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Random partitions, potential, value, and externalities

André Casajus, Yukihiko Funaki, Frank Huettner

TL;DR

This paper develops a unified, constructive foundation for extending the Shapley value to cooperative games with externalities (TUX) by interpreting a one-number potential as the expected accumulated worth of a random partition. It shows that the potential for TU games can be extended to TUX games via a path-independent restriction operator $r^p$ so that Pot^r = E_p, and that the induced $r^p$-Shapley value matches a natural random-partition Shapley value $\text{Sh}^p$; among these, only the MPW solution (arising from the Ewens-like partition $p^\star$) preserves the null-player property (or monotonicity) in the presence of externalities. The MPW construction is shown to correspond to a Chinese restaurant process, and GEN together with CI singles out $p^\star$ as the unique partition satisfying these criteria. Collectively, the results provide a rigorous, constructive route to MPW in TUX games and clarify how stochastic partition processes and restriction operators interact to yield fair, externally-aware valuations with potential-based interpretations.

Abstract

The Shapley value equals a player's contribution to the potential of a game. The potential is a most natural one-number summary of a game, which can be computed as the expected accumulated worth of a random partition of the players. This computation integrates the coalition formation of all players and readily extends to games with externalities. We investigate those potential functions for games with externalities that can be computed this way. It turns out that the potential that corresponds to the MPW solution introduced by Macho-Stadler et al. (2007, J. Econ. Theory 135, 339--356) is unique in the following sense. It is obtained as the expected accumulated worth of a random partition, it generalizes the potential for games without externalities, and it induces a solution that satisfies the null player property even in the presence of externalities.

Random partitions, potential, value, and externalities

TL;DR

This paper develops a unified, constructive foundation for extending the Shapley value to cooperative games with externalities (TUX) by interpreting a one-number potential as the expected accumulated worth of a random partition. It shows that the potential for TU games can be extended to TUX games via a path-independent restriction operator so that Pot^r = E_p, and that the induced -Shapley value matches a natural random-partition Shapley value ; among these, only the MPW solution (arising from the Ewens-like partition ) preserves the null-player property (or monotonicity) in the presence of externalities. The MPW construction is shown to correspond to a Chinese restaurant process, and GEN together with CI singles out as the unique partition satisfying these criteria. Collectively, the results provide a rigorous, constructive route to MPW in TUX games and clarify how stochastic partition processes and restriction operators interact to yield fair, externally-aware valuations with potential-based interpretations.

Abstract

The Shapley value equals a player's contribution to the potential of a game. The potential is a most natural one-number summary of a game, which can be computed as the expected accumulated worth of a random partition of the players. This computation integrates the coalition formation of all players and readily extends to games with externalities. We investigate those potential functions for games with externalities that can be computed this way. It turns out that the potential that corresponds to the MPW solution introduced by Macho-Stadler et al. (2007, J. Econ. Theory 135, 339--356) is unique in the following sense. It is obtained as the expected accumulated worth of a random partition, it generalizes the potential for games without externalities, and it induces a solution that satisfies the null player property even in the presence of externalities.
Paper Structure (26 sections, 14 theorems, 74 equations)

This paper contains 26 sections, 14 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Shapley value is a player's contribution to the expected accumulated worth of a partition
  4. Games with externalities and subgames
  5. Results
  6. Random partitions that generate the potential for TU games
  7. Potentials for TUX games equaling the expected accumulated worth of a partition
  8. The only $r^{p}$-Shapley value satisfying the null player player property is the MPW solution
  9. Discussion
  10. Potential generation and conditional independence
  11. Other prominent TUX solutions are no $p$-Shapley values
  12. Null players after removing another player
  13. Conclusion
  14. Acknowledgements
  15. Additional notation
  16. ...and 11 more sections

Key Result

Theorem 1

There exists a unique potential for TU games and we have

Theorems & Definitions (19)

  • Theorem 1: HarMas1989
  • Theorem 2: casajus-potential
  • Theorem 3: Dutta et al., 2010
  • Proposition 4
  • Corollary 5
  • Definition 6
  • Theorem 7
  • Definition 8
  • Corollary 9
  • Remark 10
  • ...and 9 more