This could have been a relatively short note about why "zero sum" is a misnomer, but I decided to elaborate some consequences. This post benefited from discussion with Sam Eisenstat.

"Zero Sum" is a misnomer.

The term intuitively suggests that an interaction is transferring resources from one person to another. For example, theft is zero-sum in the sense that it cannot create resources only transfer them. Elections are zero-sum in the sense that they only transfer power. And so on.

But this is far from the technical meaning of the term.

In order for the standard rationality assumptions used in game theory to apply, the payouts of a game must be utilities, not resources such as money, power, or personal property. Zero-sum transfer of resources is often far from zero-sum in utility.

But I'm getting ahead of myself. Let's examine the technical meaning of "zero sum" more precisely.

It's used to mean "constant sum".

The term "zero sum" is often used as a technical term, referring to games where the payouts for different players always sums to the same thing.

For example, the game rock-paper-scissors is zero sum, because it always has one winner and one loser.

More generally, constant-sum means that if you add up the utility functions of the players, you get a perfectly flat function.

"Constant sum" doesn't really make sense as a category.

It makes sense to conflate "zero sum" and "constant sum" because utility functions are equivalent under additive and positive multiplicative transforms, so we can always transform a constant-sum game down to a zero-sum game. However, by that same token, the concept of "constant sum" is meaningless: we can multiply the utility of one side or the other, and still have the same game. If you have good reflexes, you should hear "zero sum"/"constant sum" and shout "Type error! Radiation leak! You can't sum utilities without providing extra assumptions!"

Let's look at the "zero sum" game matching pennies as an example. In this game, two players have to say "heads" or "tails" simultaneously. One player is trying to match the other, while the other player is trying to be different from the one. Here's one way of writing the payoff matrix (with Alice trying to match):

	Bob
Alice		Heads	Tails
	Heads	Alice: 1 Bob: 0	Alice: 0 Bob: 1
	Tails	Alice: 0 Bob: 1	Alice: 1 Bob: 0

In that case, the game has a constant sum of 1. We can re-scale it to have a constant sum ...

Terminology / Jargon (meta)