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The Shapley value and the strength of weak players in Big Boss games

Luis A. Guardiola, Ana Meca

TL;DR

This work addresses fair yet stable allocations in Big Boss games, where a single veto player interacts with many weak players. It develops the Projected Shapley Value (PSV) allocation by projecting the Shapley value onto the τ-diagonal, thereby balancing coalitional stability with explicit acknowledgment of weak players' contributions. The PSV lies in the core segment between $\tau(v)$ and $e_1(v)$, with the key parameter $\alpha^v$ revealing a convexity-based characterization: convexity ⇔ $\alpha^v=\tfrac{1}{2}$. A statistical analysis shows that as the number of weak players grows, the PSV tends toward $e_1(v)$, enabling accurate estimation without computing the Shapley value in large games via a log-normal predictive model. Collectively, the results offer a scalable, principled approach to allocating value in hierarchical cooperative games while preserving core stability and reflecting the increasing influence of weak players.

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 $τ$-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 $τ$-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 $τ$-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.

The Shapley value and the strength of weak players in Big Boss games

TL;DR

This work addresses fair yet stable allocations in Big Boss games, where a single veto player interacts with many weak players. It develops the Projected Shapley Value (PSV) allocation by projecting the Shapley value onto the τ-diagonal, thereby balancing coalitional stability with explicit acknowledgment of weak players' contributions. The PSV lies in the core segment between and , with the key parameter revealing a convexity-based characterization: convexity ⇔ . A statistical analysis shows that as the number of weak players grows, the PSV tends toward , enabling accurate estimation without computing the Shapley value in large games via a log-normal predictive model. Collectively, the results offer a scalable, principled approach to allocating value in hierarchical cooperative games while preserving core stability and reflecting the increasing influence of weak players.

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 -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 -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 -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.

Paper Structure

This paper contains 5 sections, 7 theorems, 39 equations, 11 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries cooperative game theory
  3. The proyected Shapley value for Big Boss games
  4. Statistical analysis of the impact of increasing weak players
  5. Concluding Remarks

Key Result

Theorem 2.1

Let $v \in BBG^{N}$. Then the following assertions are equivalent:

Figures (11)

  • Figure 1: Projection of the Shapley value onto the $\tau$-diagonal
  • Figure 2: Histogram of the $\alpha ^{v}$ values for 3-player games
  • Figure 3: Histogram for 4-player game
  • Figure 4: Histogram for 5-player game
  • Figure 5: Histogram for 6-player game
  • ...and 6 more figures

Theorems & Definitions (16)

  • Theorem 2.1: muto1988big
  • Definition 3.1: $\tau$-diagonal
  • Theorem 3.2
  • Example 3.3
  • Example 3.4
  • Proposition 3.5
  • Proposition 3.6
  • Theorem 3.7
  • Corollary 3.8
  • Proposition 3.9
  • ...and 6 more