Someone, I think it was Chris, described this as a “Marxist/anarchist blog,” which I suppose is fair enough, inasmuch as I identify myself here as a Marxist and an anarchist [also an atheist, a husband, a father, a grandfather, and a violist – this last something of a reach – but whatever, as young people say.] However, that is not how I think of myself. I am a philosopher, a teacher, and, more than anything else, a life-long lover of powerful, simple ideas, so lucidly and elegantly expressed that their beauty can be seen by all. The exigencies of the present political situation have compelled me to venture very far from my true calling, but my mental health requires that I return from time to time to the realm of ideas to remind myself what I most love.
Which brings me to the subject of my musings during this morning’s walk. Can it be, I found myself wondering, that a term in a language should always be misused? I am not here merely expressing my inner pedant. Like many, I cringe when some television talking head says that this or that “begs the question,” meaning that it compels us to ask the question, not that it assumes what is to be proved. Or when another deep thinker says that it is impossible to underestimate the importance of something, meaning of course that its importance is so great that it is impossible to overestimate that importance. My favorite example of this linguistic pickiness is Harry Levin, the great Harvard Shakespeare scholar of half a century ago. When my first wife, Cynthia Griffin Wolff, mailed a complete draft of her doctoral dissertation on Samuel Richardson to Levin, who was her Director, he sent it back without comment, but he had clearly read it, because throughout the text, he had countlessly times changed “shall” to “will” and “will” to “shall.” Once those essential alterations were made, she was good to go.
No, I am talking about a made-up phrase, coined more than seventy years ago by John von Neumann – zero-sum game. [Strictly speaking, the term should be credited both to von Neumann and to his co-author, economist Oskar Morgenstern. Morgenstern was a very interesting thinker, the author, among other things, of a delightful book titled On the Accuracy of Economic Observations, which I recommend to you all, but von Neumann was one of the authentic geniuses of the twentieth century, so I shall imitate my fellow Marxists, who tend to attribute all the ideas of Marx and Engels to Marx, and speak as though Game Theory was von Neumann’s creation alone.] Everybody uses the phrase “zero-sum game,” and everybody, without exception, misuses it. Is that even linguistically possible? Here are just two examples. The first is from Barack Obama’s farewell address this past January. The second is an older misuse by Paul Krugman who, think of him what you will, is a Nobel Laureate in Economics and should know better.
Obama: “Our economy doesn't have to be a zero-sum game.”
Krugman: “Unlike war, trade is not a zero-sum game.”
I shall now explain to you exactly what “zero-sum game” means in several thousand well-chosen words. I am well aware that at this point I shall be losing virtually all of my readers, but in a desperate effort to hold a few of you before you surf away to your favorite revolutionary blog, I will simply observe that the term has its roots in the successful attempt by neo-classical economists to purge their “scientific” discipline of its radical redistributionist roots.
In 1944, von Neumann and Morgenstern published a brilliant book, Theory of Games and Economic Behavior, which created the new mathematical/economic sub-field of Game Theory. The centerpiece of the book is a powerful theorem concerning a certain sub-set of two-person games. A game is defined as a series of moves leading, by means of a termination rule, to a determinate outcome. [The positing of a termination rule eliminates the possibility of a game with an infinite number of repetitive moves. In Chess, for example the rules stipulate that if a position occurs three times, or if fifty moves are made without a piece being taken or a pawn being promoted to the eighth rank, the game is declared a draw.]
von Neumann posits that each player has a complete, transitive utility function that assigns a utility index, invariant up to an affine transformation, to each possible outcome of the game. [Invariance up to an affine transformation makes it possible to assign cardinal indices, not merely ordinal indices, to the outcomes. A familiar example of an affine transformation is the rule that allows us to figure out what the Fahrenheit equivalent is of a temperature give in degrees Celsius. The rule is Degrees F = 9/5 Degrees C + 32. It tells us that if the TV in the Paris airport, as we deplane, says it is going to be 20 degrees Celsius today, that means it will be 68 degrees F, so no jacket needed before catching a cab to the hotel.]
Now, we all remember that Jeremy Bentham brought into Political Economy the notion of a social calculation of the pleasure [or utility] and pain [or disutility] promised by a proposed law, along with the principle that we should always seek in our legislating to produce the Greatest Happiness for the Greatest Number, a principle that rapidly acquired the label Utilitarianism. What we may not so readily recall is that when Bentham proposed this now familiar principle, it was intended by him and understood by others to be a shockingly radical, not to say revolutionary, idea. Bentham stipulated that each was to count for one, which meant that the pleasures and pains of the peasants would weigh as heavily as those of the aristocrats. This was utterly unacceptable to the toffs, who protested that since their sensibilities were ever so much more refined than those of the rude masses, their delights and discomforts should carry greater weight in the social calculus [the Princess and the Pea Principle]. But there were so many peasants and so few aristocrats that no such weighting could overcome the tendency of the misery of the masses to outweigh the pleasures of the classes. It was a proposal that had the power to overturn the established order, and Bentham knew it.
Bentham’s godson, John Stuart Mill, did his best to contain the damage, arguing in Utilitarianism for a distinction between higher and lower pleasures, but that was a desperation rearguard action, akin to laying a few landmines during the retreat to Dunkirk. The real solution was advanced by the Economists, who latched onto an arcane doctrine in English Philosophy about the impossibility of interpersonal comparisons of utility. This gave rise to Indifference Maps, Pareto Preference, and all manner of highly successful defenses against the totally unacceptable suggestion that one person’s utility should be added to another’s. The firewall against the demands of the lower classes was given its theoretical imprimatur in Lionel Robbins’ classic 1932 book, Essay on the Nature and Significance of Economic Science.
Enter von Neumann. In Theory of Games and Economic Behavior, von Neumann assumed that each of the players in the two-person game had a cardinal utility function, but that in general nothing could be said about the relationship between one player’s evaluation of the outcome of a game and that of the other player. However, he proved that in one very special set of circumstances, it was possible to make such a comparison, and in fact actually to add one player’s assignment of utility to another’s. In short, one could give meaning to the notion of the sum of their evaluations, despite not making any assumptions about interpersonal comparisons of utility.
The key was the assumption that the two players had strictly opposed preference orders not merely for the finite set of outcomes of the game, but even for probability mixes of those outcomes, which can be called “lotteries.” Let me explain. There are, by the stipulation of a termination rule, a finite set of outcomes, over which each player is assumed to be able to define a utility function. Von Neumann added to this the further assumption that each player could express a complete and transitive preference over the infinitely varied ways in which one could assign probabilities to those outcomes [each assignment to sum to 1, of course]. Think of these assignments as lottery wheels, with the size of each slice of the lottery wheel corresponding to the weight being assigned to the outcome that the slice represents. Spin the wheel, and one prize will win, with the probability of that win a function of the size of the slice. Von Neumann now made a second assumption, that each player could express consistent preferences not only over prizes and lotteries of prizes, but even over lotteries of lotteries of prizes – what are called compound lotteries [The most famous example of a compound lottery is the Irish Sweepstakes, in which the prizes were not amounts of money, but betting tickets on horse races. A ticket in the Sweepstakes was a bet on a bet, as it were.]
Now, by mathematical rules quite familiar to probability theorists, compound lotteries can be reduce to simple lotteries in which the two prizes are the least and most favored outcomes. [This reduction calculation is equivalent to the assumption that the players have no pure preference for or aversion to risk itself, independent of the probabilities. That is a powerful assumption, by the way. I, for example, have an aversion to risk. Offer me the certainty of one dollar, or a fifty-fifty chance of getting nothing or two dollars, and I will take the sure dollar every time.]
Von Neumann now asks the following question: What, if anything, can we say about the utility assignments to outcomes of a game between two players who have strictly, exactly opposite preferences not only for the outcomes but also for all compound lotteries of the outcomes? His answer was simply gorgeous. First, he said, perform an affine transformation on each player’s utility function so that the least preferred outcome for that player is assigned a utility of 0 and the most preferred outcome is assigned a utility of 1. Under these very restrictive and special conditions – a two person game with a finite set of outcomes in which the players have strictly opposed preferences for compound lotteries of the outcomes – it is possible fairly easily to show that the sum of the utilities assigned by the two players to any outcome or lottery of outcomes will sum to 1. [Check my other blog for the proof.] If one then performs one final affine transformation, this time transforming player 2’s utility function so that it runs from -1 to 0 rather than from 0 to 1, then the sum of the utilities assigned by the two players to outcomes or compound lotteries of outcomes will always sum to zero.
THIS, AND ONLY THIS, IS WHAT IS MEANT BY A “ZERO-SUM GAME.
In particular, no game or game-like situation with three or more players can be a zero-sum game. Furthermore, it is a mistake to conclude that all other games are positive sum, or negative sum, or variable sum games. The concept of the sum of a game, assuming the impossibility of interpersonal comparisons of utility, is simply undefined for all such games. And this is true, regardless of what Barack Obama, Paul Krugman, and everyone else says.
So everyone always misuses the phrase “zero-sum game.”