Would it be possible to make those clearer in the post?

After the baby, when I have time to do it properly :-)

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Gains from trade: Slug versus Galaxy - how much would I give up to control you?

Would it be possible to make those clearer in the post?

After the baby, when I have time to do it properly :-)

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Gains from trade: Slug versus Galaxy - how much would I give up to control you?

Fair enough

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Gains from trade: Slug versus Galaxy - how much would I give up to control you?

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.

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 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'.

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.

Technically true: if he's linear, Bob can't lost more than $1000, because he can't gain more than $1000.

But Alice can certainly get almost everything. Say she has this: $1999.99 (or above): utility 1, $1000-$1999.99: utility 0, below $1000: utility -100. Then Alice gets $1999.99 and Bob loses 999.99.

Look at a moderately more general example, the treasure splitting game.

If the value of the hoard is large, then k is very close to 1. Alice will get the things she really likes (relative to Bob's valuation of them).

The 'central' solution is vastly favourable to Bob.

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!

I admit there is an issue with (quasi)-linear preferences if both players have similar relative valuations. However I don't see anything that argues that "the default is for them to go to the powerful player", apart from in that linear case. 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.

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.

The only choices available to them are to give some of their money to the other.

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

With linear utility on both sides, the most obvious utility function, Alice gives all her money to Bob.

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

There is no pair of utility functions under which Bob gives all his money to Alice.

Let Alice value $2100 at 1, $1000 at 0, and $0 at -1. Let Bob value $2100 at 1, $1100 at 0, and $0 at -0.5 (interpolate utility linearly between these values).

These utility functions are already normalised for the MVBS, and since they interpolate linearly, only these three points are possible solutions: Alice $2100, default ($1000,$1100), and Bob $2100. The first has a summed utility of 0.5, the second 0, the third 0 as well.

Thus Alice gets everything.

That example is artificial, but it shows that unless you posit that everyone has (equal) linear utility in every resource, there is no reason to assume the powerful player will get everything: varying marginal valuations can push the solution in one direction or the other.

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.

Since this system generally gives almost everything to the more powerful player

It does not. See this post ( http://lesswrong.com/lw/i20/even_with_default_points_systems_remain/ ): 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.

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.

If you don't know the opposing player, then you don't know what you'll find important with them and what they'll find important with you. Suppose for instance that you can produce ten million different goods, at various inefficiencies and marginal prices. Then you meet someone who only cares about good G, and only offers good H. Then the shape of your trade situation is determined entirely by each player's valuations of G and H and their ability to produce it. Even if you're extraordinarily powerful, you and they can have valuations/ability to produce of G and H that make the situation take any shape you want to (the default point is removing most of your options from consideration, so only a very few of them matter).

I don't have time to do the maths, but if your values are complicated enough, you can certainly face both A and symmetric A against (different) weaker players (and against stronger ones).

It does not. See this post ( http://lesswrong.com/lw/i20/even_with_default_points_systems_remain/ ): 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.

I will personally weight scenarios where it is normalised to a large number higher than those where it is normalised to a small number.

Yes. But you can face situation A and symmetric A both at high number normalisation scenario, and at a low one.

We need proper priors over unknown players' utilities to solve this correctly (finding a specific counter example is easy).

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Harry Potter and the Methods of Rationality discussion thread, part 24, chapter 95

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Harry Potter and the Methods of Rationality discussion thread, part 24, chapter 95

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Lotteries are a tax on people who don't understand statistics.

Not quite always

http://www.boston.com/news/local/massachusetts/articles/2011/07/31/a_lottery_game_with_a_windfall_for_a_knowing_few/