LW Folk Game Theory is in fact not real game theory. The key difference is that LW Folk Game Theory tends to assume that positive utility corresponds to “I would choose this over nothing” while negative utility corresponds to “I would choose nothing over this”, and 0 utility is the indifference point.
Real Game Theory does not make such an assumption. In real game theory, you take actions that maximize your (expected) utility. Importantly, if you just add a constant to your utility function (for every possible action / outcome), then the maximizing action is not going to change—there’s no concept of “0 is the indifference point”. So, if there are two outcomes o1,o2 that can be achieved, and no others, then the utility function U1={o1:−5,o2:−3} is identical to U2={o1:5,o2:7}. In LW Folk Game Theory, “doing nothing” is usually an action and is assigned 0 utility by convention, which prevents this from happening.
If “positive sum games” isn’t really a thing I’d have expected to run into pushback about that at some point.
Consider a two player game where for any outcome o, U1(o)+U2(o)=5. Sure sounds like a positive-sum game, right? Well, by the argument above, I can replace U2 with U′2=U2−5 and the game remains exactly the same. And now we have U1(o)+U′2(o)=0, that is, for every observation U′2(o)=−U1(o) , i.e. we’re in a zero-sum game.
As cousin_it said, really they shouldn’t be called zero-sum games, they should be called fixed-sum or constant-sum games. Two player constant-sum games are perfectly competitive, and as a result there are no threats: anything that hurts the other player helps you in exactly the same amount, and so you do it.
(As you note, if there are more than 2 players, you can get things like threats and collaboration, e.g. the weaker two players collaborate to overthrow the stronger one.)
Re: expecting pushback, I generally don’t expect LW terminology to agree particularly well with academia. The goals are different, and the terminology reflects this. LW wants to be able to compare everything to “nothing happened”, so there’s a convention that nothing happens = 0 utility. Real game theory doesn’t want to make that comparison, it prefers to have elegance and fewer assumptions.
LW “positive-sum games” means “both players are better off than if they did nothing”, or at least “one of the players is better off by an amount greater than the amount the other player is worse off”. Similarly for “negative-sum games”. This is fundamentally about comparing to “nothing happens”. Real game theory doesn’t care, it is all about action selection; and many games don’t have a “nothing happens” option. (See e.g. prisoner’s dilemma, where you must cooperate or defect, you can’t choose to leave the game.)
The thing I’m actually trying to contrast here is “the sort of strategic landscape, and orientation, where the thing to do is to fight over who wins social points, vs the sort of strategic landscape that encourages building something together.”
I usually call this competitive vs. collaborative, and games / strategies can be on a spectrum between competitive and collaborative. The maximally competitive games are two player zero sum games. The maximally collaborative games are common payoff games (where all players have the same utility function). Other games fall in between.
(where “fighting over who gets social points can still involve cooperation, but they it’s “allies at war” style cooperation that are dividing up spoils, rather than creating spoils)
Here it seems like there is both a collaborative aspect (maximizing the amount of spoils that can be shared between the two) and a competitive aspect (getting the largest fraction of the available spoils).
LW Folk Game Theory is in fact not real game theory. The key difference is that LW Folk Game Theory tends to assume that positive utility corresponds to “I would choose this over nothing” while negative utility corresponds to “I would choose nothing over this”, and 0 utility is the indifference point.
Real Game Theory does not make such an assumption. In real game theory, you take actions that maximize your (expected) utility. Importantly, if you just add a constant to your utility function (for every possible action / outcome), then the maximizing action is not going to change—there’s no concept of “0 is the indifference point”. So, if there are two outcomes o1,o2 that can be achieved, and no others, then the utility function U1={o1:−5,o2:−3} is identical to U2={o1:5,o2:7}. In LW Folk Game Theory, “doing nothing” is usually an action and is assigned 0 utility by convention, which prevents this from happening.
Consider a two player game where for any outcome o, U1(o)+U2(o)=5. Sure sounds like a positive-sum game, right? Well, by the argument above, I can replace U2 with U′2=U2−5 and the game remains exactly the same. And now we have U1(o)+U′2(o)=0, that is, for every observation U′2(o)=−U1(o) , i.e. we’re in a zero-sum game.
As cousin_it said, really they shouldn’t be called zero-sum games, they should be called fixed-sum or constant-sum games. Two player constant-sum games are perfectly competitive, and as a result there are no threats: anything that hurts the other player helps you in exactly the same amount, and so you do it.
(As you note, if there are more than 2 players, you can get things like threats and collaboration, e.g. the weaker two players collaborate to overthrow the stronger one.)
Re: expecting pushback, I generally don’t expect LW terminology to agree particularly well with academia. The goals are different, and the terminology reflects this. LW wants to be able to compare everything to “nothing happened”, so there’s a convention that nothing happens = 0 utility. Real game theory doesn’t want to make that comparison, it prefers to have elegance and fewer assumptions.
LW “positive-sum games” means “both players are better off than if they did nothing”, or at least “one of the players is better off by an amount greater than the amount the other player is worse off”. Similarly for “negative-sum games”. This is fundamentally about comparing to “nothing happens”. Real game theory doesn’t care, it is all about action selection; and many games don’t have a “nothing happens” option. (See e.g. prisoner’s dilemma, where you must cooperate or defect, you can’t choose to leave the game.)
I usually call this competitive vs. collaborative, and games / strategies can be on a spectrum between competitive and collaborative. The maximally competitive games are two player zero sum games. The maximally collaborative games are common payoff games (where all players have the same utility function). Other games fall in between.
Here it seems like there is both a collaborative aspect (maximizing the amount of spoils that can be shared between the two) and a competitive aspect (getting the largest fraction of the available spoils).