# Category Archives: Game Theory

## Combinatorial Game Theory: Surreal Numbers and the Void

###### [A pdf version is available here; LaTeX here]

Any number can be written as a tuple of games played by the void with itself.

Denote the void by the empty set ∅. We write: {∅|∅} = 0, with | as partition. ‘Tuple’ signifies ordering matters, so that {0|∅} = 1 and {∅|0} = −1. Then recursively construct the integers: {n|∅} = n + 1. Plug {∅|∅} into {0|∅} to get {{∅|∅}|∅} = 1, then this into {1|∅} to get {{{|}|∅}|∅} = 2…

So if games exist, numbers exist. Or rather: if games exist, numbers don’t have to.

Mixed orderings generate fractions, e.g. {0|1} = {{∅|∅}|{{∅|∅}|∅}} = ½. Games with infinity (written ω) or infinitesimals (ε = 1/ω) permit irrationals, and thus all reals. Further, it is valid to define {ω|∅} = ω + 1, etc. Once arithmetic operations are defined, more complex games can define and *use* such quantities as ∛ω and ω^{ω}.

Therefore: by defining numbers as games, we can construct the **surreal numbers**.^{(1)}

∗ ∗ ∗

As well as defining numbers as games, we can treat games like numbers.

{∅|∅} can be played as the *zero game*. Simply: player 1 cannot move, and loses. Any game where player 2 has a winning strategy is equivalent to the zero game. Take two games G and H, with G a 2^{nd} player win. The player with a winning strategy in H can treat the games separately, only moving in G to respond to the opponent. Player 2 wins G, but does not affect H’s outcome. Conversely, given G′ and H′, with G′ a 1^{st} player win, player 1 is last to move in G′, giving player 2 an extra move in H′, potentially altering its outcome. In terms of outcomes, we say G + H = 0 + H → G = 0.

For a game G, −G is G with the roles reversed, as by turning the board around in chess.

G = H if G + (−H) = 0, i.e. is a player 2 win, and so is equivalent to the zero game.

Two more properties are clear: G + (H + K) = (G + H) + K (associativity) and G + H = H + G (commutativity).

We can see that G + (−G) = 0 by a clever example called the Tweedledum & Tweedledee argument. In the game Blue-Red Hackenbush, players are given a drawing composed of separate edges. Each turn, player 1 removes a blue edge, plus any other edges no longer connected to the ground, and player 2 does likewise for red edges. Since in G + (−G) the number of pieces is the same, player 2 can just copy the moves of player 1 until all pieces are taken. Player 1 will not be able to move, and so will lose. Hence player 2 has a winning strategy.

Thus games are a proper mathematical object—namely, an Abelian group.^{(10)}

∗ ∗ ∗

A new notation links all this to surreal numbers. For any set of games G^{L} and G^{R}, there exists a game G = {G^{L}|G^{R}}. Intuitively, the Left player moves in any game in G^{L}, and likewise for Right in G^{R}. The zero game {∅|∅} = 0 is valid, and we may construct the surreals as before. Now we can write the surreals more easily with sets: {1, 2, … , n|} = n + 1.

## Avant-Garde Philosophy of Economics

To most people, the title of this post is a triple oxymoron. Those left thoroughly traumatized by Econ 101 in college share their skepticism with those who have dipped their toe into hybrid fields like neuroeconomics and found them to be a synthesis of the dullest parts of both disciplines. For the vast, vast majority of cases, this sentiment is quite right: ‘philosophy of economics’ tends to be divided between heterodox schools of economics whose writings have entirely decoupled from economic formalism, and—on the other side of the spectrum—baroque econophysicists with lots to say about intriguing things like ‘quantum economics’ and negative probabilities via p-adic numbers, but typically within a dry positivist framework. As for the middle-ground material, a 20-page paper typically yields two or three salvageable sentences, if even that. Yet, as anyone who follows my Twitter knows, I look *very* hard for papers that aren’t terrible—and eventually I’ve found some.

Often the ‘giants’ of economic theory (e.g. Nobel laureates like Harsanyi or Lucas) have compelling things to say about methodology, but to include them on this list seems like cheating, so we’ll instead keep to scholars who most economists have never heard of. We also—naturally—want authors who write mainly in natural language, and whose work is therefore accessible to readers who are not specialists in economic theory. Lastly, let’s strike from the list those writers who do not engage directly with economic formalism itself, but only ‘the economy’. This last qualification is the most draconian of the lot, and manages to purge the philosophers of economics (e.g. Mäki, McCloskey) who tend to be the most well-known.

The remaining authors make up the vanguard of philosophy of economics—those who alchemically permute the elements of economic theory into transdisciplinary concoctions seemingly more at home in a story by Lovecraft or Borges than in academia, and who help us ascend to levels of abstraction we never could have imagined. Their descriptions are ordered for ease of exposition, building from and often contradicting one another. For those who would like to read more, some recommended readings are provided under each entry. I hope that readers will see that people have for a long time been thinking very hard about problems in economics, and that thinking abstractly does not mean avoiding practical issues.

**M. Ali Khan**

Khan is a fascinating character, and stands out even among the other members of this list: by training he is a mathematical economist, familiar with some of the highest levels of abstraction yet achieved in economic theory, but at the same time an avid fan of continental philosophy, liberally citing sources such as De Man (a very unique choice, even within the continental crowd!), Derrida, and similar figures on the more literary side of theory, such as Ricoeur and Jameson. It may be helpful to contrast Khan to Deirdre McCloskey, who has written a couple of books on writing in economics: McCloskey uses undergraduate-level literary theory to look at economics, which (let’s face it) is a fairly impoverished framework, forcing her to cut a lot of corners and sand away various rough edges that are very much worth exploring. An example is how she considers the Duhem-Quine thesis to be in her own camp, which she proudly labels ‘postmodern’—yet, just about any philosopher you talk to will consider this completely absurd: Quine was as modernist as they come. (Moreover, in the 30 years she had between the first and second editions, it appears she has never bothered to read the source texts.) Khan, by contrast, has thoroughly done his homework and then some.

Khan’s greatest paper is titled “The Irony in/of Economic Theory,” where he claims that this ‘irony’ operates as a (perhaps unavoidable) literary trope within economic theory as a genre of writing. Khan likewise draws from rhetorical figures such as synecdoche and allegory, and it will be helpful to start at a more basic level than he does and build up from there. The prevailing view of the intersection of mathematics and literary theory is that models *are* metaphors: this is due to two books by Max Black (1962) and Mary Hesse (1963) whose main thesis was exactly this point. While this is satisfying, and readily accepted by theorists such as McCloskey, Khan does not content himself with this statement, and we’ll shortly see why.

Consider: a metaphor compares one thing to another on the basis of some kind of structural similarity, and this is a very useful account of, say, models in physics, which use mathematical formulas to adequate certain patterns and laws of nature. However, in economics it often doesn’t matter nearly as much who the particular *agents* are that are depicted by the formulas: the Prisoner’s dilemma can model the behaviour of cancer cells just as well as it can model human relations. If we change the object of a metaphor (e.g. cancer cells → people), it becomes a different metaphor; what we need is a kind of rhetorical figure where it doesn’t matter if we replace one or more of the components, provided we retain the overall framework. This is precisely what **allegory** does: in one of Aesop’s fables, say “The Tortoise and the Hare,” we can replace the tortoise by a slug and the hare by a grasshopper, but nobody would consider this to be an entirely new allegory—all that matters here is that one character is slow and the other is fast. Moreover, we can treat this allegory itself as a metaphor, as when we compare an everyday situation to Aesop’s fable (which was exactly Aesop’s point), which is why it’s easy to treat economic models simply as metaphors, even though their fundamental structure is allegorical.

The reason this is important is because Khan takes this idea to a whole new level of abstraction: in effect, he wants to connect the allegorical structure of economic models to the allegorical nature of economic texts—in particular, Paul Samuelson’s *Foundations of Economic Analysis*, which begins with the enigmatic epigraph “Mathematics is a language.” For Khan: “the *Foundations* is an allegory of economic theory and…the epigraph is a prosopopeia for this allegory” (1993: 763). Since I had to look it up too, *prosopopeia* is a rhetorical device in which a speaker or writer communicates to the audience by speaking as another person or object. Khan is quite clear that he finds Samuelson’s epigraph quite puzzling, but instead of just saying “It’s wrong” (which would be tedious) he find a way to *détourne* it that is actually quite clever. He takes as a major theme throughout the paper the ways that the same economic subject-matter can be depicted in different ways by using different mathematical formalisms. Now, it’s fairly trivial that one can do this, but Khan focuses on how in many ways certain formalisms are observationally equivalent to each other. For instance, he gives the following chart (1993: 772):

That is to say, to present probabilistic ideas using the formalism of measure theory doesn’t at all affect the content of what’s being said: it’s essentially just using the full toolbox of real analysis instead of only set notation. What interests Khan here is how these new notations change the differential relations between ideas, creating brand new forms of Derridean *différance* in the realm of meaning—which, in turn, translate into new mathematical possibilities as our broadened horizons of meaning let us develop brand new interpretations of things we didn’t notice before. Khan’s favorite example here is nonstandard analysis, which he claims ought to make up a third column in the above chart, as probabilistic and measure theoretic concepts (and much else besides) can likewise be expressed in nonstandard terms. To briefly jot down what nonstandard analysis is: using mathematical logic, it is possible to rigorously define infinitesimals in a way that is actually *usable*, rather than simply gestured to by evoking marginal quantities. While theorems using such nonstandard tools often differ greatly from ‘standard’ theorems, it is provable that any nonstandard theorem can be proved standardly, and vice versa; yet, some theorems are far easier to prove nonstandardly, whence its appeal (Dauben, 1985). In economics, for example, an agent can be modelled as an infinitesimal quantity, which is handy for general equilibrium models where we care less about particulars than about aggregate properties, and part of Khan’s own mathematical work in general equilibrium theory does precisely this.

To underscore his overall point, Khan effectively puts Samuelson’s epigraph through a prism: “Differential Calculus is a Language”, “Convex Analysis is a Language”, “Nonsmooth Analysis is a Language”, and so on. Referring to Samuelson’s original epigraph, this lets Khan “interpret the word ‘language’ as a metonymy for the collectivity of languages” (1993: 768), which lets him translate it into: “Mathematics is a Tower of Babel.” Fittingly, in order to navigate this Tower of Babel, Khan (following Derrida) adopts a term originating from architecture: namely, the distinction between keystone and cornerstone. A *keystone* is a component of a structure that is meant to be the center of attention, and clinches its aesthetic ambiance; however, a keystone has no real architectural significance, but could be removed without affecting the rest of the structure. On the other hand, a *cornerstone* is an unassuming, unnoticed element that is actually crucial for the structural integrity of the whole; take it away and the rest goes crashing down.

## The Project of Econo-fiction

I have an article up at the online magazine *Non* on what it entails to use Laruelle’s non-philosophy to talk about economics, intended as a retrospective of my essay “There is no economic world.” It contextualizes econo-fiction in terms of Laruelle’s lexicon, illustrates a philosophical quandary with viewing iterated prisoner’s dilemma experiments through the lense of ‘falsification’, and notes a few ways I’ve changed my mind since then and where I plan to go from here. While the example is deliberately simple, aimed toward readers with zero knowledge of economic theory, it shows very succinctly how the notion of ‘experiment’ in economics operates as a form of conceptual rhetoric. I’ve also included a lot of fascinating factoids I’ve discovered since then, which I plan to expand upon in upcoming posts here.

No other philosophical approach I’ve come across—not even Badiou’s—lends itself to economics as much as non-philosophy does. I’m very impressed with the way that NP can talk about the mathematical formalism in economics without overcoding it, and I’d very much like to experiment with applying NP to related disciplines. Laruelle himself hints toward new applications of his method in finance: “Philosophy is a speculation that sells short and long at the same time, that floats at once upward and downward” (2012: 331). That is, philosophy is a form of hedging. Conversely, the section containing this excerpt is entitled “Non-Philosophy Is Not a Short-Selling Speculation,” where short-selling is investing so that you make money if an asset’s price goes down. Of the continental philosophers of finance I’m familiar with, Ben Lozano’s Deleuzian approach tends to focus on the conceptual aspects of finance to the neglect of its formalism, and Élie Ayache’s brilliantly original reading of quantitative finance is in many ways quite eccentric—such as his insistence on the crucial importance of the market maker (the guys yelling at each other in retro movies about Wall Street) and that algorithms are fundamentally inferior to human traders. A Laruellean interpretation of mainstream finance would serve as a welcome foil to both.

Just the other day I discovered a form of mathematical notation that appears to open up a Laruellean interpretation of accounting, and I’m always on the lookout for quirky reinterpretations of business-related ideas. I find philosophy such a handy tool for getting myself intrinsically interested in dull (but very practical) topics and disciplines, and I’ve read a whole heap of papers over the past year, so I’m really looking forward to blogging again.

**References**

- Joncas, G. (2015). “Intro to ∄ (There is no economic world).”
*Non*. Retrieved from http://non.copyriot.com/intro-to-%E2%88%84-there-is-no-economic-world - Laruelle, F. (2012). “The Degrowth of Philosophy: Toward a Generic Ecology,” in Laruelle, F.; Mackay, R. (Ed). (2012).
*From Decision to Heresy*. Falmouth, UK: Urbanomic, pp. 327-52