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Non-abelian money

by Michael Nielsen on August 25, 2007

What would happen if we replaced the current monetary system, which is based on an abelian group [*] by a non-abelian currency system?

[*] If someone gives you x dollars, then y dollars, the result is the same as if you were given y dollars first, then x.

I’ve been puzzling about this for a few years. It raises lots of big questions. How would markets function differently? Might this lead to more efficient allocation of resources, at least in some instances? (At the very least, it’d completely change our notion of what it means to wealthy!) Might new forms of co-operation emerge? How would results in game theory change if we could use non-abelian payoffs?

More generally, it seems like this sort of idea might be used to look at all of economics through an interesting lens.

A nice toy model in this vein is to work with the group of 2 by 2 invertible matrices, with the group operation being matrix multiplication. By taking matrix logarithms, it can be shown that this model is a generalization of the current monetary system.

Electronic implementation of non-abelian money would be a snap. The social implementation might be a bit tougher, however – convincing people that their net wealth should be a matrix would be a tough sell, at least initially. Still, if non-abelian money changed some key results from economics, then in some niches it may be advantageous to make the switch, and possible to convince people that this is a good idea.

(It should, of course, be noted that there are in practice already many effects which make money act in a somewhat non-abelian fashion, e.g., inflation. From the point of view of this post, these are kludges: I’m talking about changing the underlying abstraction to a new one.)

From → ideas

  1. I’m struggling to see how non-Abelian money can possibly make sense. Remember that the primary function of money is to facilitate trade by eliminating the “double coincidence of wants”.

    Say I want to buy a car, and I make my living selling physics lessons. Without money, I would have to find someone who has a car and wants to barter it for a car’s worth of physics lessons. I’ll be spending all my time searching. But with money, the people who want to buy physics lessons don’t need to be the same people who want to sell cars, so I can eliminate the search costs and I’m better off.

    Now, it seems obvious to me that the quantity of money has to be isomorphic to the quantity of physics lessons or the quantity of cars or whatever, otherwise how can the exchange rate between physics lessons and cars be determined? But the group isomorphic to quantity of cars is just the real numbers under addition (negative numbers representing a debt, of course).

    What would trade look like with non-Abelian money? I can’t even imagine it, myself.

  2. Inflation isn’t non-abelian value of money, it’s time-variant value of money.

    Non-abelian money isn’t really “money” anymore but you could consider playing some types of multiplayer games as the exchange of non-abelian money. So depending on what people were trying to do and what the field is, strategies will vary, and game theory is what will tell you about them.

  3. Michael Nielsen permalink


    It’s simply a thought experiment, to see what might be interesting. It’s not likely to be easy to think about, precisely because it breaks some of our usual assumptions about the nature of money. But it may suggest new mechanisms for trade.

    As a related example, I was talking with my friend Ben Schumacher last week, and Ben suggested the idea of negative money — tokens whose value would actually be negative. Negative money has all kinds of cool and interesting properties. My favourites among those Ben suggested were these: (1) burglars might break into your house, and leave some cash; and (2) people would have much less incentive to counterfeit.

    Now, neither of these is a killer app, although the counterfeiting one is pretty interesting. And it’s easy to come up with long lists of criticisms of the idea.

    This type of criticism is not, in my opinion, particularly useful. It’s easy to criticise. What is hard is to find interesting new abstractions which lead to counterintuitive phenomena, properties that are interesting, but that money doesn’t ordinarily have. (1) and (2) illustrate this in spades, so I think Ben’s idea is really cool. That doesn’t mean the US Fed should start printing $-1 notes. But it might be worth tinkering around a bit further with the idea, and seeing where it goes.

    The idea of non-abelian money is similar. It’s just a fun abstraction to play with. Maybe it’ll lead to something interesting, maybe it won’t. My gut tells me it should have some interesting applications, but I don’t know what, yet.

  4. mitchell porter permalink

    Suppose you have holdings in N currencies. One can certainly write down a matrix in which element A[m,n] is the value of your mth-currency holdings in terms of the nth currency. You could then regard your net worth as formally matrix-valued, which gets you halfway to nonabelianity… And if you google, you’ll see that several people think negative money already exists.

  5. Michael Nielsen permalink

    Mitch: Under ideal conditions of no transaction costs the currency conversion matrix takes values in a commutative subgroup (actually, subalgebra) of the matrices. It was this observation, made while reading an incredibly stimulating interview with Michael Milken (I don’t recall the source), that led me to first wonder about non-abelian money.

  6. mereotelic permalink

    It may have relevance if utility (or money) is measured in terms of relevant information, order effects games with asymmetric information.

  7. Teleoplection permalink

    In an attempt to further promote the discussion of useful applications which bear upon the one Michael Nielsen has so presciently raised here (such as mechanism design and “intersubjective agreement”), I’ve started a group for those interested in contributing.

  8. Teleoplection permalink

    “Were there really such a thing as absolute intrinsic value, the resolutions given for the Kraitchik and 2-envelopes paradoxes would be final. Each player could reason strategically from an elevated multi-frame perspective to avoid the essential fallacy of these paradoxes. Unfortunately, economic uncertainty makes assessments of absolute value all but impossible; the Kraitchik rationale in effect becomes a subjective vote of confidence in one’s own opinions and projections, and all one can hope to do is allow for the dynamics of interacting subjective frames. Although the Kraitchik and 2-envelopes paradoxes deal with games whose rules seem artificial, these rules turn out to be general; interframe differentials in subjective value account for the ubiquity and validity of the Kraitchik rationale in games which locally appear to be 0-sum, but need not be so in the wider contexts to which the players are subjectively linking them…contexts that ultimately merge in the global economy, precipitating cooperation and competition leading to expectative conflicts. Indeed, relativism based on subjective value differentials expressed in a global “spacetime” of transactions or “economic events” is what allows a locally 0-sum game to be globally advantageous, contributing to an overall win-win scenario in which the economy undergoes real expansion.”

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