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Isn't property number 1 also necessary? Otherwise, you could just divide the total value of the game evenly between all the players, and this function would clearly satisfy properties 2, 3, and 5.
One property is missing: If a player is null, it receives zero. A player is null if for all not containg . The Shapley value is the only value that satisfies this property, plus 2, 3, and 5. —Preceding unsigned comment added by 193.147.86.254 ( talk) 17:26, 14 September 2007 (UTC)
That might not be correct; the null player property can be derived from linearity. Aristotles ( talk) 17:24, 9 February 2021 (UTC)
Barzilai's letter is short, not peer-reviewed, and not notable. Cretog8 ( talk) 15:30, 11 June 2008 (UTC)
The answer to the question "the Shapley value has no foundation?" posed by Fioravante Patrone en is in the Notices of the American Mathematical Society cited in the criticism section of the entry that he has repeatedly deleted. The Shapley value has no foundation because it depends on the characteristic function which is ill-defined. The characteristic/worth function in Equations (1)-(2) of the Formal Definition Section of this entry cannot be constructed without specifying whose values are being evaluated and the Shapley value cannot be defined without reference to this ill-defined function. Yet Fioravante Patrone en claims that the fact that the characteristic function cannot be constructed, which by the way he does not refute, is irrelevant to the Shapley Value entry. Could a single-reference be provided to a general procedure for the construction of the worth/characteristic function on which the Shapley value depends?
Game theory is part of Operations Research and is not the private domain of game theorists. As an OR teacher and researcher and a member of the Canadian Operational Research Society, I am aware of some of Barzilai's recent presentations such as a tutorial at the 2008 Annual Meeting of CORS, articles in the Bulletin of this society in 2007 and a Colloquium at Dalhousie University's Dept. of Mathematics. This information is publicly available and these results have not been refuted by Fioravante Patrone en or others.
It is against Wikipedia policy to attempt to discredit an editor by creating a vague impression of a conflict of interest. Uvenkata ( talk) 13:54, 20 June 2008 (UTC) [edited Cretog8 ( talk) 15:08, 20 June 2008 (UTC)] Uvenkata ( talk) 15:38, 20 June 2008 (UTC)
The fundamental point here is that Shapley's Value (Equation (3)) depends on the characteristic/worth function v(S) in Equations (1)-(2). These equations characterize the ill-defined v(S) which cannot be constructed. Therefore the Shapley value has no basis and Hart's paper directly confirms this.
This is the point of the American Mathematical Society Notices Letter which Users Fioravante Patrone en and Cretog8 have had more than enough time to refute. They cannot provide a reference to the literature where the characteristic function v(S) is constructed rather than "assumed" because this function cannot be constructed.
Arbitrarily stating "no consensus" is not a valid objection to the fact that Shapley's Value is ill-defined. If Shapley's Value is sufficiently notable to merit an entry, then the fact that it is ill-defined is also notable. The claim that criticism of this concept should not appear in its own entry is, at best, illogical.
The claim that I am Barzilai is false. Likewise, the claim of conflict of interest is unfounded. This issue is at the foundation of OR and it is in my and the public's best interest that the record be set straight. In science, errors are corrected, not concealed. Uvenkata ( talk) 15:21, 7 August 2008 (UTC)
Cretog8’s proclamations are meaningless and do not merit a response. Fioravante Patrone en defies elementary logic, contradicts himself, and ignores the facts. The points of view of both users are biased and designed to conceal from Wikipedia readers the unpleasant fact that there is no foundation for Shapley’s value. Barzilai’s Letter to the Editor, Notices of the American Mathematical Society, states that the characteristic function of a game and other game theory fundamental concepts are ill-defined. This is a very significant statement: It establishes that von Neumann and Morgenstern, Shapley, Luce & Raiffa, Hart, those game theorists quoted by Fioravante Patrone en, and Fioravante Patrone en himself (among others) have committed fundamental mathematical errors. Fioravante Patrone en’s statements cover up these errors, and specifically, the immediate and obvious implication of Barzilai’s Letter with respect to the fact that Shapley’s value is ill-defined.
Here is the full text of Barzilai’s Letter: “The assignment of values to objects such as outcomes and coalitions, i.e. the construction of value functions, is a fundamental concept of game theory. Value (or utility, or preference) is not a physical property of the objects being valued, that is, value is a subjective (or psychological, or personal) property. Therefore, the definition of value requires specifying both what is being valued and whose values are being measured. Game theory’s characteristic function assigns values to coalitions so that what is being valued by this function is clear but von Neumann and Morgenstern do not specify whose values are being measured in the construction of this function. Since it is not possible to construct a value (or utility) scale of an unspecified person or a group of persons, game theory’s characteristic function is not well-defined. Likewise, all game theory solution concepts that do not specify whose values are being measured are ill-defined.” Barzilai’s Letter was published by the American Mathematical Society because it is new, correct, and very significant.
It cannot be refuted that Shapley’s value (Equation 3) is a game theory value, and it cannot be refuted that it relies on the ill-defined von Neumann and Morgenstern’s characteristic function (Equations 1-2). Fioravante Patrone en seems to believe that when errors are published they become facts but when errors are published, including game theory errors, they do not become facts -- they become published errors. The fact is that Shapley’s value has no basis and Fioravante Patrone en cannot provide a single reference to the literature where the characteristic function of a game is constructed rather than assumed. He should be advised that the 1977 paper by Roth which he quotes contains the same errors that appear throughout game theory, including a fundamental error on its second page. He should read that page carefully.
Barzilai’s Letter applies to all non-physical variables including variables that are labelled “value,” “utility,” or “preference.” The question whose values are being constructed applies to all such variables. Fioravante Patrone en’s reference to “papers without utility” is self-contradictory: Shapley himself refers to his “value” as an evaluation by the players of a game in their utility scales and this is precisely the point of Roth’s paper which Fioravante Patrone en cites (see the quote on the first page of Roth’s paper).
There is the matter of elementary scientific integrity. Since Fioravante Patrone en knows and acknowledges that the foundations of game theory are “in a mess” he should advise Wikipedia readers that this is the case rather than suppress the facts and conceal the truth from readers. The foundations of game theory are not “in a mess” -- they do not exist. Uvenkata ( talk) 18:30, 14 August 2008 (UTC)
I think the article would benefit from an intuitive explanation of the solution concept (while I'm at it, I think it'd be good to avoid the word "fair" in the lede in favor of "cooperative solution"); the fact that each member of the coalition receives share of total surplus equal to their "marginal product" or "marginal contribution", where this is based 'as if' all the members of the coalition arrived randomly. Casting the example in the article in these terms can make this concept more accessible to the lay readers. radek ( talk) 08:03, 25 January 2010 (UTC)
v does not necesarily need to be superadditive, see for instance Multiagent Systems —Preceding unsigned comment added by 186.48.240.86 ( talk) 01:46, 23 March 2011 (UTC)
Agree Koczy ( talk) 10:11, 1 July 2011 (UTC)
The references should be linked to the text. As they are, they seem superfluous. Koczy ( talk) 10:09, 1 July 2011 (UTC)
Comments on terminology:
Suggestions on what could be further included:
Mct mht ( talk) 04:01, 17 February 2013 (UTC)
Dr. Dehez has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:
Only TU games are being considered. A section on the value of NTU games should be considered.
A bit poor on the characterization of the Shapley value. Marginalism needs to be better explained. It is linked to the monotonicity axiom introduced by Young (1985). It replaces the null player and linearity axioms. About linearity, only the additive part is actually needed.
The reference to the paper with Dehez-Tellone is one among many applications of the Shapley value, for instance bankcruptcy resolution, data and information sharing,...
I suggest to add the reference to the Handbook of Game Theory (Vol 3) and the chapters by Eyal Winter (53), Neyman (56) and Mertens (58).
We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.
Dr. Dehez has published scholarly research which seems to be relevant to this Wikipedia article:
ExpertIdeasBot ( talk) 00:33, 26 May 2015 (UTC)
Dr. Beal has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:
- In the paragraph "Properties", property 4 should not be called Zero player, since it refers to something else. Null player is the correct name. Furthermore, the characterizing result given just below can be stated as a theorem, and it can be mentioned that it is due to Shubik (1962, Management Science)
- The paragraph Addendum definitions is rather strange: it states two extra properties, but they are not related to results. Furthermore, the sentence describing "Marginalism" is misleading since the Shapley value is not the unique solution which only rests on marginal contributions. I would definitely remove this paragraph. Instead, important results on the Shapley value should be added. I have in mind the appealing characterizations by Young (1985, International Journal of Game Theory), Myerson (1980, International Journal of Game Theory) and two others in Hart and Mas-Colell (1989, Econometrica). - The article lacks of references on the applications of the Shapley value to economics and operations research, but also to other sciences. Examples are the several articles published in 2008 in volume 16 of TOP, among others. According to me this is important for Wikipedia. - The extension presented in the last part of the article is only one among many extensions of the Shapley value. For instance, the extension by Myerson (1977, Mathematics of Operations Research) has initiated a substantial literature. I believe that it would make sense to list few other extensions.
Thus my overall apreciation is that the article is rather correctly written, but gives only a very partial glimpse of the important results about the Shapley value.
We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.
We believe Dr. Beal has expertise on the topic of this article, since he has published relevant scholarly research:
ExpertIdeasBot ( talk) 15:54, 24 August 2016 (UTC)
Dr. Perez-Castrillo has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:
1/
The current sentence:
"a coalition of players cooperates, and obtains a certain overall gain from that cooperation. Since some players may contribute more to the coalition than others or may possess different bargaining power (for example threatening to destroy the whole surplus), what final distribution of generated surplus among the players should arise in any particular game?"
A more appropriate sentence:
"any coalition of players can cooperate and obtain a certain overall gain from that cooperation. Since some players may contribute more to coalitions than others or may possess different bargaining power (for example threatening to destroy the whole surplus), what final distribution of the generated surplus in the grand coalition of the players should arise in any particular game?"
2/ The current sentence:
"imagine the coalition being formed one actor at a time, with each actor demanding their contribution v(S∪{i}) − v(S) as a fair compensation, and then for each actor take the average of this contribution over the possible different permutations in which the coalition can be formed."
A more appropriate sentence:
"imagine the grand coalition being formed one actor at a time, with each actor demanding his contribution, given that the actor is called after the coalition S has already been formed, that is, v(S∪{i}) − v(S), as his compensation, and then for each actor take the average of this contribution over the possible different permutations in which the grand coalition can be formed.
3/ It is possible to take out (because it is just a repetition) the following sentence:
"Where the formula for calculating the Shapley value is:
{\displaystyle \phi _{i}(v)={\frac {1}{
— !
\sum _{R}\left[v(P_{i}^{R}\cup \left\{i\right\})-v(P_{i}^{R})\right]\,\!}
Where {\displaystyle R\,\!} R\,\! is an ordering of the players and {\displaystyle P_{i}^{R}\,\!} P_{i}^{R}\,\! is the set of players in {\displaystyle N\,\!} N\,\! which precede {\displaystyle i\,\!} i\,\! in the order {\displaystyle R\,\!} R\,\!"
}}
We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.
We believe Dr. Perez-Castrillo has expertise on the topic of this article, since he has published relevant scholarly research:
ExpertIdeasBot ( talk) 02:39, 6 September 2016 (UTC)
I am interested in the generalisation to coalitions stated in the last paragraph. However, the paper cited for this formula does not in fact appear to contain it. So I am stuck - where can I read more about that extension? Nathaniel Virgo ( talk) 14:47, 18 July 2018 (UTC)
This is based spin a “note” (not full article) in a perfectly respectable but not top journal. This is not the kind of information that belongs in the lede even if it *may be* pertinent somewhere else. Volunteer Marek 00:02, 20 May 2024 (UTC)