In a paper of 2006, Murphy attempted to use Cantor’s diagonal argument to claim that the problem of economic calculation was hyper computational, that is to say, that it could only be solved by super-Turing computing resources. We replied in 2007. The substance of our critique was as follows.
- If the economic calculation problem really was hyper computational, then no economic system, whether a market or a plan could solve it, since all such social organisations bring only finite computing resources to bear. Arguments about hypercomputation cannot, therefore, have any relevance to debates about the desirability of economic planning.
- Murphy’s claim that the problem is hyper computational involves an unwarranted assumption that the economic decision-making process must not only choose between goods that can actually be produced in the here and now but must select over all possible futures in which as yet undreamed of products become possible.
- This is grossly unrealistic since no sub-Olympian society can foresee the future. Our foresight is limited, and it remains to be shown that even where we do have foresight – for example about climate change – that the free market is effective at selecting an optimal use of resources.
- But even were the problem to be considered by the Olympians it would still be finite. There is a finite number of atoms in the Earth’s crust and a finite, though vast, number of ways in which these could be combined to make products. So the number of possible products is also finite.
A more recent paper by Ionela Baltatescu and Petre Prisecaru has attempted to defend Murphy against our criticisms. They attempt to show that there really is an infinite uncountable list of prices and that central planners must necessarily resort to this entire list in their planning activity.
If one is going to make an absurd claim, I suppose it might as well be an ambitious absurdity. Not only must planners consult infinite price lists, they must consult uncountably infinite ones! One might think it would be enough to claim a countable infinity, one that would stymie planners until the end of the universe.
They attempt to do this via an excursion into a Cantor diagonal style argument to prove the existence of an uncountable set. This argument (Hunter, 1996) is a trivial variant on the standard Cantor argument. The original Cantor argument is in terms of infinite expansion of decimal representations of real numbers between zero and one. Clearly, the Cantor argument is not affected by a change of base, to use binary fractions instead of decimal ones. It is also well known that sets can be represented by bit strings so the infinite binary strings can be considered either to be binary representations of the reals or as binary encodings of sets, hence, if one accepts the Cantor style of argument, one can interpret the diagonal method as either specifying a new real or a new set beyond the countable ones.
The problem comes in applying this to economics. The Cantor argument is about numbers and not actual materialised numbers that we can write down, but platonic numbers with infinite numbers of digits. What relevance do infinite lists of infinite bit strings have to economic life?
The authors imply, though they do not demonstrate, that the preferences of individuals can be represented as sets, and thus presumably as bit strings. Well, you can represent anything you want as bit strings, but if the strings are constrained to be binary encodings of set membership then we need a definite ordered countable domain which gives significance to the bit positions. For instance, in computing, we can readily represent sets of lower case letters as strings of 26 bits since the letters have a defined order so the first bit denotes the presence or absence of an ‘a’ the second bit the presence or absence of a ‘b’ etc.
They do not explain how this is to be done with commodities. But let us assume that the domain is the domain of all commodities present and future ordered by their time of invention. But there are all sorts of problems with defining this and mapping it onto the Cantor style argument.
- As we have argued against Murphy, the set of possible commodities is finite so the bit string to represent sets of them will also be finite. The Cantor argument though fails for finite representations.
- If the future is malleable, then the ordering of the products by time of invention is indefinite for all as yet undiscovered designs. In different futures, the products will be invented in different orders.
Furthermore, it is quite unclear how a data structure as simple as a binary encoded set is supposed to represent consumer preferences. To represent preferences you would need to associate numbers with each possible commodity, allowing for some ranking of the products. If there are n products, each of these numbers would require to be at least log2(n) bits long to perform an ordering.
Clearly so long as n is finite, so its logarithm, and the numbers could be packed one after the other into a bit string of length n log(n), but we are already some way from the representation that the authors propose. In the author’s fantasy world with infinite numbers of different possible commodities you would need an infinite number of infinite precision binary numbers to encode a single possible preference.
From context, though this is never made clear, each of these preferences is supposed to be the preference of one individual. The nearest they come to stating this is
if the planner or central planning unit is to mimic the market processes, it is bound to take into account all the preferences of individuals (which are virtually inﬁnite).
If the preferences are those of individuals, then there are only a finite number of these individuals. If you are to apply a Cantor argument, you need a table that is infinite in both directions – you need each row to be infinitely long, and you need an infinite number of rows. But with a finite number of individuals you have a finite number of rows, so the diagonal argument is inapplicable.
Note that when they introduce the preferences they just claim that they are virtually infinite. What does virtually infinite mean?
We suspect it is just a metaphor to indicate ‘very big’. But of course, if you want to dabble in Cantor diagonal arguments, virtual infinities are not good enough. You need actual infinities.
The preferences of individuals are actually finite. They only have preferences for commodities that they know about, and we can probably only make rankings on options that are immediately presented to us on a menu or shop shelf. We do not even have in our minds an ordered list of all the products currently on sale at Amazon, ranked by preference, far less a list of all the goods that could ever potentially be produced.
Think of how difficult it is to do something as simple as choosing a holiday destination from a catalogue, the more hotels or resorts you look at the longer it takes. This is not surprising since the time to sort any collection grows non linearly with the size of the collection. The best sorting algorithms grow as n log(n) the naive ones as n2 and it is impossible for humans to sort anything but a modest set without using artificial aids such as a pencil and paper to form written lists.
So the claim that human preferences are virtually infinite is not only ambiguous but positively misleading. We actually have very modest abilities to form preferences.
The authors concede that because consumers only engage in a limited number of purchases their preferences are finite:
the set of preferences of the individuals on the market is a ﬁnite set. In a society based on collective forms of property, the preferences of the individuals are not really limited. They may express how many preferences they want to, and central planning unit must take into account all these preferences
Why the difference when it comes to central planning?
In planned economies, consumers have the same psychological limitations, the same limited time and limited attention span, and choices they make, whether by voting or more plausibly by buying, will also be limited.
The authors recycle the von Mises argument that you need a market to give prices to intermediate goods if you are going to make a rational choice of techniques. They write:
In a planned economy, a central planning unit still has the problem of deﬁning the prices of intermediate goods, and it still must work taking into account an inﬁnite uncountable set of prices for the intermediate goods.
This objection was long ago demolished by Kantorovich who showed that given a specification of the mix of final products in the plan, it was possible by purely in-natura calculation to arrive at an optimal plan. Kantorovich’s algorithm uses what he calls resolving multipliers which have a certain similarity to prices – in that they are a set of numbers associated with products. But these numbers are arrived at by purely mathematical operations from the known resources and techniques available. There are only a finite number of intermediate products and a finite set of resolving multipliers required. For a review of how Kantorovich solved the problem see here.