zero-sum game

DEFINITION: A “zero-sum game” is a type of two-sided game—in the mathematical, game-theoretic sense of the word “game”—in which a positive outcome for one side corresponds to an equivalent but negative outcome for the other side.

In colloquial speech, a zero-sum game is known as a “win-lose” situation.

The opposite of a zero-sum game is a non-zero-sum game.

ETYMOLOGY: The phrase “zero-sum” derives from the fact that the sum of the outcomes of the game for the two sides always equals zero.

For example, in a zero-sum game, if Side A achieves an outcome of +n, then Side B must achieve an outcome of –n. And, of course, +n + –n = 0.

USAGE: The classic example of a zero-sum situation is cutting up a cake.

To make the zero-sum character of dividing up a cake apparent, we need to make two assumptions:

  • a specified number of pieces—say, corresponding to the number of guests at a party
  • a rule for cutting the pieces—namely, that they should all be of the same size

It is clear that if we divide the cake by the specified number of pieces—let us call this number z—and impose the constraint that all the pieces be of the same size, then the size of each piece will be fixed by the basic rules. Let us call this fixed size m.

Given these assumptions, then, if one piece > m by some fixed amount j, then the recipient of that m + j sized piece of cake is by definition the “winner” in a zero-sum game.

The definition also allows us to calculate exactly the size of each of the remaining z – 1 pieces, always assuming equality.

Since each piece will clearly be < m by some amount k < j, each of the other guests at the party will receive an m k sized slice of cake.

More exactly, each of the other guest will get a slice of cake of a size determined as follows:

m – k = (zm – (m + j)) / (z – 1)

This may seem like much ado about very little when it comes to dividing up a cake.

However, in political economy, the notion of a zero-sum game is very important because it is often lurking in the background of bitter policy disputes as an unexamined presupposition.

For example, the very language in which policy proposals are couched—especially, in terms of the words “distribution” or “redistribution”—rest upon the unspoken assumption that we are talking about a zero-sum game.

Thus, progressives tend to look upon income, not as something earned through human effort, but as something that is just there—representing a fixed amount of money at any given moment in time.

Since this fixed amount of money will not be equally “distributed”—which is simply to say that some people will be have more money than others—it will be argued that justice demands “redistribution,” which amounts to taking money away from some people and giving it to others.

In short, progressives see economic output as a sort of big cake or pie. Indeed, sometimes the actual language of someone’s not getting his “fair share of the pie” is even used.

To be sure, if it were true that the economy was a zero-sum game, then such a way of speaking and thinking would make sense.

But it is not true.

In fact, the assumption underlying progressive rhetoric about “redistribution” is demonstrably false.

The consequences of the economic activity of hundreds of millions of individuals simply bears no resemblance to a zero-sum game.

Not only is it possible that the fortunes of all those who are productively engaged in economic activity rise or fall together. It is the usual case.