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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 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'.

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.

You're right, I made a false statement because I was in a rush. What I meant to say was that as long as Bob's utility was linear, whatever utility function Alice has there is no way to get all the money.

Are you enforcing that choice? Because it's not a natural one.

It simplifies the scenario, and suggests.

Linear utility is not the most obviously correct utility function: diminishing marginal returns, for instance.

Why is diminishing marginal returns any more obvious that accelerating marginal returns. The former happens to be the human attitude to the thing humans most commonly gamble with (money) but there is no reason to privilege it in general. If Alice and Bob have accelerating returns then in general the money will always be given to Bob, if they have linear returns, it will always be given to Bob, if they have Diminishing returns, it could go either way. This does not seem fair to me.

varying marginal valuations can push the solution in one direction or the other.

This is true, but the default is for them to go to the powerful player.

Look at a moderately more general example, the treasure splitting game. In this version, if Alice and Bob work together, they can get a large treasure haul, consisting of a variety of different desirable objects. We will suppose that if they work separately, Bob is capable of getting a much smaller haul for himself, while Alice can get nothing, mkaing Bob more powerful.

In this game, Alice's value for the whole treasure gets sent to 1, Bob's value for the whole treasure gets sent to a constant more than 1, call it k. For any given object in the treasure, we can work out what proportional of the total value each thinks it is, if Alice's number is at least k times Bob's, then she gets it, otherwise Bob does. This means, if their valuations are identical or even roughly similar, Bob gets everything. There are ways for Alice to get some of it if she values it more, but there are symmetric solutions that favour Bob just as much. The 'central' solution is vastly favourable to Bob.

It does not. See this post ( ): any player can lie about their utility to force their preferred outcome to be chosen (as long as it's admissible). The weaker player can thus lie to get the maximum possible out of the stronger player. This means that there are weaker players with utility functions that would naturally give them the maximum possible. We can't assume either the weaker player or the stronger one will come out ahead in a trade, without knowing more.

Alice has $1000. Bob has $1100. The only choices available to them are to give some of their money to the other. With linear utility on both sides, the most obvious utility function, Alice gives all her money to Bob. There is no pair of utility functions under which Bob gives all his money to Alice.

If situation A is one where I am more powerful, then I will always face it at high-normalisation, and always face its complement at low normalisation. Since this system generally gives almost everything to the more powerful player, if I make the elementary error of adding the differently normalised utilities I will come up with an overly rosy view of my future prospects.

You x+y > 2h proof is flawed, since my utility may be normalised differently in different scenarios, but this does not mean I will personally weight scenarios where it is normalised to a large number higher than those where it is normalised to a small number. I would give an example if I had more time.

I didn't interpret the quote as implying that it would actually work, but rather as implying that (the Author thinks) Hanson's 'people don't actually care' arguments are often quite superficial.

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