Talk:Cooperative game theory

Myerson Value
Is this the appropriate place to discuss the Myerson Value? The Myerson Value is the Shapley Value for a network-restricted cooperative game, where the feasible coalitions are only those that are connected under a given network. It was one of the first solution concepts for a network formation game, but is definately a cooperative concept. The Network page doesn't really seem to have a section on strategic network formation, so I don't want to totally revise that page. DOD 16:51, 4 December 2005 (UTC)
 * Yeah, we really don't have anything on network formation stuff. If you wanted to start a Network formation (game theory) or something like that you could do it.  You could also start the article Myerson value or put it here.  Any of those would be great, thanks for your help! --best, kevin [kzollman][talk] 17:33, 4 December 2005 (UTC)

Core
The definition of the Core seems to be wrong, or at least inaccurate. There is no reason why the grand coalition must be in the core (note the core may be empty) and the definition is not with respect to the grand coalition. radek 07:27, 15 January 2006 (UTC)
 * The condition is correct. If the core is empty, the condition must be applied to the empty set, and all statements (and their opposits!) are true for the empty set. Hence the GC forms. It is truem though that nowadays we usually consider a modified version of the core, where this condition is ignored. I believe, this is the D-core, but that should be checked. Koczy 23:38, 30 August 2007 (UTC)

Cooperation in parlour games
The Spiel des Jahres article links here for cooperative game, which I think is incorrect. In "board game theory", a game between coalitions of players wouldn't even be considered cooperative...

Regular Volleyball isn't a cooperative game?
A cooperative game is a game where groups of players ("coalitions") may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players.

This is exactly what Volleyball is, a competition between coalitions of players. But the article says: "In a cooperative version of volleyball, the emphasis is on keeping the ball in the air for as long as possible.", stating implicity that regular volleyball isn't a cooperative game.

Football/Soccer, Basketball, Baseball, etc. are all cooperative games, because they are a competition between coalitions of players!

And the article even says "Recreational games are rarely cooperative". BW A A :(

--200.216.8.76 19:14, 11 December 2006 (UTC)

By my reading NONE of the examples are cooperative games and these should all be removed; they are games in which players have correlated preferences over the outcomes; this gives them an incentive to cooperate but does not make this a cooperative game. In none of these cases (with the exception perhaps of some role playing games) can players sign binding agreements. In the team sports examples the team membership is given and coalitions can not be formed (and enforced) between arbitrary sets of players.

I am removing the section "Some examples of recreational cooperative games" -- it is unrelated to cooperative game theory. Admiller (talk) 09:19, 10 March 2008 (UTC)

Coordination game relationships
This article need show relationships with coordination game article.

Many concern modeled by coordination games are also, usually, called "coordination dilemmas"; and people make mistakes using this terms.


 * I am sorry, what is confusing? In cooperative games communication is free, so coordination is never an issue. Koczy 23:31, 30 August 2007 (UTC)

Superadditivity
Is it possible to explain the rationale behind the axiom of superadditivity, since it seems a bit counter-intuitive when $$A \cap B \neq \emptyset$$? Or is it forbidden for a player to belong to more then one coalition? —Preceding unsigned comment added by 82.73.205.214 (talk) 07:00, 3 September 2007 (UTC)
 * Thanks for pointing out. The condition applies for non-intersecting sets only. (See Owen, p213). Koczy 21:25, 5 September 2007 (UTC)

Re: introduction
Competition among coalitions of players? I think this would describe hybrid games, but here I would say it is individuals who "compete" (I really do not find this word appropriate here), except in a different way. Koczy 21:31, 5 September 2007 (UTC)

A cooperative game is a game where groups of players ("coalitions") may enforce ...
The definition given in the lead is hardly universal. I've seen board games where all participants should cooperate for best results marketed as "cooperative games". I think the article should either describe more meanings of the term, or state clearly the context in which the term has the meaning stated.

Intuitive Examples Needed
Someone needs to write intuitive examples of these concepts, illustrations of the formal statements. ngeorgak 11:59 12 Nov 2012 (UTC)

Dr. Echenique's comment on this article
Dr. Echenique has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

"Focuses on TU games. I would have treated TU and NTU games symmetrically."

We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

Dr. Echenique has published scholarly research which seems to be relevant to this Wikipedia article:


 * Reference : Federico Echenique, 2003. "A Short And Constructive Proof of Tarski's Fixed-Point Theorem," GE, Growth, Math methods 0305001, EconWPA.

ExpertIdeasBot (talk) 02:48, 28 May 2016 (UTC)

Dr. Herings's comment on this article
Dr. Herings has reviewed this Wikipedia page, and provided us with the following comments to improve its quality:

"General comments: 1. It would be natural to include Peleg, B., and P. Sudhölter (2010), "Introduction to the Theory of Cooperative Games," Theory and Decision Library, Series C: Game Theory, Mathematical Programming and Operations Research, Kluwer Academic Publishers, Norwell, Massachusetts. as a reference for cooperative game theory.

2. This article is only about cooperative games with transferable utility (and contains no material related to the more general non-transferable utility games), so the title is misleading.

In game theory, a cooperative game is a game where groups of players ("coalitions") may enforce cooperative behaviour, hence the game is a competition between coalitions of players, rather than between individual players. An example is a coordination game, when players choose the strategies by a consensus decision-making process.

Comments: Not many game theorists would think of a cooperative game as a competition between coalitions of players. A coordination game is not the first example of a cooperative game that comes to mind. In fact, the term strategies in connection with coordination game suggests a non-cooperative rather than a cooperative game.

A cooperative game is given by specifying a value for every coalition. Formally, the game (coalitional game) consists of a finite set of players N {\displaystyle N}  N, called the grand coalition, and a characteristic function v : 2 N   → R    {\displaystyle v:2^{N}\to \mathbb {R} }  v:2^{N}\to {\mathbb {R}} [1] from the set of all possible coalitions of players to a set of payments that satisfies v ( ∅ ) = 0   {\displaystyle v(\emptyset )=0}  v(\emptyset )=0.

Comment: It should be noticed that this is the definition of a TU-game. In cooperative game theory one disguishes transferable utility (TU) games and non-transferable utility (NTU) games.

It is sometimes assumed that a simple game is nonempty or that it does not contain an empty set.

Comment: What is meant is that it is sometimes assumed that the collection of winning coalitions is nonempty or that it does not contain an empty set.

A simple game W  {\displaystyle W}  W is strong if the complement of any losing coalition is winning, that is, if S ∉ W   {\displaystyle S\notin W}  S \notin W implesN ∖ S ∈ W   {\displaystyle N\setminus S\in W}  N\setminus S \in W.

Comment: imples should be implies

(If v  {\displaystyle v}  v is a colitional simple game that is proper and strong, v ( S ) = 1 − v ( N ∖ S )   {\displaystyle v(S)=1-v(N\setminus S)}  v(S)=1-v(N\setminus S) for any S   {\displaystyle S}  S.)

Comment: colitional should be coalitional

The Nakamura number of a simple game is the minimal number of winning coalitions with empty intersection.

Comment: Replace by: When a simple game is not collegial, its Nakamura number is the size of the smallest collection of winning coalitions having empty intersection.

More generally, a complete investigation of the relation among the four conventional axioms (monotonicity, properness, strongness, and non-weakness), finiteness, and algorithmic computability[3] has been made (Kumabe and Mihara, 2011[4]), whose results are summarized in the Table "Existence of Simple Games" below.

Existence of Simple Games[5]

Type

Finite Non-comp

Finite Computable

Infinite Non-comp

Infinite Computable

1111 no yes yes yes 1110 no yes no no 1101 no yes yes yes 1100 no yes yes yes 1011 no yes yes yes 1010 no no no no 1001 no yes yes yes 1000 no no no no 0111 no yes yes yes 0110 no no no no 0101 no yes yes yes 0100 no yes yes yes 0011 no yes yes yes 0010 no no no no 0001 no yes yes yes 0000 no no no no

Comment: It is not possible to follow this part without additional explanation.

A solution concept is a vector x ∈ R N     {\displaystyle x\in \mathbb {R} ^{N}}  x\in {\mathbb {R}}^{N} that represents the allocation to each player.

Comment: A solution concept is not a single vector, but rather a function that associates vectors to a set of cooperative games.

Zero Allocation to Null Players: The allocation to a null player is zero. A null player i  {\displaystyle i}  i satisfies v ( S ∪ { i } ) = v ( S ), ∀   S ⊆ N ∖ { i }   {\displaystyle v(S\cup \{i\})=v(S),\forall ~S\subseteq N\setminus \{i\}}  v(S\cup \{i\})=v(S),\forall ~S\subseteq N\setminus \{i\}.

Comment: Replace by: A player $ i $ such that, for every $ S \subseteq N \setminus \{i\}, $ $ v(S \cup \{i\}) = v(S), $ is called a null player. The allocation to a null player is zero.

For simple games, there is another notion of the core, when each player is assumed to have preferences on a set X  {\displaystyle X}  X of alternatives.

Comment: This is not another notion of the core. It is a special case of the core as defined for an NTU-game.

The Nakamura number of a simple game is the minimal number of winning coalitions with empty intersection.

Comment: Replace by: When a simple game is not collegial, its Nakamura number is the size of the smallest collection of winning coalitions having empty intersection.

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We hope Wikipedians on this talk page can take advantage of these comments and improve the quality of the article accordingly.

We believe Dr. Herings has expertise on the topic of this article, since he has published relevant scholarly research:


 * Reference 1: Peter Csoka & P. Jean-Jacques Herings, 2014. "Risk Allocation under Liquidity Constraints," Working Papers 2014.47, Fondazione Eni Enrico Mattei.


 * Reference 2: Brink J.R. van den & Talman A.J.J. & Herings P.J.J. & Laan G. van der, 2013. "The average tree permission value for games with a permission tree," Research Memorandum 001, Maastricht University, Graduate School of Business and Economics (GSBE).

ExpertIdeasBot (talk) 16:26, 11 July 2016 (UTC)

Rewritten introduction
I completely re-wrote the introduction. It wasn't very clear and making the distinction between a "cooperative game" and the "cooperative game theory", which are somewhat different in some contexts (cooperative games can be modeled under non-cooperative game theory). I also renamed the article into "cooperative game theory", as I believe this its main focus and a more general concept.

I tried to incorporate some of the comments made by Dr. Herings (see above on talk page). I also rewrote the "non-cooperative game" page in a symmetrical fashion.

I hope I had the right understand of the topic. If there is anything you feel is inaccurate, please feel free to edit or to let me know as I am not an expert in the field.

7804j (talk) 21:15, 15 September 2016 (UTC)