The Bondareva-Shapley Theorem characterizes when the core of a cooperative game with transferable payoffs is nonempty, meaning some way of dividing the grand coalition's total value leaves no smaller coalition better off acting alone, showing this holds exactly when the game is balanced in a precise technical sense. Named for Olga Bondareva and Lloyd Shapley, who proved it independently, it is a foundational existence result of cooperative game theory.
Facts
StatementThe core of a cooperative game with transferable payoffs is non-empty if and only if the game is balanced. 2 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
In Branch
Source Bondareva-Shapley theorem - Wikipedia
Proved By
Source Bondareva-Shapley theorem - Wikipedia
Source Bondareva-Shapley theorem - Wikipedia
Sources
1. Wikipedia: Bondareva-Shapley theorem
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
Specifically, the game's core is non-empty if and only if the game is balanced.
2. Bondareva-Shapley theorem - Wikipedia
Theorem section
The game's core is non-empty if and only if the game is balanced.
References section
Bondareva, Olga N. (1963). Some applications of linear programming methods to the theory of cooperative games (In Russian). Problemy Kybernetiki.
- In Branch: Game Theory, Lead sentence
Proved By: Olga Bondareva, Lead sentence, attribution
formulated independently by Olga Bondareva and Lloyd Shapley
Proved By: Lloyd Shapley, Lead sentence, attribution
formulated independently by Olga Bondareva and Lloyd Shapley
In Group: Equilibria, Bargaining and Cooperative Games, Lead paragraph
The Bondareva-Shapley theorem, in game theory, describes a necessary and sufficient condition for the non-emptiness of the core of a cooperative game in characteristic function form.
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