Maybe I’m too close to the subject, but I feel there are two major new results in here, neither of which I knew before, and neither of which I’ve found in the literature . They are:
1) NBS and KSBS are not pareto optimal when used in situations where the game is uncertain, or iterated independently.
2) Solutions that are Pareto-Optimal are all of a particular format, that involve linear isomorphisms between the two utilities (technically, classes of affine isomorphisms with the same linear part).
The last sentence doesn’t admit the whole post was pointless; it admits the approach doesn’t completely solve the bargaining problem. But neither do NBS and KSBS—if they did, there wouldn’t be two of them, and they would be Pareto optimal.
If you are providing original research, please provide references to relevant existing research and indicate what you are doing that has not been done before.
I am providing original research, I haven’t found any trace of this in the litterature (though I didn’t do anything like a full litterature review), so can’t provide references, and the whole of the results are my idea, so, as far as I’m aware, nothing of what I am doing has been done before.
I feel there are two major new results in here, neither of which I knew before, and neither of which I’ve found in the literature . They are:
1) NBS and KSBS are not pareto optimal when used in situations where the game is uncertain, or iterated independently.
But you don’t show this, you simply claim it without proof or explanation when you write:
If we wanted to decide on our bargaining theory in advance, what would we do? … Obviously not use the KSBS; it gives 1⁄4 of an aircraft carrier to each player.
No explanation, no calculations. Maybe if you had written “a quarter carrier plus 3⁄4 euros” a reader could reconstruct your thinking. (And you don’t provide even an example of the “iterated independently” part of the claim. Presumably, you are “iterating” with changes to the game payoffs between each iteration.) It also is extremely puzzling that in this posting you are saying that NBS and KSBS are not Pareto optimal when in the last posting, it seemed that they were Pareto by definition. What has changed?
If you had analyzed this, your posting here might have been more clear and more interesting. You would have pointed out that if our protagonists wait to bargain until the coin has been flipped and it is by-that-time known which bargainer has the chance at a carrier, then you get a post-flip Pareto-optimal KSBS or NBS bargain which is not pre-flip Pareto optimal. That is, the numerical meaning of Pareto-optimality changes when the coin is flipped. Or, more precisely, the meaning changes when the bargainers learn the results of the coin flip—when their best model of the world changes.
Or, in the local jargon, to use the phrase Pareto-optimal as if it were an objective property of a given bargain is to commit the “mind projection fallacy”.
To summarize my disagreement with this one of your claims—you have not shown anything wrong with KSBS or NBS—you have merely shown that the “optimal” bargain depends upon what the bargainers know, and that sometimes what you know hurts you.
2) Solutions that are Pareto-Optimal are all of a particular format, that involve linear isomorphisms between the two utilities (technically, classes of affine isomorphisms with the same linear part).
Perhaps because my undergrad degree was in economics, this item struck me as so trivial that it didn’t seem even worth mentioning. But maybe you are correct that it is worth spelling out in detail. However, even here there are interesting points that you could have made, but didn’t.
The first point is that your μ factor, as well as U1 and U2, are not pure real numbers, they are what a scientist or engineer would call dimensioned quantities. Say that U1 is denominated in apples, and U2 is denominated in oranges. So it is mathematical nonsense to even try to add U1 to U2 (as naive utilitarianism requires) unless a conversion factor μ is provided (denominated in apples/orange).
The second is to point out more clearly that every bargaining solution (including KSBS and NBS) is equivalent to a choice of a μ such that U1 +μU2 gets maximized. Your real claim here is the insistence upon dynamic consistency. Choose μ before the coin gets flipped, you advise, and then stick with that choice even after you know the true state of the world.
And then, if you are familiar with the work of John Rawls, point out that this advice is roughly equivalent to Rawls’s “Veil of Ignorance”.
Now that might have been interesting.
Another direction you might have gone with this series is to continue the standard textbook development of bargaining theory—first covering Nash’s 1953 paper in which he shows how to select a disagreement point taking into account the credible threats which each bargainer can make against the other, and then continuing to the Harsanyi/Seldon theory for games with incomplete information, and then continuing on through the modern theory of mechanism design. Smart people have been working hard on these kinds of problems for more than 50 years, so there is little a smart amateur can add unless he first becomes familiar with existing results. My main complaint about your attempt is that you quite clearly are not familiar. This stuff is not rocket science. Papers and tutorials are available online. Go get them.
Actually, this is misstated. Misstated as badly as the equivalent claim that in some games it helps to be irrational.
What helps is not being irrational, the thing that helps is being thought irrational (even if the only way to be thought irrational is to actually be irrational).
And in this case, similarly, it is not what you know that hurts you, it is what the other guy knows.
It also is extremely puzzling that in this posting you are saying that NBS and KSBS are not Pareto optimal when in the last posting, it seemed that they were Pareto by definition. What has changed?
You explained that yourself: they are Pareto Optimal in a single game, but are not when used as sub-solutions to a game of many parts.
Well, you seem to have understood everything pretty well, without the need for extra information. And yes, I know about comparing utility functions, and yes I know about Rawl’s Veil of Ignorance, and its relevance to this example; I just didn’t want to clutter up a post that was long enough already.
The insistence on dynamic consistency is to tie it in with UDT agents, who are dynamically consistent. And the standard solutions to games of incomplete information do not seem to be dynamically consistent, so I did not feel they are relevant here.
The first point is that your μ factor, as well as U1 and U2, are not pure real numbers, they are what a scientist or engineer would call dimensioned quantities.
U1 and U2 are equivalent classes of functions from a measurable set of possible worlds to the reals, where the equivalence classes are defined by affine transformations on the image set. μ is a functional that takes an element of u1 of U1 and an element u2 of U2 and maps them to a utility function equivalence class U3. It has certain properties under affine transformations of u1 and u2, and certain other properties if U1 and U2 and replaced with U1′ and U2′, and these properties are enough to uniquely characterise μ, up to a choice of elements in the real projective line with some restrictions.
But U1+μU2 is an intuitive summary of what’s going on.
Well, you seem to have understood everything pretty well, without the need for extra information.
Actually, I didn’t understand at all on first reading. I only came up with the “dynamic consistency” interpretation on a third draft of the grandparent, as I struggled to explain my earlier complaint more fully.
I didn’t actually put in dynamic consistency by hand—it just seems that anything that is Pareto optimal in expected utility for GG requires dynamical consistency.
anything that is Pareto optimal in expected utility for GG requires dynamical consistency.
Which is a cool result.
In my opinion, it is an intuitively obvious, but philosophically suspect, result.
Obvious because of course you travel farthest if you continue in a straight line, refusing to change course in mid stream.
Suspect because you have received new information in midstream suggesting that your original course is no longer the direction you want to go. So isn’t an insistence on consistency a kind of foolishness, a “hobgoblin of little minds”?
But then, arguing for consistency, it could be pointed out that we are allowed to take new information into account in adjusting our tactics so as to achieve optimal results—maximizing the acquisition of joint utility in accordance with our original goals. The only thing we are not allowed to do is to use the new information to adjust our notion of fairness.
But then, arguing against consistency, we must ask “Why not adjust our notion of fairness?” After all, fairness is not some standard bestowed upon us from heaven—it is something we constructed ourselves for an entirely practical purpose—fairness exists so as to bind together agents with divergent purposes so they can cooperate.
So, if the arrival of new information suggests that a new bargain should be struck, why not strike a new bargain? Otherwise we can get into a situation in which one or the other of the agents no longer has any reason to cooperate except for a commitment made back in his younger and more ignorant days.
So you can call the result “cool” if you wish. I’m going to call it puzzling and provocative. Yes, I know that the “rules” of cooperative game theory include free enforcement of commitments. What puzzles me, though, is why the agents are willing to commit themselves to a course of action which may seem foolish later.
In your example, they are, in a sense, trading commitments—the commitments are a kind of IOU (or, since they are conditional, a kind of lottery ticket). In effect, someone who makes a commitment is printing money, which can then be used in trade. An interesting viewpoint on bargaining—a viewpoint worth exploring, I think.
The last part of your response was unworthy? Don’t apologize, I had it coming.
The last part of my (“not rocket science”) response was unworthy? Well, I’ll apologize, if you insist, but I really think that you did a good job with the first (tutorial) post, but a rather confused and confusing job with the second post, when you thought you were sharing original research.
Well, the posts were actually written for the purpose of the second post, and the new results therein. The first one was tacked on as an afterthought, when I realised it would be nice to explain the background to people.
Once again, my ability to predict which post people on less wrong will like fails spectacularly.
Maybe I’m too close to the subject, but I feel there are two major new results in here, neither of which I knew before, and neither of which I’ve found in the literature . They are:
1) NBS and KSBS are not pareto optimal when used in situations where the game is uncertain, or iterated independently.
2) Solutions that are Pareto-Optimal are all of a particular format, that involve linear isomorphisms between the two utilities (technically, classes of affine isomorphisms with the same linear part).
The last sentence doesn’t admit the whole post was pointless; it admits the approach doesn’t completely solve the bargaining problem. But neither do NBS and KSBS—if they did, there wouldn’t be two of them, and they would be Pareto optimal.
I am providing original research, I haven’t found any trace of this in the litterature (though I didn’t do anything like a full litterature review), so can’t provide references, and the whole of the results are my idea, so, as far as I’m aware, nothing of what I am doing has been done before.
But you don’t show this, you simply claim it without proof or explanation when you write:
No explanation, no calculations. Maybe if you had written “a quarter carrier plus 3⁄4 euros” a reader could reconstruct your thinking. (And you don’t provide even an example of the “iterated independently” part of the claim. Presumably, you are “iterating” with changes to the game payoffs between each iteration.) It also is extremely puzzling that in this posting you are saying that NBS and KSBS are not Pareto optimal when in the last posting, it seemed that they were Pareto by definition. What has changed?
If you had analyzed this, your posting here might have been more clear and more interesting. You would have pointed out that if our protagonists wait to bargain until the coin has been flipped and it is by-that-time known which bargainer has the chance at a carrier, then you get a post-flip Pareto-optimal KSBS or NBS bargain which is not pre-flip Pareto optimal. That is, the numerical meaning of Pareto-optimality changes when the coin is flipped. Or, more precisely, the meaning changes when the bargainers learn the results of the coin flip—when their best model of the world changes.
Or, in the local jargon, to use the phrase Pareto-optimal as if it were an objective property of a given bargain is to commit the “mind projection fallacy”.
To summarize my disagreement with this one of your claims—you have not shown anything wrong with KSBS or NBS—you have merely shown that the “optimal” bargain depends upon what the bargainers know, and that sometimes what you know hurts you.
Perhaps because my undergrad degree was in economics, this item struck me as so trivial that it didn’t seem even worth mentioning. But maybe you are correct that it is worth spelling out in detail. However, even here there are interesting points that you could have made, but didn’t.
The first point is that your μ factor, as well as U1 and U2, are not pure real numbers, they are what a scientist or engineer would call dimensioned quantities. Say that U1 is denominated in apples, and U2 is denominated in oranges. So it is mathematical nonsense to even try to add U1 to U2 (as naive utilitarianism requires) unless a conversion factor μ is provided (denominated in apples/orange).
The second is to point out more clearly that every bargaining solution (including KSBS and NBS) is equivalent to a choice of a μ such that U1 +μU2 gets maximized. Your real claim here is the insistence upon dynamic consistency. Choose μ before the coin gets flipped, you advise, and then stick with that choice even after you know the true state of the world.
And then, if you are familiar with the work of John Rawls, point out that this advice is roughly equivalent to Rawls’s “Veil of Ignorance”.
Now that might have been interesting.
Another direction you might have gone with this series is to continue the standard textbook development of bargaining theory—first covering Nash’s 1953 paper in which he shows how to select a disagreement point taking into account the credible threats which each bargainer can make against the other, and then continuing to the Harsanyi/Seldon theory for games with incomplete information, and then continuing on through the modern theory of mechanism design. Smart people have been working hard on these kinds of problems for more than 50 years, so there is little a smart amateur can add unless he first becomes familiar with existing results. My main complaint about your attempt is that you quite clearly are not familiar. This stuff is not rocket science. Papers and tutorials are available online. Go get them.
Actually, this is misstated. Misstated as badly as the equivalent claim that in some games it helps to be irrational.
What helps is not being irrational, the thing that helps is being thought irrational (even if the only way to be thought irrational is to actually be irrational).
And in this case, similarly, it is not what you know that hurts you, it is what the other guy knows.
Hmm. Did you see:
“Information Hazards: A Typology of Potential Harms from Knowledge”
http://www.nickbostrom.com/information-hazards.pdf
...?
You explained that yourself: they are Pareto Optimal in a single game, but are not when used as sub-solutions to a game of many parts.
Well, you seem to have understood everything pretty well, without the need for extra information. And yes, I know about comparing utility functions, and yes I know about Rawl’s Veil of Ignorance, and its relevance to this example; I just didn’t want to clutter up a post that was long enough already.
The insistence on dynamic consistency is to tie it in with UDT agents, who are dynamically consistent. And the standard solutions to games of incomplete information do not seem to be dynamically consistent, so I did not feel they are relevant here.
U1 and U2 are equivalent classes of functions from a measurable set of possible worlds to the reals, where the equivalence classes are defined by affine transformations on the image set. μ is a functional that takes an element of u1 of U1 and an element u2 of U2 and maps them to a utility function equivalence class U3. It has certain properties under affine transformations of u1 and u2, and certain other properties if U1 and U2 and replaced with U1′ and U2′, and these properties are enough to uniquely characterise μ, up to a choice of elements in the real projective line with some restrictions.
But U1+μU2 is an intuitive summary of what’s going on.
Actually, I didn’t understand at all on first reading. I only came up with the “dynamic consistency” interpretation on a third draft of the grandparent, as I struggled to explain my earlier complaint more fully.
I didn’t actually put in dynamic consistency by hand—it just seems that anything that is Pareto optimal in expected utility for GG requires dynamical consistency.
Which is a cool result.
In my opinion, it is an intuitively obvious, but philosophically suspect, result.
Obvious because of course you travel farthest if you continue in a straight line, refusing to change course in mid stream.
Suspect because you have received new information in midstream suggesting that your original course is no longer the direction you want to go. So isn’t an insistence on consistency a kind of foolishness, a “hobgoblin of little minds”?
But then, arguing for consistency, it could be pointed out that we are allowed to take new information into account in adjusting our tactics so as to achieve optimal results—maximizing the acquisition of joint utility in accordance with our original goals. The only thing we are not allowed to do is to use the new information to adjust our notion of fairness.
But then, arguing against consistency, we must ask “Why not adjust our notion of fairness?” After all, fairness is not some standard bestowed upon us from heaven—it is something we constructed ourselves for an entirely practical purpose—fairness exists so as to bind together agents with divergent purposes so they can cooperate.
So, if the arrival of new information suggests that a new bargain should be struck, why not strike a new bargain? Otherwise we can get into a situation in which one or the other of the agents no longer has any reason to cooperate except for a commitment made back in his younger and more ignorant days.
So you can call the result “cool” if you wish. I’m going to call it puzzling and provocative. Yes, I know that the “rules” of cooperative game theory include free enforcement of commitments. What puzzles me, though, is why the agents are willing to commit themselves to a course of action which may seem foolish later.
In your example, they are, in a sense, trading commitments—the commitments are a kind of IOU (or, since they are conditional, a kind of lottery ticket). In effect, someone who makes a commitment is printing money, which can then be used in trade. An interesting viewpoint on bargaining—a viewpoint worth exploring, I think.
Sorry, that last part of the response was unworthy; but if you’re being condescending to me, I feel the immense urge to be condescending back.
The last part of your response was unworthy? Don’t apologize, I had it coming.
The last part of my (“not rocket science”) response was unworthy? Well, I’ll apologize, if you insist, but I really think that you did a good job with the first (tutorial) post, but a rather confused and confusing job with the second post, when you thought you were sharing original research.
Well, the posts were actually written for the purpose of the second post, and the new results therein. The first one was tacked on as an afterthought, when I realised it would be nice to explain the background to people.
Once again, my ability to predict which post people on less wrong will like fails spectacularly.