Timeline for An equivariant social choice in Mathematical economics
Current License: CC BY-SA 3.0
49 events
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| Jul 30, 2017 at 10:04 | vote | accept | Ali Taghavi | ||
| Oct 26, 2016 at 15:03 | comment | added | Steven Landsburg | Or if you want a more interesting manifold---suppose a circular beach surrounds a lake and the govt is deciding where on that beach to erect a lifeguard station. | |
| Oct 22, 2016 at 9:41 | comment | added | Mark Grant | @AliTaghavi: Regarding the assumption that $M$ is a manifold: That depends on the context. For voters in an election, for example, there is usually a finite number of preferences. But it is easy to imagine situations where the set of preferences naturally form a metric space, or even a manifold (as in Steven Landsburg's explanation regarding govt spending on Maths above). | |
| Oct 22, 2016 at 9:39 | comment | added | Ali Taghavi | And finally I think the following question is natural: Assume that M is a manifold such that it admits a n-social choice for every n, is it true to say that for every n there is a social choice with the following stronger condition: $f(\overbrace{x_{1},x_{1},x_{1},\ldots,x_{1}}^{k-times},x_{2},x_{3},\ldots, x_{n-k})=x_{1} $ if $k > [n/2]$? This is natural expectation from a fair choice. The answer is affirmative for $M=\mathbb{R}^{n}$. | |
| Oct 22, 2016 at 9:28 | comment | added | Ali Taghavi | As another question: Is there a social choice on $\mathbb{R}$, as preference manifold, with two voters such that f is smooth without singularity but the corresponding level sets(foliation) is not topological equivalent to the social choice corresponding to the natural mean $x+y/2$? | |
| Oct 22, 2016 at 9:24 | comment | added | Ali Taghavi | @MarkGrant After all I am interested in the variety of social choice: I explain: We say that two social choice f and g are similar if there is a homeomorphism $\phi$ on $M\times M\times \ldots \times M$ such that $f=g\circ \phi$. This socially says that the two selection have same nature. Questions: Are there two real analytic social choice on $\mathbb{R}$ which are not similar(Lets we have two voter) | |
| Oct 22, 2016 at 9:18 | comment | added | Ali Taghavi | But I have some questions about the naturality of the classical social choice. If it would be a natural modeling of the voter-preference, so any such f should NOT be defined on the configuration space! because the result of a selection is meaning less if all people chose different preference. Moreover what is the justification of manifold consideration? Is not more natural to consider a discrete preference space? | |
| Oct 22, 2016 at 9:14 | comment | added | Ali Taghavi | @MarkGrant Any way I confess that I did not present necessary social motivation for such consideration. | |
| Oct 22, 2016 at 9:12 | comment | added | Ali Taghavi | @MarkGrant To be honest, my motivation was Client-brand systems not Voter -preference system. So the different brands gives different goods. This is why I excluded repeated point to have the configuration space.After all I asked the same question in the post: what is the social justification for the group action? | |
| Oct 18, 2016 at 7:13 | comment | added | Mark Grant | @AliTaghavi: Yes, that's what I mean. | |
| Oct 17, 2016 at 22:46 | comment | added | Ali Taghavi | @MarkGrant Do you mean : what is the social justification for this group action on M? | |
| Oct 16, 2016 at 15:06 | comment | added | Steven Landsburg | @MarkGrant: "What's unclear to me is why the manifold of preferences $M$ should come with an action of the symmetric group". What's even more unclear is why that symmetric group should just happen to be of the same size as the number of voters. If a new voter is born, do the relevant symmetries of $M$ suddenly change? | |
| Oct 16, 2016 at 14:14 | comment | added | Steven Landsburg | Or: If all Trump supporters and all Clinton supporters can unanimously agree that Todd Trimble is their second choice, why should the planner be forbidden by assumption from adopting a mechanism that makes Todd Trimble the winner? (In this case $M$ is the set of orderings of the candidates.) | |
| Oct 16, 2016 at 14:06 | comment | added | Steven Landsburg | The condition is unnatural because social choice is supposed to model compromise, and the condition rules out all compromises by fiat. If I want the new bus station built in my backyard, and you want the new bus station built in your backyard, why would we want to force the social planner to choose one of those options without allowing him to build the new bus station, say, somewhere half way in between? | |
| Oct 16, 2016 at 9:51 | comment | added | Mark Grant | Dear @AliTaghavi, what is unclear to me is why the manifold of preferences $M$ should come with an action of the symmetric group? | |
| Oct 16, 2016 at 6:27 | comment | added | Ali Taghavi | Could I remove and remedy some unclear points of my post? | |
| Oct 16, 2016 at 6:22 | comment | added | Ali Taghavi | @StevenLandsburg Thank you very much for your explanation: I realize what are you saying. At the first part of the post I just recall the classical notion of social choice IN THE MATHEMATICAL language. By "Perhaps the following.."I tried to give a personal understanding of the economics interpretations of this classical math notion. Then I tried to show that this interpretation is more natural provided we remove repeated points that is x_{i}=x_{j}. This leads me to the new formulation in terms of configuration space. But can I ask you why do you think the condition is unnatural? | |
| Oct 16, 2016 at 5:49 | comment | added | Steven Landsburg | No, my point was certainly not that any natural notion of social choice should satisfy that condition. (Let's call it the "Taghavi condition"). My point was that a) in the first part of your post, you seem not to be assuming the Taghavi condition, b) In the part beginning "Perhaps the following..." you suddenly seem to be assuming it, c) It is in any event a completely unnatural condition to impose, but d) you're still free to impose it if you want to, but e) the important thing is to stop contradicting yourself and tell us whether you're assuming it or not. | |
| Oct 16, 2016 at 5:39 | comment | added | Ali Taghavi | But it is not the case, any such f can not satisfy that condition. | |
| Oct 16, 2016 at 5:36 | comment | added | Ali Taghavi | By "Very interesting point" as a response to your first comment I mean: "If the classical notion of the social choice is very natural, then any such f should satisfies the condition you mentioned in your first comment. | |
| Oct 16, 2016 at 5:34 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Oct 16, 2016 at 5:32 | comment | added | Ali Taghavi | @StevenLandsburg Prof. Landsburg My apology if I did not understand your comment, completely.I would appreciate if you say me what is unclear. I tried to replace the classical notion of social choice by a new one: My suggestion is the following: A continuous map from configuration space to the ambient space which is equivariant(With respect to the obvious action of the symmetric group on the configuration space and a given action on M | |
| Oct 16, 2016 at 4:49 | comment | added | Steven Landsburg | In view of the fact that you've completely ignored my comment, please remove the edit saying you've responded to it. | |
| Oct 15, 2016 at 12:34 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Oct 15, 2016 at 10:53 | answer | added | Mark Grant | timeline score: 4 | |
| Oct 14, 2016 at 19:51 | comment | added | Michael Greinecker | @StevenLandsburg The "homotopic to dictatorial rule"-formulation is the one Chichilnisky originally used when she started that literature within social choice theory. The issue is not emphasized in the article cited by her though, so I was barking up the wrong tree. | |
| Oct 14, 2016 at 18:41 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Oct 14, 2016 at 18:31 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Oct 14, 2016 at 17:30 | comment | added | Steven Landsburg | @MichaelGreinecker: Suppose $n=2$, $M$ is the positive real numbers, $x$ is the amount I'd like our government to spend on mathematical research and $y$ is the amount you'd like our government to spend on mathematical research. Then it seems entirely reasonable to hope that the amount $f(x,y)$ that the government actually spends is a continuous function of $x$ and $y$. Once you admit that continuity is a reasonable requirement, you are forced to consider the topology on $M$. That seems like a pretty sensible motivation. | |
| Oct 14, 2016 at 17:09 | comment | added | Steven Landsburg | @MichaelGreinecker: If "the original topological social choice problem" means the first several lines of the post, before it goes off the rails at "a client has to choose....", then I don't understand what you're objecting to. The formulation of the problem contains nothing remotely similar to the phrase (or concept) "homotopic to a dictatorial rule". | |
| Oct 14, 2016 at 17:05 | comment | added | Michael Greinecker | I would deny that the original topological social choice problem has a sensible economic motivation, "homotopic to a dictatorial rule" is a fairly useless concept. | |
| Oct 14, 2016 at 16:50 | comment | added | Ali Taghavi | @StevenLandsburg By real modeling I mean a modeling which is constructed based on the real social situations, not just a pure mathematical construction without real application to the real social situation. By this definition, I would like to know whether you count the social choice modeling, as described in various papers and I introduced in the first lines of my question , as a real modeling? | |
| Oct 14, 2016 at 16:39 | comment | added | Steven Landsburg | @AliTaghavi: I don't know what a "real modeling" means. I don't know of any literature that makes the assumption you''re making, and I can't think of any reason why it should. | |
| Oct 14, 2016 at 16:31 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Oct 14, 2016 at 16:19 | history | edited | Ali Taghavi | CC BY-SA 3.0 |
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| Oct 14, 2016 at 16:08 | comment | added | Ali Taghavi | @StevenLandsburg No I am not saying that there is an $i$ with that property. But I think your question is very interesting. Because if the modeling is a real modeling, the preference function must satisfy the condition you mentioned, that is there is a $i$ with $f((x_[1},x_{2},\ldots,x_{n}))=x_{i}$ | |
| Oct 14, 2016 at 16:02 | comment | added | Ali Taghavi | ...any way my modeling is based on the following story: A client enters a shopping center consisting of $n$ different brands(WITH DIFFERENT PRODUCTS). He/She would be offered to choose $x_{i}$ from the $i-th$ brands, however she/he is not obligated to apply the offers. Perhaps the ordering can have some social justification. | |
| Oct 14, 2016 at 15:58 | comment | added | Steven Landsburg | @AliTaghavi: I asked a yes-or-no question. "Very interesting point" is not an answer to a yes-or-no question. Then you say "Such assumption is not included....", but you've left unedited the part where you appear to have made this assumption. So I have no idea what you're trying to say. | |
| Oct 14, 2016 at 15:42 | comment | added | Ali Taghavi | @usul Thank you for your comment. I think the philosophy of Math modeling is a bit complicated. In this particular case, the social choice, the comment of Landsburg is important. | |
| Oct 14, 2016 at 15:34 | comment | added | Ali Taghavi | @StevenLandsburg Very interesting point. Such assumption is not included in no modeling of "Social choice", neither in Eckmann paper, nor other references. | |
| Oct 14, 2016 at 13:17 | comment | added | Ali Taghavi | @GerryMyerson Prof. Myerson No, F_n(M) is the set of n ordered objects where objects are in M. | |
| Oct 14, 2016 at 13:05 | review | Close votes | |||
| Oct 14, 2016 at 15:18 | |||||
| Oct 13, 2016 at 23:25 | comment | added | usul | I think Ali must have gotten the following standard model a bit twisted? $M$ is a space of possible preferences, and there are $n$ agents each of whom, $i$, has a preference $x_i \in M$. A social choice function $f: M^n \to M$ maps $(x_1,\dots,x_n)$ to a single group preference $f(x_1,\dots,x_n) \in M$. Condition (1) is unanimity (if everyone prefers $x$, the group chooses $x$) and condition (2) is anonymity (rearranging the agents' names doesn't matter). But I don't see how to reconcile this with the client-item story or the ordered configuration space. | |
| Oct 13, 2016 at 22:05 | comment | added | Steven Landsburg | I do not understand the bit about a client "choosing one item among $n$ items". Are you saying that for every $(x_1,\ldots,x_n)$ there exists an $i$ such that $f(x_1,\ldots,x_n)=x_i$ ? | |
| Oct 13, 2016 at 21:43 | history | edited | Gerry Myerson | CC BY-SA 3.0 |
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| Oct 13, 2016 at 21:42 | comment | added | Gerry Myerson | In your first display, $M$ is the set whose elements are the $n$ objects? | |
| S Oct 13, 2016 at 20:38 | history | suggested | J.J. Green | CC BY-SA 3.0 |
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| Oct 13, 2016 at 20:20 | review | Suggested edits | |||
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| Oct 13, 2016 at 20:00 | history | asked | Ali Taghavi | CC BY-SA 3.0 |