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.