Cost Share Equilibrium

• Cost Share Equilibrium is a robust allocation method where no coalition can rearrange payments to benefit every member, refining classical Nash equilibria.
• LP duality and integrality gap conditions are pivotal in efficiently computing strong equilibria in strategic cost sharing games.
• This concept guarantees optimal efficiency with a price of anarchy of 1, blending insights from cooperative game theory and combinatorial optimization.

A cost share equilibrium is a central concept in strategic cost sharing games, representing a robust allocation of costs among self-interested agents where no group of agents can jointly rearrange their payments to strictly improve the cost for every member. This notion generalizes and refines classical Nash equilibrium by requiring resilience to coordinated coalitional deviations, and serves as a bridge between cooperative game theory and the algorithmic paper of combinatorial optimization problems arising in cost sharing scenarios such as set cover, facility location, and network design (Hoefer, 2010).

1. Definition and Fundamental Properties

A cost share equilibrium in the context of strategic cost sharing games is formally defined as a strong equilibrium: a strategy profile (vector of contributions by all agents to the purchase of shared resources) in which no coalition of agents can jointly change their allocations in a way that strictly decreases every individual cost within the coalition. Formally, a state $s$ is a strong equilibrium (SE) if for every coalition $C$ and every alternative profile $s'_C$ for $C$, for all $i \in C$,

$c_i(s'_C, s_{-C}) \geq c_i(s)$

with strict inequality for at least one member if a deviation exists. SEs are thus robust to both individual and arbitrary group deviations, in contrast to Nash equilibria which privilege resistance only to individual deviations.

The key distinction between SE and NE is this robustness: while NE admits unilateral optimality (no single agent can improve), SE enforces no-benefit-to-all for every possible coalition, rendering the outcome resilient to collusion.

2. Relationship to the Core in Cooperative Game Theory

The existence and structure of cost share equilibria are intimately connected to the core—the principal solution concept in cooperative game theory for transferable utility cost games. The core comprises cost allocations in which no coalition can break away to achieve lower total payment for its members (given the subset's induced resource requirements and costs).

The paper establishes that any SE yields an allocation in the core: the payment vector in an SE covers exactly the costs, and for every coalition $C$, the total payment of $C$ is at most the minimum resource cost that $C$ could achieve if acting alone. Therefore, SE can be viewed as a "strategic refinement" of the core; the allocation must not only be undominated by any coalition but be robust in the explicit strategic game-theoretic sense.

This connection is particularly strong when the underlying combinatorial optimization (social cost minimization) problem has an integrality gap of one, as this guarantees both core non-emptiness and efficient computation of SE via LP dual solutions.

3. Combinatorial Optimization Game Structures

Cost share equilibrium has powerful implications for games built on combinatorial optimization frameworks. In the class of set cover, vertex cover, facility location, and network design games, the agents correspond to requests or elements, and the shared resources correspond to sets, facilities, or network components with assigned costs.

In such settings, social optima (minimum overall resource cost realizations covering all agents' requirements) play a foundational role. The existence and computation of SEs are fundamentally tied to the ability to share the cost of these optimal solutions among the agents in such a way that SE conditions hold. The modular structure of these problems facilitates reductions to linear and integer programming models, providing a route for rigorous analysis.

4. Linear Programming and Duality: Existence and Characterization

The crux of the existence theory for cost share equilibrium lies in LP duality. For games like set cover, the standard integer program (covering constraints, binary purchase variables) admits a linear programming relaxation:

$$\min \sum_{S \in \mathcal{S}} c(S) x_S \qquad \text{subject to}~ \sum_{S : e \in S} x_S \geq 1~\forall e,~ x_S \geq 0.$$

The dual, with variables $\gamma_e$ for elements, is:

$$\max \sum_{e} \gamma_e, \quad \sum_{e \in S} \gamma_e \leq c(S)~\forall S,~\gamma_e \geq 0.$$

The presence of an integral optimal solution to the LP (integrality gap 1) implies that complementary slackness produces dual variables (interpretable as cost shares) satisfying both resource purchase and robustness constraints required by SE. The dual variables form an SE payment vector whose structure matches that of a core allocation.

Therefore, the integrality gap acts as a necessary and sufficient condition for the existence of strong equilibria in many canonical cost sharing games. When this gap exceeds 1, SE may not exist, but approximate SEs (with relaxed efficiency and stability constraints) can still be constructed.

5. Efficiency: Price of Anarchy and Implications

A highlighted result is that the strong price of anarchy (PoA)—the ratio between the cost in the worst SE and the true social optimum—is always 1 in these strategic cost sharing games. This follows because any coalition can, if a suboptimal resource set is purchased, switch to the optimal solution and re-split the costs for universal improvement, contradicting SE. Therefore, every SE induces a purchase of an optimal set of resources.

This is in contrast to the price of anarchy for Nash equilibrium, which can be as large as $\Theta(n)$, especially when resources can be overbought or inefficiently coordinated. Thus, SE delivers maximal efficiency among all stable arrangements, tightly aligning individual and collective incentives.

Approximate strong equilibria (($\alpha, \beta$)-SE), where the cost for every player is at most $\alpha$ times the cost under a deviation and purchased resources cost at most $\beta$ times the optimum, broaden the set of computable equilibria; practical algorithms for set cover, facility location, and network design yield $(f, f)$- or $(2,2)$-SEs, where $f$ is the maximum frequency of an element over sets.

6. Algorithmic Computation and Applications

Computation of cost share equilibria (when they exist) can be effected efficiently (i.e., in polynomial time) in all settings where the LP formulation has an integrality gap of 1. The process involves solving the LP, extracting dual variables, and constructing payment vectors that satisfy SE conditions via complementary slackness. In cases where only approximate SEs are achievable, primal–dual and combinatorial approximation algorithms provide payments that guarantee near-optimality and “near-stability”.

Cost share equilibrium concepts have direct applicability to numerous real-world systems: network design and maintenance, collaborative facility placement, sensor coverage, and decentralized resource investment problems. The robust efficiency property of SE provides practical assurance of both fairness (no group is overcharged relative to their needs) and global cost-optimality in decentralized environments.

7. Summary and Significance

The cost share equilibrium, as formalized via strong equilibrium in strategic cost sharing games (Hoefer, 2010), operationalizes the convergence of fairness (core allocations), strategic robustness (resilience to arbitrary coalition deviation), and algorithmic tractability (via LP duality). Its existence, tight connection to the core, and maximum efficiency property (PoA = 1) position it as a canonical solution in both theory and practice. Furthermore, the development of approximate strong equilibria extends these desirable features to settings where integral solutions are not available, allowing for widespread deployment in resource-sharing and combinatorial optimization-based applications.