# Talk:Shapley value

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## Characterized by properties 2, 3, and 5?

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.

## Properties 2, 3 and 5 do NOT characterize the Shapley value

One property is missing: If a player is null, it receives zero. A player ${\displaystyle i}$ is null if ${\displaystyle v(S\cup \{i\})=v(S)}$ for all ${\displaystyle S}$ not containg ${\displaystyle i}$. 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)

You are right. Just take v(N) and divide it evenly among the players. This is another solution, different from the Shapley value, that satisfies 2, 3, and 5. --Fioravante Patrone en (talk) 01:19, 29 April 2008 (UTC)

## Barzilai's letter

Barzilai's letter is short, not peer-reviewed, and not notable. Cretog8 (talk) 15:30, 11 June 2008 (UTC)

What does length have any thing to do with relevance or accuracy? This is a valid criticism of the Shapley value. It is accurate, peer-reviewed, and notable. Uvenkata (talk) 08:42, 12 June 2008 (UTC)
not notable at all. --Fioravante Patrone en (talk) 11:59, 12 June 2008 (UTC)
The immediate implication of Barzilai's Letter to the Editor, Notices of the American Mathematical Society, April 2008, is that the Shapley Value has no foundation. The claims that an argument that nullifies Shapley's Value is not notable or not relevant to Shapley's Value defy mathematical logic and common sense.Uvenkata (talk) 11:19, 17 June 2008 (UTC)
the Shapley value has no foundation?
do you have an idea of the references about Shapley value, including criticism that you can find since the book of Luce and Raiffa (1957, 51 years before Barzilai)? Maybe you could have a look at these references: the edited book by Roth (1988), and the chapers of the Handbook of Game Theory by: Hart, McLean, Mertens, Monderer and Samet, Neyman whose bibliographical details you may find here: [1]. You may also have a look at the bibliography online by Hart: [2]
do you have an idea about the debate which is going on (since its "birth") on the foundations of game theory? I liked so much the opinion of Ken Binmore (1997) that I quoted it in my GT book: the foundations of Game Theory are now in such a mess. Do you think that researchers in game theory had to wait for the letter by Barzilai for some criticism?
resuming. The page by Barzilai is so deep, so new, that it deserves a citation in the page? No. --Fioravante Patrone en (talk) 12:26, 17 June 2008 (UTC)
Let's leave this out as WP:OR. If the insight is really important, then others will presumably pick it up and run with it, and we can cite them citing the letter.Cretog8 (talk) 14:27, 17 June 2008 (UTC) [edited Cretog8 (talk) 15:08, 20 June 2008 (UTC)]
But anyway I am negatively impressed by the fact that -all- of the visible contributions of this user are about Barzilai. The suggestion by Cretog8 seems to me a valuable one to decide about the relevance of such an insight. --Fioravante Patrone en (talk) 15:03, 17 June 2008 (UTC) [edited Cretog8 (talk) 15:08, 20 June 2008 (UTC)]
With regards to the observation that "Do you think that researchers in game theory had to wait for the letter by Barzilai for some criticism? ", I would like to comment that if the emperor has no clothes, he has no clothes, even if a two-year old makes that observation If others have also made that observation, their observations could also be noted, so that the encyclopaedia presents all the facts, including criticisms of a particular theory.

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)

discredit? Vague impression? I quoted facts.
a technical comment. By chance, I am professor of operations research, in my "real life". Neverthless, game theory is not part of operaitons research. This is a very old fashioned point of view, not that of a child who sees a naked emperor
you want a reference? By chance I have a paper. There is no utility there (and if you look aruond, you will find more papers without "utility": see, for example, the use of Shapley value for reliability theory). The characteristic function of a game is not "obliged" to rely on utility functions. Small children have grown enough. An intriguing reading could be also: Roth, A. E. [1977], "The Shapley Value as a von Neumann-Morgenstern Utility," Econometrica 45, 657-664. I have found it much deeper than the one page letter that you consider so revolutionary. --Fioravante Patrone en (talk) 17:34, 20 June 2008 (UTC)

It appears I made a procedural mistake in some of my comments. I've removed that mistake and some follow-ups in comments by Fioravante Patrone en and Uvenkata which might address things in a way which they shouldn't be addressed. I hope this post-censoring is OK with you both, if not you may revert and we can see how to continue.Cretog8 (talk) 15:08, 20 June 2008 (UTC)
In the process of undoing what has been termed as a procedural mistake, a part that addressed a comment by Fioravante Patrone en was also removed. I have edited the discussion again as a result.Uvenkata (talk) 15:41, 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)

sorry, you got already enough answers from my side, but apparently you are not interested in answers. Let me just add that science or the public interest will not be protected by the inclusion of Barzilai's letter here. Some modesty would be welcome. --Fioravante Patrone en (talk) 16:29, 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)

please refrain from continuously insert your propaganda about Barzilai. You found no support on it up to now. Respect the existing opinions and wait for addinional ones, please. --Fioravante Patrone en (talk) 06:24, 15 August 2008 (UTC)
I like Barzilai for his great contributions with Jon Borwein, etc., but any claim that von Neumann, Shapley, Aumann, Milnor, Dubey, Lucas, Owen, etc. were spouting nonsense doesn't need further discussion. Just keep deleting it. 15:28, 1 July 2011 (UTC)

## a non-technical explanation

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)

## Incorrect v function definition

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)

## References

The references should be linked to the text. As they are, they seem superfluous.Koczy (talk) 10:09, 1 July 2011 (UTC)

Maybe "coalition game" should say "coalition/cooperative game with transferable utility". The Shapley value, for superadditive TU games, is a special case of a solution concept for NTU cooperative games.
"Linearity" is more clear, and stronger, than "additivity" IMO.

Suggestions on what could be further included:

A sketch of the short proof of uniqueness, which uses linearity in an essential way, would be nice.
Cute and immediate example: a special case of Nash bargaining can be viewed as a TU cooperative game. Shapley value and the Nash solution agree in that case.
Comparison with the core: how about simple examples where the Shapley value lies and does not lie in the core? The Glove game already shows the latter case.

Mct mht (talk) 04:01, 17 February 2013 (UTC)

I am going to go ahead and change "additivity" to linearity. Saying a value is additive is misleading. The term "additive" is alluding to the fact that a game is a measure-like object (in the finite case, imposing additivity makes the game a measure defined on the discrete sigma algebra). A value is a linear functional on this (Banach) space of measures. Mct mht (talk) 08:23, 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:

• Reference : Pierre Dehez & Daniela Tellone, 2009. "Data Games : Sharing public goods with exclusion," Working Papers of BETA 2009-31, Bureau d'Economie Theorique et Appliquee, UDS, Strasbourg.

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:

• Reference 1: Sylvain Beal & Andre Casajus & Frank Huettner, 2015. "Efficient extensions of the Myerson value," Working Papers 2015-01, CRESE.
• Reference 2: Beal, Sylvain & Remila, Eric & Solal, Philippe, 2012. "Axioms of invariance for TU-games," MPRA Paper 41530, University Library of Munich, Germany.

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:

• Reference 1: Effrosyni Diamantoudi & Ines Macho-Stadler & David Perez-Castrillo & Licun Xue, 2011. "Sharing the surplus in games with externalities within and across issues," UFAE and IAE Working Papers 880.11, Unitat de Fonaments de l'Analisi Economica (UAB) and Institut d'Analisi Economica (CSIC).
• Reference 2: David Perez-Castrillo & David Wettstein, 2005. "Implementation of the Ordinal Shapley Value for a three-agent economy," UFAE and IAE Working Papers 647.05, Unitat de Fonaments de l'Analisi Economica (UAB) and Institut d'Analisi Economica (CSIC).

ExpertIdeasBot (talk) 02:39, 6 September 2016 (UTC)