If 2W>X+Y then the socially optimal outcome is Work/Work, and a means to coordinate on that outcome would be socially useful. If 2W<X+Y, the socially optimal outcome is for one player to Work while the other Shirks, but with no obvious choice for which one of you it should be.
(Under PD / too many cooks:)
(I’m a little surprised that this is the only case where I’ve wanted to rename the game depending on the social preference of the outcomes. That said, the only other games where X+Y isn’t forced to be greater or less than 2X are the Farmer’s Dilemma and the Abundant Commons, and those are the ones I’d most expect to want to split in future.)
To my personal taste, “chicken” is the game where 2W<X+Y; I think of chicken as fundamentally being a game where there isn’t a fair and socially optimal outcome w/o correlated randomness (ie in correlated equilibria, we have the “stop light” solution where one player gets a signal to go straight, with fair odds of who gets to go). It becomes “hawk/dove” when 2W≥X+Y, representing the idea that fighting can’t increase the amount of resources (at best the hawk strategy doesn’t destroy anything unless there’s another hawk; realistically it burns some resources either way, just a lot more when there are two hawks). However, I realize this isn’t standard—it seems like a lot of people consider both games to be “chicken”.
(By the way, I was surprised you labelled the section “farmer’s dilemma”—just from my experience, “chicken” is far more common, and “hawk/dove” is also quite common, but not as common as “chicken”; whereas I’d never actually heard “farmer’s dilemma” before.)
[I also want to briefly point out that it’s questionable to assume that the utility functions of the two players are comparable, and hence, that social welfare is necessarily captured by the sum of the two payoffs. Utility functions are always “up to a scalar”, so really “symmetric game” should mean something more complicated … but all that being said, I think you handled this the right way in the end, because we don’t want to deal with that extra complexity in a classification like this.]
To my personal taste, “chicken” is the game where 2W<X+Y; I think of chicken as fundamentally being a game where there isn’t a fair and socially optimal outcome w/o correlated randomness (ie in correlated equilibria, we have the “stop light” solution where one player gets a signal to go straight, with fair odds of who gets to go). It becomes “hawk/dove” when 2W≥X+Y, representing the idea that fighting can’t increase the amount of resources (at best the hawk strategy doesn’t destroy anything unless there’s another hawk; realistically it burns some resources either way, just a lot more when there are two hawks).
Hm. I agree that Hawk/Dove feels weird with 2W<X+Y. Chicken… I think I don’t have strong feelings on the question. To the extent that I do, it’s hard to disentangle from “actually, the outcome where you crash into each other really is super super bad”, but that’s not actually necessarily true. If I think in terms of “if you both swerve, then you’re both cowards”, then yeah, 2W<X+Y does feel natural. For the Farmer’s Dilemma I feel like it can go both ways—there can be gains from working together, but also inefficiencies if e.g. you only have one bulldozer.
Farmer’s Dilemma is definitely the less common name. (I didn’t think I coined it myself but couldn’t find any other references to it. From memory, I wrote most of the linked article without realizing it was the same as Chicken.) But I also just like it better as a name, and I have the vague sense that people equipped with that name will have an easier time recognizing instances of the problem. It’s probably relevant that Order Without Law talks about problems that it calls Prisoner’s Dilemmas, that I think are often Farmer’s Dilemmas, in the context of… well, ranching, but close enough.
Yep, very interesting :) The different names/stories definitely make me think about the games in very different ways, such that it feels quite non-obvious that the Farmer’s Dilemma is Chicken (or Hawk/Dove), intuitively.
(Actually, my mental tag for this is going to be the problem of cleaning common spaces/dishes in an apartment—it’s better if someone does it, but no one wants to do it individually.)
Also, I’ve probably heard people refer to the commons game as a PD, too. I think PD informally becomes a tag for a coming-apart of the individual and social good.
(Under “farmer’s dilemma”:)
(Under PD / too many cooks:)
To my personal taste, “chicken” is the game where 2W<X+Y; I think of chicken as fundamentally being a game where there isn’t a fair and socially optimal outcome w/o correlated randomness (ie in correlated equilibria, we have the “stop light” solution where one player gets a signal to go straight, with fair odds of who gets to go). It becomes “hawk/dove” when 2W≥X+Y, representing the idea that fighting can’t increase the amount of resources (at best the hawk strategy doesn’t destroy anything unless there’s another hawk; realistically it burns some resources either way, just a lot more when there are two hawks). However, I realize this isn’t standard—it seems like a lot of people consider both games to be “chicken”.
(By the way, I was surprised you labelled the section “farmer’s dilemma”—just from my experience, “chicken” is far more common, and “hawk/dove” is also quite common, but not as common as “chicken”; whereas I’d never actually heard “farmer’s dilemma” before.)
[I also want to briefly point out that it’s questionable to assume that the utility functions of the two players are comparable, and hence, that social welfare is necessarily captured by the sum of the two payoffs. Utility functions are always “up to a scalar”, so really “symmetric game” should mean something more complicated … but all that being said, I think you handled this the right way in the end, because we don’t want to deal with that extra complexity in a classification like this.]
Hm. I agree that Hawk/Dove feels weird with 2W<X+Y. Chicken… I think I don’t have strong feelings on the question. To the extent that I do, it’s hard to disentangle from “actually, the outcome where you crash into each other really is super super bad”, but that’s not actually necessarily true. If I think in terms of “if you both swerve, then you’re both cowards”, then yeah, 2W<X+Y does feel natural. For the Farmer’s Dilemma I feel like it can go both ways—there can be gains from working together, but also inefficiencies if e.g. you only have one bulldozer.
Farmer’s Dilemma is definitely the less common name. (I didn’t think I coined it myself but couldn’t find any other references to it. From memory, I wrote most of the linked article without realizing it was the same as Chicken.) But I also just like it better as a name, and I have the vague sense that people equipped with that name will have an easier time recognizing instances of the problem. It’s probably relevant that Order Without Law talks about problems that it calls Prisoner’s Dilemmas, that I think are often Farmer’s Dilemmas, in the context of… well, ranching, but close enough.
Yep, very interesting :) The different names/stories definitely make me think about the games in very different ways, such that it feels quite non-obvious that the Farmer’s Dilemma is Chicken (or Hawk/Dove), intuitively.
(Actually, my mental tag for this is going to be the problem of cleaning common spaces/dishes in an apartment—it’s better if someone does it, but no one wants to do it individually.)
Also, I’ve probably heard people refer to the commons game as a PD, too. I think PD informally becomes a tag for a coming-apart of the individual and social good.