Balanced contributions, consistency, and value for games with externalities

TL;DR

This paper extends the Shapley value to games with externalities (TUX games) by introducing balanced contributions for externalities (BC$^{\text{X}}$) and two generalized consistency notions (SC$^{\text{X}}$ and HMC$^{\text{X}}$). It shows that the MPW solution, defined as $\mathrm{MPW}(w)=\mathrm{Sh}(\bar{v}_w)$, is characterized uniquely by BC$^{\text{X}}$ with EF$^{\text{X}}$, and also by 2S$^{\text{X}}$ with either SC$^{\text{X}}$ or HMC$^{\text{X}}$. The results rely on a restriction operator $r^{\star}$ and the commutativity between averaging and restriction, linking the MPW solution to the classical Shapley value via the average game. The findings advance fair allocation in settings with externalities and offer theoretical foundations for implementations in mechanism design and network-based applications.

Abstract

We consider fair and consistent extensions of the Shapley value for games with externalities. Based on the restriction identified by Casajus et al. (2024, Games Econ. Behavior 147, 88-146), we define balanced contributions, Sobolev's consistency, and Hart and Mas-Colell's consistency for games with externalities, and we show that these properties lead to characterizations of the generalization of the Shapley value introduced by Macho-Stadler et al. (2007, J. Econ. Theory 135, 339-356), that parallel important characterizations of the Shapley value.

Theorems & Definitions (14)

• Example 1
• Theorem 2: Myerson, 1980
• Example 3
• Theorem 4
• Lemma 5
• Theorem 6: Sobolev, 1975
• Theorem 7: Hart and Mas-Colell, 1989
• Theorem 8
• Theorem 9
• Proposition 10