The Equilibrium Theory of Games

The other day I ran into a good friend from Tlön, who told me the most fascinating story about his discovery of a new theory of games.

I owe my discovery of the nature of equilibrium in card games to an odd conjunction of mirrors and an encyclopedia. The mirror was in our library, and the encyclopedia was called Encyclopedia Equilibria (London, 1942, Enlarged ed. 1983). The mirror was an abomination, for in its reflection, one could see their opponents cards, and thus it led me to a crisis in belief. The encyclopedia, however, was even more of an anomaly, containing a fallaciously named article the "Foundations of an Economic Analysis of Games" whose contents we learned about while hearing the history of a visitor to our weekly card game.

We had gathered together for our weekly tribute to Bacchus, stealing ourselves away in the library in an grand attempt to free ourselves through the copious use of wine and madness. The game of choice was loosely poker, but in a desire to satiate our fierce intellect, variations of a suitably clever nature were de rigueur and widely worshiped. Tonight's game brought together the usual crowd, a former theoretical physicist turned new media mogul, a government employee who, though not Russian, had achieved the title of Tsar, myself, who you know quite well, an altruistic mechanic who spent his summers on the road fixing those who'd broken down on their summer jaunts, and a new player, a friend of the mogul, introducing himself only as a ferryman by the name of Doe.

As the night grew long and after a game involving a slight variation on Pineapple, we began to notice a trend that our visitor, who had kept quite to himself except for small gestures and smaller chat apparently designed to help him blend in, had won the vast majority of the games we had played. Now normally we might subscribe his success to the whims of Fortuna, and yet, we had played a number of games that the regulars had devised and had deeply analyzed over a series of years. Flying Nun Chuck, for example, was a game we had dreamed up under the influence of the flicker of a mute second rate Kung Fu movie on the television installed in the library wall. We, the four, had played the game thousands of times, and in secret all of us had calculated to serve depth the odds involved in the game. Yet the ferryman had cleaned up all five times we had played the game.

Our curiosity thus peaked, we inquired, softly, where the ferryman had learned to play cards in such a fine fashion. "School," was his reply, which he threw at us with a hint of puzzlement, as if everyone had learned such skills in this manner. A game of Lincoln later (which the ferryman won) we probed more about his school: was there a thriving culture of card playing at the school? "Well, in so much as we studied the classics and modern theory of cards, yes, there was such a culture. But enough about my classes, let us try a game of Lame Brain Pete. Do you know the rules?"

At which point the four regulars, who were all looking at vastly reduced stacks sitting beside nearly empty glasses of wine, gave each other the eye which only old friendship could comprehend, and I interjected "Why don't we take a break and try this new bottle of Montrachet 1975 that the mogul brought along out on the deck?"

Jackets donned we thus braved the gentle spring cool to step out onto the deck, the sparkling city holding our focus as we sipped a fine vintage. "You were very lucky, ferryman, to have taken classes about card games in school," said the mogul. "I myself learned way too much about Hamiltonian dynamics, which I have only sparse occasion to employ in my new media empire."

"But of course, in Ciao Chug this was the main point of life," the ferryman replied. "One can only come to know the mind of God through games of chance, you know."

A long pause later, "But of course, I'm now a heretic, having disavowed the great book. In truth, I am far fallen. I've not cracked open Equilibria for years."

Our puzzlement clearly plaster across our faces, the ferryman, seemed, for the first time, to notice that his opponents were actual people and stuttered, "But of course you know of the good book, don't you? The book which describes in detail how balance is achieved in God's great plan?" Flustered when we answered in the negative, "But there was a copy right here in your library!"

A few seconds later the ferryman reappeared surrounded by his condensed, now rapid, breath, with a finely bound volume of the reference mold. It was a book that I had never seen despite the claim that it was from my library. On the spine was a strangely calligraphic title, Encyclopedia Equilibria.

Thumbing through the volume with the mechanic and the Tsar perched on shoulder, the ferryman explained. "The Encyclopedia Equilibria contains the story of the discovery the idea of equilibrium. Through a series of rational arguments it shows how one can approach the one true point of balance in the universe. This is the doctrine of tatonnement, of a path to greater harmony and value in the world. If you live your life according to the ideas described in Equilibria, then all that causes trouble in your life will dissolve away and you will be at peace with the world."

"While I was in the monasterial academy of Ciao Chug, I took on a double major. On the one hand, I studied the theory of nonlinear dynamics, at the time a new and exciting field. But my other major was dedicated to a single chapter in Equilibra, 'Foundations of an Economic Analysis of Games.' This chapter, as you probably know, was subject to much debate in the theological analysis of Equilibria."

Again puzzled looks forced the ferryman onward. "You see this chapter showed how to apply the concepts of balance to games. This was, of course, a quite serious matter, since the entire monastery engaged each day in card games as a way to get closer to balance with the universe. The method to our play was dictated by our detailed analysis of the games. In these analysis, it was required that we apply a rational strategy. For each game we developed a complete set of strategies which everyone should employ. These strategies were our daily chants to get closer to God."

"My own studies were focused on a close reading of the 'Games' chapter of the good book. For most of my years at Ciao Chug, I was, I suppose, very content. All flowed in ways which only increased my own personal parsimony. I had analyzed Pineapple and found an even simpler method for explaining the rational strategy, and achieved a note of distinction among my peers."

"But a chance meeting soon changed my world. Late, one day, to the daily game of Hold'em, I stumbled upon the eldest brother of the monastery engaged in kissing a rather old statue located on the monastery grounds. Shocked at such a sight, I climbed up beside him and asked him what he was doing. `My mind. My mind. My mind,' was all he could reply. Bewildered and concerned for the elder, I ran to the gambling hall, summoning the other brothers, who quickly took the crazy elder away."

"A few weeks later, the crazy elder had returned to the fold. I watched him closely, of course, curious about the affliction which had caused his lunacy. I did not detect any hint of behavior beyond the norm, except for one small anomaly. As part of my major, I was required to keep track of the results of our daily games. Every week I tallied the games, check for balance by careful examination of the statistics of the results. These statistics had never, and still, even after the crazy elder's odd affair, never showed any deviation from those derivable from the good book. But I had been watching the elder, and it seemed to me that he was winning quite a large, but not extraordinarily large, number of the games. This observation piqued my curiosity and led me deep into the records of the individual brother's games, and a detailed analysis of the games of the crazy elder."

"And it was in this study that I discovered my own downfall. While the aggregate displayed no deviation from the good book, the crazy elder was winning much more than balance should dictate. How could this be? Well a doubt once hatched is nearly impossible to place in a cage, and so I began a detailed analysis of all brothers and their results in the past few years of game playing. And what did I discovered? That the elder was the only brother who deviated from the balance, sans one other brother. And that other brother was me."

"Shocked at this discovery I fled the monastery for the high hills of Uqbar. For days I fasted, and went over the rational arguments which had, for so long kept me in such great comfort. I wondered how it could be that I played the rational strategy, but was, somehow, the only one to come out with a worse result than every else. How was it that the crazy elder won? How was it that the total result of all of this was nil? I slept and thought, slept and thought, until one night, probably delirious, I awoke with the startling observation. Perhaps, there was no equilibrium theory of games. Perhaps it was possible to buck the balance by the use of irrational strategies. But if going out of balance could lead to more winning, then the book was wrong. The book was a but a panacea. This would, of course, explain why the crazy elder could do so well. That was the easy conclusion. But the idea that awoke me in shock, was my reason for why I was the one singled out to balance out the crazy elder. My conclusion: I was the only one playing the rational strategy which would lead to balance."

"It would have been natural for me, at that point, to flee Ciao Chug. But instead I did something rather devious. I resumed my life as normal with the exception that I no longer used the strategy devised from careful study of the book. Instead I began to experiment with radical new strategies for the card games. Quickly I found a strategy with which I began to win many more games than any other brother. Indeed it soon became apparent that this was so. Brothers began avoiding me and whispering behind my back. This all came to an explosive end, one night, when I did not lose a single game. I was summoned before the Council of Economic Policy and promptly excommunicated."

The night had drawn long and the tolerable spring cold had turn biting. "So this is how you were so consistently beating us?" asked the mechanic. "By using the strategies which you developed to beat the other brothers."

The ferryman, stood and stretched. He glanced dreamily toward the city lights below. He then slowly opened his jacket to reveal a handgun stashed in the inner pocket. "No," he said, "I beat you by reading your cards in the reflection of the television."


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Is this from Orbis Tertius? Was your friend a blind old man by any chance?

By AB Casares (not verified) on 12 May 2009 #permalink

Actually, there is something to the notion that deviating from the equilibrium strategy can throw off an equilibrium-playing opponent. For example, in heads-up Texas hold'em limit poker, where the button is the small blind, it is a mistake for the small blind to simply call the big blind. He should either raise or fold. However, if the big blind is playing an equilibrium strategy, he won't know what to do if the small blind calls.

Borges walks around here too?

s/peaked our curiosity/piqued our curiousity/

It pretty much drives me crazy when people talk about [poker; but really any game theoretic] equilibria without offering an equilibration mechanism.

s/Jackets dawned/Jackets donned/

s/wims of Fortuna/whims of Fortuna

(If you've been to Fortuna, you'd know it's "winds.")

I looked for you guys in the library, but I must have taken the wrong fork on the garden path.

Mark: yeah it's easy to get lost in the library, what with it being unending but not infinite.