Partition function form games with probabilistic beliefs

Technical Summary: Partition Function Form Games with Probabilistic Beliefs

Problem Statement
The paper addresses the definition and non-emptiness of the core in cooperative games with partition function form (PFF). Unlike characteristic function form games, where a coalition's worth is independent of the actions of non-members, PFF games involve inter-coalitional externalities: the payoff of a coalition $S$ depends on the partition $\pi$ of the entire player set $N$.

The central challenge in PFF games is that the core is not uniquely defined without specifying the conjectures a deviating coalition holds regarding the behavior of outsiders. Existing literature relies on deterministic conjectures (e.g., $\alpha$-conjectures where outsiders minimize the deviator's payoff, $\gamma$-conjectures where outsiders form singletons, or $\delta$-conjectures where outsiders form a single coalition). The authors propose a framework where coalitions hold probabilistic beliefs over the set of possible partitions formed by outsiders. These beliefs need not be consistent with actual equilibrium choices, reflecting a form of bounded rationality or heuristic reasoning, particularly relevant in games with large player sets where the number of partitions is vast.

Methodology
The authors analyze symmetric partition function form games, where the worth of a coalition depends only on its size and the structure of the partition. They focus on two distinct environments:

1. Positive Externalities: The merging of outside coalitions increases the worth of a coalition $S$.
2. Negative Externalities: The merging of outside coalitions decreases the worth of $S$.

The core analytical tool is mathematical induction on the number of players ($n$). The authors define specific recursive restrictions on how a coalition's probabilistic beliefs ($h_{n,S}$) must evolve as the game size increases from $n$ to $n+1$. These restrictions ensure that the expected worth of a coalition in a larger game does not grow too rapidly relative to the per-capita value of the grand coalition.

Key definitions include:

• Admissible Beliefs: A sequence of beliefs where, as $n$ increases, the coalition assigns relatively lower probability weights to partitions with few members (under positive externalities). This is formalized by linking the expected worth in game $n+1$ to that in game $n$ via the inequality $V^{\tilde{h}_{n+1}}(S) = \frac{n}{n+1}V^{h_n}(S)$.
• R-admissible Beliefs: A variation used when specific conditions on the $\gamma$-conjecture worth are met, allowing for a different recursive structure.
• Inductive Step: The proof demonstrates that if an equal split allocation is stable (in the core) for a game of size $n$, it remains stable for size $n+1$, provided the beliefs satisfy the defined recursive constraints.

Key Results
The paper establishes sufficient conditions for the non-emptiness of the core in symmetric PFF games with probabilistic beliefs.

1. Positive Externalities:

   • Proposition 1: If the worth under $\gamma$-conjectures satisfies $V^{\gamma}_{n+1}(S) \leq \frac{n}{n+1}V^{\gamma}_n(S)$, and coalitions hold admissible beliefs (assigning lower probability to favorable partitions as $n$ grows), the core is non-empty. The proof relies on showing that the expected worth of any coalition $S$ in an $(n+1)$-player game, $V^{h_{n+1}}(S)$, is bounded such that the equal split allocation remains unblockable.
   • Proposition 2: If the condition $V^{\gamma}_{n+1}(S) > \frac{n}{n+1}V^{\gamma}_n(S)$ holds, the core is still non-empty provided coalitions hold R-admissible beliefs. This case requires a specific restriction on the set of distributions to ensure the inductive step holds.
   • In both cases, the authors note that for singleton coalitions ($|S|=1$), the probability assigned to the scenario where all outsiders form a single coalition must be "sufficiently low" to prevent deviation.

2. Negative Externalities:

   • The analysis is presented as a mirror image of the positive externality case. Under negative externalities, the merging of outsiders hurts the coalition. Consequently, the recursive belief restrictions are inverted: as $n$ increases, the coalition must assign relatively lower probabilities to partitions with many members (which are the most favorable under negative externalities). The inductive logic remains identical, ensuring the non-emptiness of the core under these inverted belief constraints.

Significance and Claims
The paper claims two primary contributions:

1. Generalization of Core Definitions: The framework generalizes existing deterministic conjectures. For instance, the $\gamma$-core and $\delta$-core are shown to be special cases of the proposed probabilistic approach where the probability mass is concentrated entirely on singleton or grand coalition partitions, respectively.
2. Bounded Rationality in Cooperative Games: The authors posit that probabilistic beliefs serve as a "rule of thumb" for coalitions that cannot accurately deduce the equilibrium partition of outsiders, especially in large games. This approach offers a way to model coalition formation where agents possess cognitive limitations regarding the complexity of the partition space.

The paper explicitly avoids proposing new applications or experimental validations, instead focusing on the theoretical conditions required to guarantee the existence of a stable allocation (the core) in the presence of externalities and uncertain beliefs. It builds upon previous work by Lekeas & Stamatopoulos (2014) and Stamatopoulos (2021), extending their analysis of admissible beliefs from specific oligopolistic and environmental contexts to general symmetric environments.