The problem is with the word “acts”. Some lotteries might not be achievable by any act, so this phrasing makes it sound like the VNM only applies to the subset of lotteries that is actually possible to achieve. And I realize that you’re using the word “act” more specifically than this, but typically, people consider doing the same thing in a different context to be the same “act”, even though its consequences may depend on the context. So when I first read the paragraph I quoted after only skimming the rest, it sounded like it was claiming that the VNM system can only describe deontological preferences over actions that don’t take context into account, which is, of course, ridiculous.
Also, while it is true that the VNM system defines utility over lotteries, it is fairly trivial to modify it to use utility over outcomes (see first section of this post)
Perhaps worth noting that earlier in the document we defined acts as functions from world states to outcomes so this seems to resolve the second concern somewhat (if the context is different then presumably this is represented by the world states being different and so there will be different functions in play and hence different acts).
In terms of the first concern, while VNM may define preferences over all lotteries, there’s a sense where in any specific decision scenario, VNM is only appealed to in order to rank the achievable lotteries and not all of them. Of course, however, it’s important to note as well that this is only part of the story.
Anyway, I changed this for the next update so as to improve clarity.
Perhaps worth noting that earlier in the document we defined acts as functions from world states to outcomes so this seems to resolve the second concern somewhat.
What? That’s what I thought “acts” meant the first time, before I read the document more thoroughly and decided that you must mean that acts are lotteries. If you are using “act” to refer to functions from world states to outcomes, then the statement that the VNM system only applies to acts is simply false, rather than misleading.
Okay, so I’ve been reading over Peterson’s book An Introduction to Decision Theory and he uses much the same language as that used in the FAQ with one difference: he’s careful to talk about risky acts rather than just acts (when he talks about VNM, I mean, he does simply talk about acts at some other point). This seems to be a pretty common way of talking about it (people other than Peterson use this language).
Anyway, Peterson explicitly defines a “lottery” as an act (which he defines as a function from world states to outcomes) whose outcome is risky (which is to say, is determined randomly but with known probability) [I presume by the act’s outcome he means the outcome that will actually occur if that act is selected].
Would including something more explicit like this resolve your concerns or do you think that Peterson does things wrong as well (or do you think I’m misunderstanding what Peterson is doing)?
Either Peterson does things wrong, you’re misunderstanding Peterson, or I’m misunderstanding you. When I have time, I’ll look at that book to try to figure out which, unless you manage to sort things out for me before I get to it.
Peterson defines an act “as a function from a set of states to a set of outcomes”
The rest of the details are contained in this quote: “The key idea in von Neumann and Morgenstern’s theory is to ask the decision maker to state a set of preferences over risky acts. These acts are called lotteries, because the outcome of each act is assumed to be randomly determined by events (with known probabilities) that cannot be controlled by the decision maker”.
However, I don’t particularly see the need to get caught up in the details of what some particular people said: mostly I just want a clear way of saying what needs to be said.
Perhaps the best thing to do is (a) be more explicit about what lotteries are in the VNM system; and (b) be less explicit about how lotteries and acts interact. Use of the more neutral word “options” might help here [where options are the things the agent is choosing between].
Specifically, I could explicitly note that lotteries are the options on the VNM account (which is not to say that all lotteries are options but rather that all options are lotteries on this account), outline everything in terms of lotteries and then, when talking about the issue of action guidance, I can note that VNM, at least in the standard formulation, requires that an agent already has preferences over options and note that this might seem undesirable.
If I understand correctly, Peterson is defining “acts” and “risky acts” as completely separate things (functions from states to outcomes, and lotteries over outcomes, respectively). If that’s true, it clears up the confusion, but that seems like extraordinarily bad terminology.
Okay, well I’ve rewritten this for the next update in a way that hopefully resolves the issues.
If you have time, once the update is posted I’d love to know whether you think the rewrite is successful. In any case, thanks for taking the time to comment so far.
Perhaps worth noting that earlier in the document we defined acts as functions from world states to outcomes
I could not find a definition of “world state” in the document. All you say is
a state is a part of the world that is not an act (that can be performed now by the decision maker) or an outcome.
which is by no means a good definition. It tells you what a state is not, but not what it is. It even fails at that, given that it uses the term “part of the world” without it being previously defined.
The problem is with the word “acts”. Some lotteries might not be achievable by any act, so this phrasing makes it sound like the VNM only applies to the subset of lotteries that is actually possible to achieve. And I realize that you’re using the word “act” more specifically than this, but typically, people consider doing the same thing in a different context to be the same “act”, even though its consequences may depend on the context. So when I first read the paragraph I quoted after only skimming the rest, it sounded like it was claiming that the VNM system can only describe deontological preferences over actions that don’t take context into account, which is, of course, ridiculous.
Also, while it is true that the VNM system defines utility over lotteries, it is fairly trivial to modify it to use utility over outcomes (see first section of this post)
Thanks for the clarification.
Perhaps worth noting that earlier in the document we defined acts as functions from world states to outcomes so this seems to resolve the second concern somewhat (if the context is different then presumably this is represented by the world states being different and so there will be different functions in play and hence different acts).
In terms of the first concern, while VNM may define preferences over all lotteries, there’s a sense where in any specific decision scenario, VNM is only appealed to in order to rank the achievable lotteries and not all of them. Of course, however, it’s important to note as well that this is only part of the story.
Anyway, I changed this for the next update so as to improve clarity.
What? That’s what I thought “acts” meant the first time, before I read the document more thoroughly and decided that you must mean that acts are lotteries. If you are using “act” to refer to functions from world states to outcomes, then the statement that the VNM system only applies to acts is simply false, rather than misleading.
Okay, so I’ve been reading over Peterson’s book An Introduction to Decision Theory and he uses much the same language as that used in the FAQ with one difference: he’s careful to talk about risky acts rather than just acts (when he talks about VNM, I mean, he does simply talk about acts at some other point). This seems to be a pretty common way of talking about it (people other than Peterson use this language).
Anyway, Peterson explicitly defines a “lottery” as an act (which he defines as a function from world states to outcomes) whose outcome is risky (which is to say, is determined randomly but with known probability) [I presume by the act’s outcome he means the outcome that will actually occur if that act is selected].
Would including something more explicit like this resolve your concerns or do you think that Peterson does things wrong as well (or do you think I’m misunderstanding what Peterson is doing)?
Either Peterson does things wrong, you’re misunderstanding Peterson, or I’m misunderstanding you. When I have time, I’ll look at that book to try to figure out which, unless you manage to sort things out for me before I get to it.
Some quotes might help.
Peterson defines an act “as a function from a set of states to a set of outcomes”
The rest of the details are contained in this quote: “The key idea in von Neumann and Morgenstern’s theory is to ask the decision maker to state a set of preferences over risky acts. These acts are called lotteries, because the outcome of each act is assumed to be randomly determined by events (with known probabilities) that cannot be controlled by the decision maker”.
The terminology of risky acts is more widespread than Peterson: http://staff.science.uva.nl/~stephane/Teaching/UncDec/vNM.pdf
However, I don’t particularly see the need to get caught up in the details of what some particular people said: mostly I just want a clear way of saying what needs to be said.
Perhaps the best thing to do is (a) be more explicit about what lotteries are in the VNM system; and (b) be less explicit about how lotteries and acts interact. Use of the more neutral word “options” might help here [where options are the things the agent is choosing between].
Specifically, I could explicitly note that lotteries are the options on the VNM account (which is not to say that all lotteries are options but rather that all options are lotteries on this account), outline everything in terms of lotteries and then, when talking about the issue of action guidance, I can note that VNM, at least in the standard formulation, requires that an agent already has preferences over options and note that this might seem undesirable.
If I understand correctly, Peterson is defining “acts” and “risky acts” as completely separate things (functions from states to outcomes, and lotteries over outcomes, respectively). If that’s true, it clears up the confusion, but that seems like extraordinarily bad terminology.
Okay, well I’ve rewritten this for the next update in a way that hopefully resolves the issues.
If you have time, once the update is posted I’d love to know whether you think the rewrite is successful. In any case, thanks for taking the time to comment so far.
Point conceded (both your point and shminux’s). Edited for the next update.
I could not find a definition of “world state” in the document. All you say is
which is by no means a good definition. It tells you what a state is not, but not what it is. It even fails at that, given that it uses the term “part of the world” without it being previously defined.