In the real world, agent’s marginals vary a lot, and the gains from trade are huge, so this isn’t likely to come up.
I doubt this claim, particularly the second part.
True, many interactions have gains from trade, but I suspect the weight of these interactions is overstated in most people’s minds by the fact that they are the sort of thing that spring to mind when you talk about making deals.
Probably the most common form of interaction I have with people is when we walk past each-other in the street and neither of us hands the other the contents of their wallet. I admit I am using the word ‘interaction’ quite strangely here, but you have given no reason why this shouldn’t count as a game for the purposes of bargaining solutions, we certainly both stand to gain more than the default outcome if we could control the other). My reaction to all but a tiny portion of humanity is to not even think about them, and in a great many cases there is not much to be gained by thinking about them.
I suspect the same is true of marginal preferences, in games with small amounts at stake, preferences should be roughly linear, and where desirable objects are fungible, as they often are, will be very similar accross agents.
In the default, Alice gets nothing. If k is small, she’ll likely get a good chunk of the stuff. If k is large, that means that Bob can generate most of the value on his own: Alice isn’t contributing much at all, but will still get something if she really cares about it. I don’t see this as ultra-unfavourable to Alice!
If k is moderately large, e.g. 1.5 at least, then Alice will probably get less than half of the remaining treasure (i.e. treasure Bob couldn’t have acquired on his own) even by her own valuation. Of course the are individual differences, but it seems pretty clear to me that compared to other bargaining solutions, this one is quite strongly biased towards the powerful.
This question isn’t precisely answerable without a good prior over games, and any such prior is essentially arbitrary, but I hope I have made it clear that it is at the very least not obvious that there is any degree of symmetry between the powerful and the weak. This renders the x+y > 2h ‘proof’ in your post bogus, as x and y are normalised differently, so adding them is meaningless.
Your “walking by in the street” example is interesting. But the point of weighting your utilities is to split the gains from every single future transaction and interaction with them. Since you’re both part of the same economic system, they will have (implicit or explicit) interactions in the future. Though I don’t yet know the best way of normalising multiple agents utilities, which we’d need to make this fully rigorous.
And seeing how much world GDP is dependent on trade, I’d say the gains from trade are immense! I note your treasure hunting example has rather large gains from trade...
So, what we do know:
1) If everyone has utility equally linear in every resource (which we know is false), then the more powerful player wins everything (note that this one of the rare cases where there is an unarguable “most powerful player”)
It’s a proof based on premises of uncertain validity. So it certainly proves something, in some situations—the question is whether these situations are narrow, or broad.
Would it be possible to make those clearer in the post?
I had thought, from the way you phrased it, that the assumption was that for any game, I would be equally likelly to encounter a game with the choices and power levels of the original game reversed. This struck me as plausible, or at least a good point to start from.
What you in fact seem to need, is that I am equally likely to encounter a game with the outcome under this scheme reversed, but the power levels kept the same. This continues to strike me as a very substansive and almost certainly false assertion about the games I am likely to face.
I doubt this claim, particularly the second part.
True, many interactions have gains from trade, but I suspect the weight of these interactions is overstated in most people’s minds by the fact that they are the sort of thing that spring to mind when you talk about making deals.
Probably the most common form of interaction I have with people is when we walk past each-other in the street and neither of us hands the other the contents of their wallet. I admit I am using the word ‘interaction’ quite strangely here, but you have given no reason why this shouldn’t count as a game for the purposes of bargaining solutions, we certainly both stand to gain more than the default outcome if we could control the other). My reaction to all but a tiny portion of humanity is to not even think about them, and in a great many cases there is not much to be gained by thinking about them.
I suspect the same is true of marginal preferences, in games with small amounts at stake, preferences should be roughly linear, and where desirable objects are fungible, as they often are, will be very similar accross agents.
If k is moderately large, e.g. 1.5 at least, then Alice will probably get less than half of the remaining treasure (i.e. treasure Bob couldn’t have acquired on his own) even by her own valuation. Of course the are individual differences, but it seems pretty clear to me that compared to other bargaining solutions, this one is quite strongly biased towards the powerful.
This question isn’t precisely answerable without a good prior over games, and any such prior is essentially arbitrary, but I hope I have made it clear that it is at the very least not obvious that there is any degree of symmetry between the powerful and the weak. This renders the x+y > 2h ‘proof’ in your post bogus, as x and y are normalised differently, so adding them is meaningless.
Your “walking by in the street” example is interesting. But the point of weighting your utilities is to split the gains from every single future transaction and interaction with them. Since you’re both part of the same economic system, they will have (implicit or explicit) interactions in the future. Though I don’t yet know the best way of normalising multiple agents utilities, which we’d need to make this fully rigorous.
And seeing how much world GDP is dependent on trade, I’d say the gains from trade are immense! I note your treasure hunting example has rather large gains from trade...
So, what we do know:
1) If everyone has utility equally linear in every resource (which we know is false), then the more powerful player wins everything (note that this one of the rare cases where there is an unarguable “most powerful player”)
2) In general, to within the usual constraints of not losing more than you can win, any player can get anything out of the deal (http://lesswrong.com/r/discussion/lw/i20/even_with_default_points_systems_remain/ , but you consider these utilities naturally occurring, rather than the product of lying)
I don’t therefore see strong evidence I should reject my informal proof at this point.
I think you and I have very different understandings of the word ‘proof’.
It’s a proof based on premises of uncertain validity. So it certainly proves something, in some situations—the question is whether these situations are narrow, or broad.
Would it be possible to make those clearer in the post?
I had thought, from the way you phrased it, that the assumption was that for any game, I would be equally likelly to encounter a game with the choices and power levels of the original game reversed. This struck me as plausible, or at least a good point to start from.
What you in fact seem to need, is that I am equally likely to encounter a game with the outcome under this scheme reversed, but the power levels kept the same. This continues to strike me as a very substansive and almost certainly false assertion about the games I am likely to face.
After the baby, when I have time to do it properly :-)
Fair enough