Multiplayer cooperative games - Revision history

Glyn00: /* Shapley value */

2021-01-24T19:49:02Z

‎Shapley value

Glyn00: /* Shapley value */

2021-01-24T19:48:30Z

‎Shapley value

Glyn00: /* Shapley value */

2021-01-24T19:46:09Z

‎Shapley value

Glyn00: /* Shapley value */

2021-01-24T19:45:13Z

‎Shapley value

Glyn00: /* Shapley value */

2021-01-24T19:43:21Z

‎Shapley value

Glyn00: /* Cooperative games */

2021-01-24T19:38:54Z

‎Cooperative games

Glyn00: /* Superadditivity and Cohesiveness */

2021-01-24T19:30:23Z

‎Superadditivity and Cohesiveness

Glyn00: /* Coalitions */

2021-01-24T19:27:36Z

‎Coalitions

Glyn00: /* Coalitions */

2021-01-24T19:26:34Z

‎Coalitions

Glyn00: /* Superadditivity and Cohesiveness */

2021-01-24T19:23:27Z

‎Superadditivity and Cohesiveness

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 *n- is total number of the players
-*the sum extends over all subsets S of N not containing player i
+*the sum extends over all subsets <math>S</math> of <math>N</math> not containing player <math>i</math>
 =Multiplayer cooperative games=

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 The Shapley value always occurs and it is unique. It can be driven as the only solution that meets the axioms of symmetry, additivity, and a dummy player axiom that says a player that does not add anything to any coalition obtains what he can achieve for himself.
-The number of player which is given a coalitional game, according to the Shapley value, <math>(v,N)</math> is given by the formula :
+The number of player which is given a coalitional game, according to the Shapley value, <math>(v,N)</math> is given by the formula :<ref name="Shapley"> Shapley value [online]. Wikipedia. Dostupné z: https://en.wikipedia.org/wiki/Shapley_value</ref>
 :<math>\varphi_i(v)=\sum_{S \subseteq N \setminus
-\{i\}} \frac{|S|!\; (n-|S|-1)!}{n!}(v(S\cup\{i\})-v(S))</math>
+\{i\}} \frac{|S|!\; (n-|S|-1)!}{n!}(v(S\cup\{i\})-v(S))</math><ref name="Shapley"> Shapley value [online]. Wikipedia. Dostupné z: https://en.wikipedia.org/wiki/Shapley_value</ref>
 =Multiplayer cooperative games=

← Older revision		Revision as of 19:46, 24 January 2021
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	The number of player which is given a coalitional game, according to the Shapley value, <math>(v,N)</math> is given by the formula :		The number of player which is given a coalitional game, according to the Shapley value, <math>(v,N)</math> is given by the formula :
		+
		+	:<math>\varphi_i(v)=\sum_{S \subseteq N \setminus
		+	\{i\}} \frac{\|S\|!\; (n-\|S\|-1)!}{n!}(v(S\cup\{i\})-v(S))</math>

	=Multiplayer cooperative games=		=Multiplayer cooperative games=

← Older revision		Revision as of 19:45, 24 January 2021
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	The Shapley value always occurs and it is unique. It can be driven as the only solution that meets the axioms of symmetry, additivity, and a dummy player axiom that says a player that does not add anything to any coalition obtains what he can achieve for himself.		The Shapley value always occurs and it is unique. It can be driven as the only solution that meets the axioms of symmetry, additivity, and a dummy player axiom that says a player that does not add anything to any coalition obtains what he can achieve for himself.
		+
		+	The number of player which is given a coalitional game, according to the Shapley value, <math>(v,N)</math> is given by the formula :

	=Multiplayer cooperative games=		=Multiplayer cooperative games=

← Older revision		Revision as of 19:43, 24 January 2021
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	The Shapley value is a solution concept in cooperative game theory, which was introduced it in 1951. To each cooperative game it gives a exclusive distribution of a total surplus produced by the coalition of all players. The Shapley value is characterized by a collection of necessary properties.<ref name="Shapley"> Shapley value [online]. Wikipedia. Dostupné z: https://en.wikipedia.org/wiki/Shapley_value</ref>		The Shapley value is a solution concept in cooperative game theory, which was introduced it in 1951. To each cooperative game it gives a exclusive distribution of a total surplus produced by the coalition of all players. The Shapley value is characterized by a collection of necessary properties.<ref name="Shapley"> Shapley value [online]. Wikipedia. Dostupné z: https://en.wikipedia.org/wiki/Shapley_value</ref>
		+
		+
		+	The Shapley value always occurs and it is unique. It can be driven as the only solution that meets the axioms of symmetry, additivity, and a dummy player axiom that says a player that does not add anything to any coalition obtains what he can achieve for himself.

	=Multiplayer cooperative games=		=Multiplayer cooperative games=

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 In other words, it is the set of possible payoff vectors for the grand coalition that no coalition can break. If such coalition S exists, we shall say that S can improve upon or block x, and x is deemed unstable.
 That is, in a context where any coalition can get together, when S has a blocking move, coalition S will create and walk back the grand coalition and its payoffs ,<math>Xs</math> to get to a better payoff for each of the members of the coalition, a plan that is enforceable for them.
 =Multiplayer cooperative games=

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 ===Cohesiveness===
 Cohesiveness is a special case of the stronger assumption, superadditivity, which requires
-<math> v(S</math> ∪ <math> T) </math> ≥ <math>v(S) + v(T)</math> for all disjoint coalitions S and T.
+<math> v(S</math> ∪ <math> T) </math> ≥ <math>v(S) + v(T)</math> for all disjoint coalitions S and T. <ref name="TU2"> Cooperative Game Theory [online]. Vrije Universiteit Amsterdam. Dostupné z: https://www.cs.vu.nl/~eliens/download/paper-cooperative-game-theory.pdf</ref>
-Cohesiveness guarantees that the equilibrium coalition is the one that should form. The worth of other coalitions will influence how <math>v(N)</math> will be divided among the players.
+Cohesiveness guarantees that the equilibrium coalition is the one that should form. The worth of other coalitions will influence how <math>v(N)</math> will be divided among the players. <ref name="TU2"> Cooperative Game Theory [online]. Vrije Universiteit Amsterdam. Dostupné z: https://www.cs.vu.nl/~eliens/download/paper-cooperative-game-theory.pdf</ref>
 ==The Core==

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 Here is the graph example for 4 agents coalition.
 [[File:Coalition Graph for 4 agents.png]]
 === Characteristic function ===

← Older revision		Revision as of 19:26, 24 January 2021
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	* Players {5}, {6} are one-member coalition, it means that these players do not cooperate with anyone. For typical conflict situations, a prerequisite is a game with a free disjunctive coalition structure. This coalition structure indicates that any coalition is allowed and a player can only be a member of one coalition.		* Players {5}, {6} are one-member coalition, it means that these players do not cooperate with anyone. For typical conflict situations, a prerequisite is a game with a free disjunctive coalition structure. This coalition structure indicates that any coalition is allowed and a player can only be a member of one coalition.

		+	Here is the graph example for 4 agents coalition.
		+	[[File:Coalition Graph for 4 agents.png]]
	=== Characteristic function ===		=== Characteristic function ===

← Older revision		Revision as of 19:23, 24 January 2021
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−	* Example: v(C) = <math>\|C\|^2</math>: – <math>v(C U D)</math> = <math>(\|C\|+\|D\|)^2</math> ≥ <math>\|C\|^2</math>+<math>\|D\|^2</math> = <math>v(C) + v(D)</math> <ref name="TU"> Cooperative Game Theory [online]. ~~Czech Technical University In Prague~~. Dostupné z: https://cw.~~fel~~.~~cvut~~.cz/~~b181~~/~~_media~~/~~courses/be4m36mas/mas2017~~-~~cooperative_games~~.pdf</ref>	+	* Example: v(C) = <math>\|C\|^2</math>: – <math>v(C U D)</math> = <math>(\|C\|+\|D\|)^2</math> ≥ <math>\|C\|^2</math>+<math>\|D\|^2</math> = <math>v(C) + v(D)</math> <ref name="TU2"> Cooperative Game Theory [online]. Vrije Universiteit Amsterdam. Dostupné z: https://www.cs.vu.nl/~eliens/download/paper-cooperative-game-theory.pdf</ref>

	* In superadditive games, two coalitions can always merge without losing money; hence, we can assume that players form the grand coalition		* In superadditive games, two coalitions can always merge without losing money; hence, we can assume that players form the grand coalition