Well, it is back to lockdown here at Carolina Meadows. The dining halls are closed, in person events are canceled, strict mask regulations have been reimposed, and my golden years are turning out to be fool’s gold. Susie and I have not been to a movie theater in two years, we have stopped going out to dinner at local restaurants, I have started again doing a little cooking just to vary our evening meals a bit, and I am struggling with the imperfect curbside pickup arrangements of our local supermarket. What to do? Five mornings a week I spend half an hour peddling on my exercycle in an effort to hold off the advance of my Parkinson’s. I am approaching 4000 straight wins in FreeCell (hat tip to David Palmeter) and there are only so many jigsaw puzzles I can tackle. So, faute de mieux, I have decided to spend a little time on my blog addressing my own personal bugabears: the seemingly universal misunderstanding of the phrases “zero-sum game” and “Prisoner’s Dilemma.” I shall leave to another post the confusion surrounding the so-called “free rider problem”
Why on earth am I doing this? The answer is simple. I am desperately trying to avoid going crazy. Some people fend off incipient madness by binge eating. Some drink. Some take pills. Still others have soothing rituals that they perform before meals or on going to bed. I find it reassuring to tell myself old familiar stories about ideas. So here goes. You can, if you wish, view what follows as a particularly public form of psychotherapy.
Let me begin with the term “zero-sum game,” which was introduced into our discourse by John von Neumann and Oskar Morgenstern in their classic work The Theory of Games and Economic Behavior. A rather lengthy back story is required in order to explain the total significance of the now familiar expression. In 1789 Jeremy Bentham officially published An Introduction to the Principles of Morals and Legislation (it had actually been printed some years earlier), in which he introduced the theory of utilitarianism. So much time has passed since then and utilitarianism has become so familiar a part of our philosophical, moral, and political discussions that it is difficult to recall just how revolutionary Bentham’s book was. The problem was simple. Bentham said that the goal of the state should be to pursue the greatest happiness for the greatest number, adding that each person in the calculation should count for one. This was in its original form totally unacceptable for the simple reason that there were vastly more peasants and workers then there were gentlemen and gentle ladies. Oh, you could try to adjust things a bit by observing that the upper classes had much more refined sensibilities and therefore suffered more pain from even the slightest deprivation than the lower classes suffered as a consequence of their miserable existence. But there were so bloody many peasants and so few gentlemen and gentle ladies that no matter how you did the calculation, it turned out that Bentham’s theory implied making some major changes in society, changes that would benefit the lower classes at the expense of the upper classes.
John Stuart Mill had a go at correcting things by distinguishing qualities or classes of pleasures. Socrates dissatisfied is more valuable than a pig satisfied, he famously observed. But even so, there were so many of them, those pigs, those unwashed masses, those peasants and laborers.
Eventually, philosophers, in service to the economists, found a manageable solution: the problem of other minds. They decided that it was impossible for one person directly to compare his or her pain or pleasure with that of any other person. Hence it was impossible to add the pleasures and pains of different people in order to form a judgment about the relative desirability of competing social policies. To indulge for a moment in the jargon that became popular among those who talk about this sort of problem, the most you could say was that each person has a utility function that is invariant under affine transformations. This meant that neither the zero point nor the size of each unit of pleasure or pain was invariant. To calculate a conversion formula that would permit us to add one person’s pleasures and pains to that of another person would require two points of interpersonal comparison and since these did not exist, nothing at all could be concluded about how one person’s pleasures and pains compared with that of another.
You could of course always make judgments about the relative desirability of two policies when everybody agreed which one was to be preferred (or, to be more precise, if everyone preferred the first to the second or was indifferent between them and at least one person strictly preferred the first to the second, then you could conclude that the first was socially to be preferred to the second.) This permitted a partial ordering of available alternatives, labeled “Pareto preferred” in honor of Vilfredo Pareto, who thought the idea up.
And so what is usually referred to these days as Welfare Economics, even though the one thing that it is incapable of actually talking about is human welfare, came into existence, flourished, and even won for its ablest practitioners a number of Nobel prizes.
Okay, back to zero-sum games. Von Neumann began his work by considering the simplest possible games – two-person games in which the preferences of the two participants for the possible outcomes of the game exhibit a very precise mathematical characteristic. The first person’s preferences for the various possible outcomes are the exact, precise opposite of the second person’s preferences. Indeed not only is it the case that the first person playing the game prefers or is indifferent to one possible outcome of the game over a second possible outcome if and only if the second person prefers or is indifferent to the second possible outcome over the first. It must also be the case that the first person prefers or is indifferent to one probability combination of the possible outcomes over a second probability combination of the possible outcomes if and only if the second person prefers or is indifferent to the second probability combination over the first. Persons exhibiting this rather unusual preference structure are said to have strictly opposed preferences.
Since each person’s preferences are assumed to be invariant under affine transformations, one could without loss of information convert each person’s preferences to a scale running from 0 to 1 and since the two persons are assumed to have strictly opposed preferences, it follows that the possibility rated zero by the first player would be rated 1 by the second. It was then possible for Von Neumann to prove fairly easily (I am skipping over a lot of mathematics here) that the sum of the utility index assigned by player one to any outcome or probability combination of outcomes added to the quite independent utility index assigned by player two to that outcome or probability combination of outcomes must always be equal to one. And since the players’ utility indices are invariant under affine transformations, one could revise one player’s index to run from -1 to 0. The result would be then that the sum of the two utility indices assigned by the players to an outcome or probability combination of outcomes would always add up to zero.
And that, and only that, is what is meant by the phrase zero-sum game!
Certain thing should be obvious right off the bat. First of all, the concept of a zero-sum game is only defined for two-person games. It has simply no meaning for games with more than two players. Second, there is no special thing called a positive sum game or a negative sum game. There are only constant-sum games and games for which the concept of a sum is undefined. Third, the condition of strictly opposed preference orders is extremely restrictive. For example, the game that consists of a negotiation between a buyer and seller for a piece of property is almost certainly not a constant sum game because presumably, although the buyer’s interests are opposed to the seller’s interests, both would prefer coming to an agreement rather than having the negotiation breakdown, in which case their preferences are not strictly opposed.
So why do von Neumann and Morgenstern devote so much attention to zero-sum games? The answer is that for that special and rather simple case, von Neumann can prove an elegant and very powerful theorem, namely that every two-person zero-sum game has a unique solution. (The proof is really classy but there are limits to what I can do on a blog.)
I think it is fair to say that almost nobody who uses the phrase “zero-sum game” has the foggiest idea of any of this.
There, I feel better.
Tomorrow, Prisoner’s Dilemma.