On the relational dependency coalitional games

Abstract

Cooperating game theory is becoming increasingly popular in AI, data science, and game theory applications in sharing and circular economy. Social media shows us the impact of many influencers on millions (even hundreds of millions) of followers, which raised the need to have a new model of coalition game, in which the influence or dependence of players on others are not equal, some have more than others. In this paper, we introduce a relational dependency game with new properties of minimal winning coalitions and maximal losing coalitions and their in-depth relationship with different approaches from simple games. In this new model, unlike simple games, all winning coalitions have the same payoff but losing coalitions have different payoffs, which coincides with Leo Tolstoy’s philosophy: all happy families are alike, but each unhappy family is unhappy in its way. The algorithm to find a minimal winning coalition among maximal losing coalitions is addressed in this paper. In this new model, unlike a simple game, we present the relational dependency coalition game model in which players depend on or do not depend on one another when they share a common interest in achieving a specific goal or outcome. The players must find a minimal winning coalition on which all players of the game depend on achieving the highest payoff. Closure operations and choice functions arise naturally in this game when there is a one-to-one correspondence between the winning coalition/losing coalition and the closure operation/choice function. And the game becomes more complex when relational independence lives with dependency among players. How to have a structural representation of relational independence along with dependency and how to describe a minimal winning collation on a simple hypergraph is also addressed in the paper.