Path Cooperative Games

• Path Cooperative Games are cooperative network models where success is determined by forming connected paths (such as between a source and sink) in graphs.
• They employ unified frameworks like stochastic path integrals and combinatorial Hodge theory to derive value allocation methods mirroring classical solution concepts.
• These games have practical applications in revenue sharing, network reliability, and resource allocation with efficient algorithms based on shortest-path and series-parallel decompositions.

A Path Cooperative Game (PC-Game) is a class of cooperative games defined on networks or more general graphs, in which successful coalitions are characterized by the ability to establish connectivity—typically, existence of a path between designated nodes (such as source-sink pairs)—through subsets of agents represented by edges or vertices. The conceptual framework integrates combinatorial optimization, cooperative game theory, network flow, and discrete Hodge theory, supporting a range of solution concepts including the Shapley value, least core, and nucleolus. PC-Games serve as a natural model for distributive dilemmas in networked systems, such as profit allocation in transportation, communications, and revenue sharing platforms.

1. Formal Model: Definitions and Graphical Abstractions

At the core, a Path Cooperative Game is specified by a finite, connected, directed (or undirected) graph $G = (E, \mathcal{E})$, with vertex set $E$ (which in the general formalism may represent arbitrary "cooperation states") and edge set $\mathcal{E} \subset E \times E$ indicating admissible transitions in the cooperative process. Every edge $(S, T) \in \mathcal{E}$ is assigned a strictly positive weight $\lambda(S, T) > 0$, and for non-edges, $\lambda(S, T) = 0$. The foundational value function is $v: E \rightarrow \mathbb{R}$ with $v(\emptyset) = 0$.

A prototypical PC-Game is constructed by specifying $N$ players and, for each player $i$, an antisymmetric edge-contribution function $f_i: \mathcal{E} \rightarrow \mathbb{R}$ with $f_i(T, S) = -f_i(S, T)$. In settings where $E = 2^{[N]}$ (the Boolean hypercube) and $\lambda \equiv 1$, this construction recovers the classical Shapley cooperative game.

Specializations of the model include:

• Edge-Path Cooperative Game (EPCG): Each edge is a player; $v(S)$ indicates the existence of an $s$–$t$ path in the subgraph induced by $S \subseteq E$.
• Vertex-Path Cooperative Game (VPCG): Vertices (besides $s, t$) are players; $v(S)$ tracks connectivity in the subgraph induced by $S$ plus terminals.

Cost-based generalizations associate each player with a nonnegative cost $c_i$, and define the characteristic function for a coalition $S$ as $r - \min\{\sum_{i \in T} c_i : T \subseteq S, \text{$T$enables an$s$–$t$path}\}$ (when $v(S) = 1$) (Aziz et al., 2011).

2. Value Allocation: Stochastic Path Integrals and Hodge Decomposition

The central value allocation mechanism in PC-Games is established by two fundamentally equivalent perspectives:

• Stochastic Path-Integral Formulation: By defining a time-reversible Markov chain over the cooperation state space $E$ (transition probabilities proportional to $\lambda(S, T)$), a player $i$'s value $V_i(S)$ is given by the expected sum of $f_i$-contributions along random paths from $\emptyset$ to $S$:

$$V_i(S) = \mathbb{E}\left[\sum_{n=1}^{\tau_{\emptyset, S}} f_i(X_{n-1}, X_n)\right]$$

where $X_n$ denotes the state at step $n$, and $\tau_{\emptyset, S}$ is the hitting time (Lim, 2021).

• Combinatorial Hodge Theory and Poisson Equation: Edge-flows $f_i$ decompose orthogonally as $f_i = d v_i + \eta_i$, where $d$ is the discrete gradient and $\eta_i \in \ker(d^*)$. The unique solution $v_i$ with $v_i(\emptyset) = 0$ satisfies the discrete Poisson equation $L v_i = d^* f_i$, with $L = d^* d$ the graph Laplacian. The equivalence $V_i(S) = v_i(S)$ forms the crux theorem, showing that the stochastic and Hodge-theoretic approaches coincide (Lim, 2021).

For the Boolean hypercube, these formulations reproduce the Shapley value, and beyond, they afford generalizations to arbitrary networks and state spaces (allowing edges that can both add and remove players, reflect technological upgrades, etc.).

3. Classical Solution Concepts: The Core, Least Core, and Nucleolus

Several key cooperative game-theoretic solution concepts are adapted to PC-Games:

• Core: The set of imputations $x \in \mathbb{R}^N$ satisfying efficiency ($x(N) = v(N)$) and coalition rationality ($x(S) \geq v(S)$ for all $S \subseteq N$). In standard PC-Games (e.g., $s$–$t$ path games), the core is typically empty unless there exists a single-arc cut (Fang et al., 2015, Aziz et al., 2011).
• Least Core: A polytope of imputations allowing a uniform relaxation parameter $\epsilon^*$, maximizing $\epsilon$ such that $x(S) \geq v(S) - \epsilon$ for all $S \subseteq N$ and $x(N) = v(N)$. For edge-path games, the least core is completely characterized: $\epsilon^* = 1 - 1/c_E$, where $c_E$ is the minimum $s$–$t$ cut size (Aziz et al., 2011).
• Nucleolus: The unique imputation minimizing the lexicographically ordered excess vector $e(S) = x(S) - v(S)$. For path games, the nucleolus can be computed in strongly polynomial time (notably in undirected series-parallel graphs, via structural decomposition) (Fang et al., 2015, Aziz et al., 2011).

The least core and nucleolus, in these games, have intimate duality relationships with classical network flow (primal-dual LP) theory and the packing/covering structure of $s$–$t$ paths and cuts.

4. Algorithmic Methods and Complexity

Solution computation in PC-Games leverages classic combinatorial algorithms:

• Separation Oracles and the Ellipsoid Method: For the least core and nucleolus, despite exponential constraint counts, polynomial-time algorithms are realized by separation via shortest-path (for minimal winning coalitions) and max-flow/min-cut (for path/cut dual games) subroutines. For cost-based games, shortest path or minimum cut computations on vertex/edge-weighted subgraphs suffice (Aziz et al., 2011).
• Series-Parallel Decomposition: For undirected series-parallel graphs, the nucleolus is computed inductively on the decomposition tree; series and parallel compositions yield explicit update rules for allocations based on cut sizes (Aziz et al., 2011).
• Probabilistic Estimation in Reliability Extensions: The Shapley value under reliability (random failure) extensions is intractable in exact form but admits efficient Monte Carlo estimation using randomized permutations, survivor sampling, and s–t-reachability checks. Error bounds are governed by concentration inequalities; sample complexity is $O(\epsilon^{-2} \ln(1/\delta))$ (Bachrach et al., 2012).

5. Reliability Extensions and Robustness

Reliability extensions introduce independent survival probabilities $r_i$ for players (edges/vertices). The value function becomes the probability that survivors in a coalition enable an $s$–$t$ path:

$v_{\text{rel}}(C) = \Pr\left\{\text{surviving edges in } C \text{ form an $s$–$t$ path}\right\}$

This modifies the structural properties of the game:

• The Shapley value can be estimated unbiasedly via random sampling.
• Convex base games retain a non-empty core under any survival profile.
• Veto agents and core stability are preserved, and, in some cases, reliability extensions can stabilize otherwise unstable base games and create non-empty cores (Bachrach et al., 2012).

Polynomial-time algorithms exist for core testing and allocation in games with a bounded number of types and survival probabilities.

6. Applications and Broader Implications

PC-Games and their solution methodologies are applicable to a diverse array of domains:

• Revenue management and cost allocation in networked services, where guaranteed connectivity or flow must be bought from distributed agents.
• Network reliability and robust profit sharing under component failures, modeling real-world infrastructure uncertainties (Bachrach et al., 2012).
• Distributed systems and platform design, such as networked marketplaces and communication networks, where fair resource utilization requires path-based cooperation (Lim, 2021).
• Generalization of Shapley value and Nash/Kohlberg–Neyman solutions, via the Hodge theoretic framework, to non-standard coalition spaces and dynamic reallocation scenarios.

These themes indicate PC-Games serve as an organizing model wherever connectivity, flow, or compliance emerge as collective, distributed, and stochastic phenomena across a network.

7. Illustrative Example: Glove Game and Beyond

As a specific case, consider the 3-player "glove" game: $N = 3$, where value accrues only if the left-glove owner (player 1) is present with at least one right-glove owner (players 2 or 3). On the unweighted hypercube, the standard path-integral or Hodge-theoretic approach yields Shapley values $v_1(123) = 1/2$, $v_2(123) = v_3(123) = 1/4$. Allowing for more general edge sets, such as multi-player jumps or non-monotone reversals, and edge-contributions $f_i$ that split incremental value among newcomers, the allocation framework remains consistent and "fair" through the unified path-integral / Poisson equation machinery (Lim, 2021).

These methods can be instantiated with general cooperation networks, arbitrary coalition spaces, and generalized value functions, positioning PC-Games as a foundational instrument for theoretical and applied research in cooperative multiagent networks.