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.
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.