here, too, the problem is that an infinite set of possibilities intrudes in an unexpected way. In matching pennies that way is self reference: you think that I think that you think. In any event, recent results show that the observer’s ability to compute a Nash equilibrium is not the contestant’s ability to compute a best reply. See, again, Prasad’s statement quoted earlier.

The real point on matching pennies can be expressed without the use of mathematical symbols. If we both flip fair coins, neither one of us has an incentive to change strategy. But because any reply to the coin flip is as good as any other, neither party has an incentive to stick with Nash equilibrium either. In such a situation, why should either party have confidence that the other will go along with the Nash equilibrium? The Nash equilibrium is not sticky.

7.Peter Klein | 5 March 2009 at 10:34 amRoger, I’m a little confused by your response to Steve. Isn’t his point simply that Nash equilibrium only requires that agents have a best-response _function_, not a unique best response? In the penny-matching game there is a unique Nash equilibrium because there is only one pair of choices (each playing heads with probability 0.5) that represents a set of mutual best responses. But the Nash concept itself does not say anything about agents’ abilities to _compute_ this particular strategy (any more than Walrasian equilibrium requires agents to know, ex ante, the equilibrium set of prices). Am I missing something?

8.Roger Koppl | 5 March 2009 at 10:56 amHi Peter,

It’s easy to compute the Nash equilibrium in matching pennies. Each contestant can do it. Thus, I don’t think I can accept your interpretation. The point is that knowledge of the Nash equilibrium does not help you formulate a best reply in matching pennies.

Let’s say that I think that you think I’ll flip a fair coin. I think that you think I’ll play Nash. If I really thought that, I would have no reason to expect you to play Nash. If I’m flipping a fair coin, then any strategy on your side has the same expected payoff for you. You have no incentive to pick the Nash strategy. If I don’t have any confidence you will pick Nash, however, I can’t be sure my coin flip is a best reply. If you pick heads for sure, for example, then Nash is not a best reply on my side. The situation is the same for you. Neither of us can have confidence the other will play along with the Nash solution. Best reply strategies are not computable.

I might respond to the problem by retreating to maximin, which does imply flipping a fair coin. But then I’m not computing a best reply strategy.

9.spostrel | 5 March 2009 at 9:42 pmHeavy weather is being made here over very little. It was noticed back in the 1950s that if your opponent is playing a Nash equilibrium mixed strategy then you are indifferent across all the pure strategies available to you. In fact, this is a basic solution technique. So your best response function becomes a correspondence in response to that particular rival mixed strategy.

But there is no problem with not having a unique response when every response is just as good as every other one. Best does not imply unique.

Furthermore, none of this has anything to do with computability, or observers vs. participants, or constructivism (which, contrary to Roger’s initial disavowal immediately popped into his first reference). There are real serious problems with the Nash program–for example, Nash players typically do worse than non-Nash players in the game of picking a number between 0 and 100 and trying to match 2/3 of the crowd average–but this computability stuff seems to miss the point entirely.

10.Roger Koppl | 6 March 2009 at 7:42 amI think the conversation is winding down, Steve, as we seem to be moving in a circle.

The demonstration in question can be found in Koppl, R. & Rosser J. B. Jr. (2002). All that I have to say has already crossed your mind. Metroeconomica 53, 339-360. A draft is available here:

http://cob.jmu.edu/rosserjb/MetroRevised%20LBS2.doc

I think our proof is solid and likely to hold up to critical scrutiny.

At the beginning of our exchange you said I made an absurd claim, namely, that the Nash equilibrium is not computable in matching pennies. Now you seem to say that I’m making a trivial claim. I suppose that’s progress of a sort. After the first paragraph of your last comment you say, “none of this has anything to do with computability.” Well, that’s basically right, which would seem to explain why you think I’m saying something trivial. There is, however, a link to computability: some of the relatively banal considerations you raise are helpful in conveying the intuition behind the proof that best-reply strategies are non-computable in this game.

You say “this computability stuff seems to miss the point entirely.” I could express agreement or disagreement if I knew which point you had in mind! Indeed, we haven’t progressed far enough in the discussion to decide whether “this computability stuff” is a mere curiosity or something that matters for economics. Naturally, I think it matters. We haven’t discussed that, however.

Closing remark: Best reply, best-reply function, and best-reply strategy are three different things. Please have a look at the Metroeonomica paper before expressing further dismissive thoughts on that matter.