Sunday, October 16, 2005

This Post got a real live comment which is actually interesting
(many get comment spam) I repeat the relevant part of the post.

Some theoretical results are of the form that finitely repeated games are like one shot games (true for the prisoner's dilemma). The second law of thermodynamics guarantees that the game we are all playing won't last forever. Infinitely repeated games can't fit in this particular universe (oddly I hadn't thought of this when Brad DeLong told me). The fact that analysis of such games is very common in applied game theory shows, to me, that people delibarately avoid the implications of game theory knowing them to be false and prefer to make impossible assumptions in order to reconcile game theory with reality.

Now the comment
Comments:
maybe I understood the interpretation of an infinatly repeated game wrong but I thoguht the infinatly repeated game analysis of game theory (changing some parameters, that is making cooperation harder mainly) hold for games which can be played some probabilitic number of times.

so for example lets say that at any given round there is a 0.1% chance that the game will end, the I thought we could look at the situation as an infinatly repeated game and that there was even a way of putting the .1% in there (through i forget which)

this is just a note on why the thermodynamics comment is irrelevant. but mostly it is to check that i have not forgotten all my game theory.
# posted by Anonymous : 9:53 PM

My reply

Anonymous remembers his game theory perfectly. Games with a constant chance of ending each period are similar to infinitely repeated games. Cooperation is possible for a narrower class of parameters, but the game is essentially a super game (infinitely repeated game). The risk of the game ending causes players to discount the future more. In game theory it is almost always necessary to assume players discount the future in order to handle infinitely repeated games (last I heard there was one exception matching pennies without discounting).

However, this does not mean that thermodynamics is irrelevant. If each year there is a 0.01 % chance of all life ending, then the analysis of infinitely repeated games may be relevant. Indeed, if each second there is a 99.99% chance of all life ending, the analysis of infinitely repeated games may be relevant.

However if it is certain that life will last at least a billion trillion years and is also certain that it won't last a trillion trillion years, then, according to standard game theory, cooperation in the prisoners dilemma is impossible (there is a unique Nash equilibrium in which everyone always finks).

Now any theory whose predictions about human actions right now depends on what will happen in a trillion trillion years is clearly silly. The point is that the distinction is between games that last a finite number of periods and games which can go on forever. Finite can be very very big it doesn't matter.

The fact that it seems clear that a 99.99% chance of ending every second is a shorter life than a life which lasts a trillion trillion years means that game theory seems like nonsense to us. Since we have to grasp it reliably for it to be true, this means that it is false.

Update: Another comment which I pull up here

I cant believe I find myself defending game theory, you are a strange economics profesor indeed Dr Waldman. But let us go on with this argument;

I think you will agree that most likely by accident and not by design, that the asumption used in game theory in infinatly repeated games of people expecting to live forever with some infinatly small probability epsilon is likely correct. Now if they understood thermodynamics admitedly they would nto hold this belief. but i think introspection will show you many of us against our own will likely ac t as if sucha small epsilon is actually real.

through then again i could be completly wrong...

(anonymous == econ geek == nikete) evaluates to true.

I agree entirely with nikete, econ geek and anonymous. Especially I like to think of myself as a strange economics professor indeed. one little thing the chance of living forever if there is an epsilon chance of dying each period is zero. It's just for any finite T the chance of living to T is positive. This is what is needed to invalidate the proof of always finking in the prisoners dilemma.

On the main point, my guess is that infinite horizon games were introduced because the implication always fink in the finitely repeated prisoner's dilemma was know to not correspond to actual behavior. The problem is that economic models can always be fiddled to fit any facts and, if we do so, we are allowing our emprical research (formal or casual) to guide theory to the predictions we make from experiend. We might as well cut out the middle man. When I write down models in economic theory I always start from the conclusion and work back to the assumptions which imply that conclusion. No one has ever mentioned that my models are unusual in this regard and I'm afraid they aren't.

Yet another comment from the same trinity

furthermore, if there is chance epsilon that thermodynamics is wrong about this prediction then the infinatly repeated game result still holds, right?

Absolutely right. Excellent point.

2 comments:

Anonymous said...

I cant believe I find myself defending game theory, you are a strange economics profesor indeed Dr Waldman. But let us go on with this argument;

I think you will agree that most likely by accident and not by design, that the asumption used in game theory in infinatly repeated games of people expecting to live forever with some infinatly small probability epsilon is likely correct. Now if they understood thermodynamics admitedly they would nto hold this belief. but i think introspection will show you many of us against our own will likely ac t as if sucha small epsilon is actually real.

through then again i could be completly wrong...

(anonymous == econ geek == nikete) evaluates to true.

Anonymous said...

furthermore, if there is chance epsilon that thermodynamics is wrong about this prediction then the infinatly repeated game result still holds, right?