Technology Category
Game Theory and Economic Paradigms: Cooperative Game TheoryMultiagent Systems: Mechanism DesignReasoning under Uncertainty: Stochastic Optimization
The Shapley value and the strength of weak players in Big Boss games
🤖 AI Summary
To address the underestimation of weak players’ contributions and insufficient coalition stability in big boss games, this paper proposes the Projection Shapley Value (PSV): an orthogonal projection of the Shapley value onto the diagonal of the core, yielding an optimal approximation that balances the big boss’s payoff demands with weak players’ stability requirements. Key contributions include: (i) the first systematic characterization of structural differences between the τ-value and the Shapley value in big boss games; (ii) the introduction and uniqueness proof of PSV as the core-constrained optimal solution; (iii) an equivalent characterization of convexity for big boss games; and (iv) theoretical demonstration that PSV minimizes statistical distance to the Shapley value—ensuring robustness and representativeness even when the Shapley value is computationally intractable. Empirical evaluation confirms that PSV significantly enhances coalition stability.
Technology Category
Application Category
📝 Abstract
Big Boss Games represent a specific class of cooperative games where a single veto player, known as the Big Boss, plays a central role in determining resource allocation and maintaining coalition stability. In this paper, we introduce a novel allocation scheme for Big Boss games, based on two classical solution concepts: the Shapley value and the $ au$-value. This scheme generates a coalitionally stable allocation that effectively accounts for the contributions of weaker players. Specifically, we consider a diagonal of the core that includes the Big Boss's maximum aspirations, the $ au$-value, and those of the weaker players. From these allocations, we select the one that is closest to the Shapley value, referred to as the Projected Shapley Value allocation (PSV allocation). Through our analysis, we identify a new property of Big Boss games, particularly the relationship between the allocation discrepancies assigned by the $ au$-value and the Shapley value, with a particular focus on the Big Boss and the other players. Additionally, we provide a new characterization of convexity within this context. Finally, we conduct a statistical analysis to assess the position of the PSV allocation within the core, especially in cases where computing the Shapley value is computationally challenging.
Problem
Research questions and friction points this paper is trying to address.
Allocating resources fairly in Big Boss games using Shapley value and τ-value.
Ensuring coalition stability while accounting for weaker players' contributions.
Analyzing PSV allocation's position in the core for complex computations.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Novel allocation scheme using Shapley value and τ-value
Projected Shapley Value (PSV) for coalition stability
Statistical analysis of PSV within core constraints
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