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Just to be nit picky, it's not really a Prisoner's Dilemma since the equilibrium strategies are not strictly dominant. It's just an inefficient equilibrium.

And there's a huge literature in development economics on "urban bias" though it mostly attributes the "cities are too big" result to government policy.

There's a bit of a simplification on that, consider if the utility function works as follows (in R notation, mostly it reads as expected):

util <- function( x, y ) { 1 - sqrt(( x - 0.6 )^2 + ( y - 0.4 )^2 )}

So for the agent controlling the x variable, the "Reaction Function" is always simply x = 0.6 regardless of y value.

Thus, in a symmetric situation, equilibrium is at x* = y* = 0.6 and it's very simple to demonstrate that neither agent can individually improve their situation from that point. Here's the graphical illustration of the same function (sorry but I felt compelled to put X on the bottom axis, because we have to have standards in this world).

Clearly both agents would be better off if they could achieve a compromise equilibrium at x = y = 0.5 which would be easy enough if they just coordinated their actions between each other with some sort of contractual instrument.

Making the jump from this abstract model to inner city living is quite something though... when you consider actually many people are perfectly happy in small houses with minimal facilities. E.g. you don't have sufficient kitchen to cook in, and you don't have space to store food, but that doesn't matter because there are a dozen good and cheap cafes within 5 minutes walk and plenty of other shops that are open long hours. That's not to say everyone wants to live like this, and that's fine because there are various other options to choose from.

notsneaky: I was wondering about my calling it PD. Fair point.

Tel: those are circular indifference curves, I think, so the reaction function is horizontal, so there's no strategic complementarity.

Yes, they are circular, and I would say "reaction function is vertical" because we all know X is on the bottom :-P

Agreed, no strategic complementarity in my example above, but that does not significantly change the point being illustrated anyhow. Consider a simple affine transform:

trans.X <- function( x, y ) { 1.1 * x + 0.3 * y - 0.1 }
trans.Y <- function( x, y ) { 0.3 * x + 2 * y - 0.5 }
util <- function( x, y ) { 1 - sqrt(( x - 0.6 )^2 + ( y - 0.4 )^2 )}
util.trans <- function( x, y ) { util( trans.X( x, y ), trans.Y( x, y ))}

You can have your elliptical indifference curves if you want, but structurally I'd argue it's pretty much the same problem. At least, I don't see where this transform introduced any new and interesting behaviour into the model. Admittedly, the red line cannot be sketched in with a simple eyeball solution any more, and I'm too lazy to spend the time to calculate it out properly. However, if you kind of imagine where that red line would be, you still get the same end-result that the two parties need to coordinate with a contractual instrument if they want to optimize their situation (i.e. the Nash equilibrium leaves room for Pareto improvement).

Here's the graphic FWIW (I'm on a roll posting graphics to your site) with same colours as above via transformed coordinates:

As mentioned above, the fact that this sort of situation can in principle exist, doesn't directly lead to any conclusions about density of city living. It might give us clues about where to go looking, I'm not one of those anti-model type Austrians, I just want to see a bit more real-world data demonstrating this.

notsneaky: I was wondering about my calling it PD. Fair point.

I would argue that the local region close to the Nash equilibrium would be a similar shape to a Prisoner's Dilemma, and the far flung regions in the corners are unlikely to be visited in any practical situation, so we can ignore those regions completely. Thus, the final analysis can borrow heavily from work already done in PD without running into difficulty.

Tel, no, with a PD a person would want to play x* no matter what anyone else plays. There'd be no strategic complementarity (since if the other guy raises their x you'd still want to play x*, rather than also raising it as with sc). It's a different game, although the colloquial use of "Prisoner's Dilemma" to mean "any inefficient outcome of strategic interaction" is pretty common. But since Nick's post is about clarifying the confusion between strategic complementarity and positive externalities, I thought it'd be alright for me to get nit picky and clear up that confusion too.

Or to put it in terms of the graph, with a strongly dominant strategy like in the PD, the best response function is a horizontal (or vertical, for other players) line.

notsneaky: just a semantic dispute with you, but: would we really want to insist that a game is not called "PD" unless the players have dominant strategies? Because dominant strategies are very very rare, except in very simple games where there are only 2 (or 3) strategies.

Well, the usual presentation of the PD involves explaining that the player has an incentive to confess regardless of what other player does. You're right though that the strong dominance in PD comes from it usually being presented as a 2x2 game, in which case to have a unique inefficient equilibrium you need these strategies to be SD (otherwise you either get 2 equilibriua, a coordination game not a PD, or the outcome is efficient)

I guess the answer depends on what you want to illustrate with the PD. One way to think of it is that it shows that you can get an inefficient outcome *even with* strongly dominant strategies.

Also if you don't insist on them SDs defining PD then you can just use the phrase "inefficient equilibrium" and it works just as well, and the only reason to say "Prisoner's Dilemma" is ... aesthetics?

Putting it another way - is "Prisoner's Dilemma" synonymous with "inefficient equilibrium" or is it a special case of the latter that we want to emphasize for some reason.

notsneaky: I'm not sure. I have tended to use "PD" perhaps as just a metaphor for any game that has a unique inefficient equilibrium, just because strict dominance seems so unlikely in more general games. But I can understand people wanting a stricter definition.

This might - and I'm just speculatin' - have something to do with what happens as you increase the number of players in a game. Take Cournot competition. You increase the number of firms, you approach perfect competition. If the Cournot equilibrium involved strictly dominant strategies rather than just being an inefficient equilibrium, that wouldn't happen since with strictly dominant strategies everyone would keep playing the same inefficient strategy.

One can of course think of games where increasing the number of players does not increase efficiency. But maybe it's a necessary not a sufficient condition kind of thing. If the inefficient equilibrium involves strictly dominant strategies competition won't ever help. If the inefficient equilibrium involves just best responses, then increasing the number of players might help.

I'm just making stuff up though.

Nick:
Enjoyed your post as usual. Yesterday I realized that despite my many years studying economics, I had been combining the two concepts into one by using using the apparently dreaded word "network externality."

This morning the distinction returned to being hazy and elusive.

Let me try it out with respect to payments systems. Suppose I am a member of a 10 person payment system. You join. That to me, an existing member, is a positive externality. I know longer have to rush over with a pile of bills when I need to pay you.

Now Frances is a non-member. Does your joining create a "strategic complementary for her? Before you joined, she would gain access to ten members if she joined. Now she gains access to eleven members.

John: thanks!

I *think* if the choice is between joining and not-joining a payment system, we get strategic complementarity plus a positive externality for existing members when a new member joins, with no effect on non-members.

A different, more difficult case is where there are two different systems, or two different monies, like gold and silver. Menger would say there is very strong strategic complemantarity (the reaction function is not just upward-sloping, but has a slope greater than one in the neighbourhood of 50% using gold and 50% using silver). So we get two stable Nash equilibria, and an unstable equilibrium with 50% using each. In this case, if one person switches from silver to gold, that creates a positive externality for other gold-users and a negative externality for other silver-users.

"You can't have some areas of higher than average population density without other areas of lower than average population density."

Yes, but that doesn't really seem to grapple with the fact that, e.g., the world as a whole has become much more urbanized over time, especially in recent decades. If every Single person on earth moves into a city and robots do our farming that is an important shift, even though the average population density of the earth is the same, and the increase in density in some places is balanced by the decrease in others.

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