Computing Nash Equilibria in Nonconvex Games via Outer Approximations

Core Concepts

The Cut-and-Play algorithm computes Nash equilibria in simultaneous non-cooperative games where players decide via nonconvex and possibly unbounded optimization problems with separable payoff functions.

Abstract

The paper introduces the Cut-and-Play (CnP) algorithm, a practically-efficient method for computing equilibria in non-cooperative games where players solve nonconvex optimization problems. The key ideas are: The authors prove that when the player's objective functions are separable, the game admits an equivalent convex representation, even if the players' feasible sets are inherently nonconvex. This allows them to exploit a sequence of polyhedral convexifications of the original nonconvex game. CnP blends concepts such as relaxation (approximation), cutting planes, branching, and complementarity problems to find an exact Nash equilibrium (up to a machine epsilon) or prove its non-existence. It does not require the players' optimization problems to be convex or continuous, and can handle unbounded strategy sets. The authors present computational results on two families of challenging nonconvex games: Integer Programming Games and Nash games among Stackelberg Players. CnP outperforms existing game-specific algorithms in terms of computing times and social welfare.

Stats

The game is formulated as a Separable-Payoff Game (SPG), where each player i solves a minimization problem of the form: min_{x_i} {f^i(x_i; x_{-i}) := c_i^T x_i + sum_{j=1}^{m_i} g_j^i(x_{-i}) x_i_j} subject to x_i in X_i.

Quotes

"We introduce Cut-and-Play, a practically-efficient algorithm for computing Nash equilibria in simultaneous non-cooperative games where players decide via nonconvex and possibly unbounded optimization problems with separable payoff functions." "CnP overcomes some well-known algorithmic limitations. Specifically, our algorithm does not: (a.) require that the players' optimization problems are convex or continuous (b.) compute only pure (i.e., deterministic) equilibria (c.) rely on alternative or weaker concepts of equilibria."

Key Insights Distilled From

The Cut-and-Play Algorithm: Computing Nash Equilibria via Outer Approximations

by Margarida Ca... at arxiv.org 05-06-2024

https://arxiv.org/pdf/2111.05726.pdf

Deeper Inquiries

How can the Cut-and-Play algorithm be extended to handle games with non-separable payoff functions?

The Cut-and-Play algorithm can be extended to handle games with non-separable payoff functions by incorporating techniques to deal with the complexity introduced by such functions. One approach could involve reformulating the non-separable payoff functions into a form that allows for convex approximations. This reformulation may involve decomposing the non-separable functions into simpler components or applying approximation methods to represent them in a more tractable manner. Additionally, the algorithm can be adapted to handle non-separable payoff functions by incorporating more sophisticated optimization techniques that can handle non-convex and non-separable functions. This may involve using advanced optimization solvers that can handle non-convex optimization problems efficiently. Furthermore, the algorithm can be enhanced to iteratively refine the approximations of the non-separable payoff functions, similar to how it refines the convex approximations in the current version of the algorithm. By iteratively improving the approximations of the non-separable payoff functions, the algorithm can converge towards an equilibrium solution even in the presence of complex payoff structures.

What are the limitations of the computational convexifiability assumption, and how can it be relaxed in practice?

The computational convexifiability assumption imposes restrictions on the computational tractability of obtaining convex approximations of the feasible sets in the Cut-and-Play algorithm. One limitation is that not all sets may be easily computationally convexifiable, especially in cases where the sets are highly complex or have intricate geometries. In such scenarios, the algorithm may struggle to converge or may require a large number of refinements to reach a satisfactory approximation. To relax the computational convexifiability assumption in practice, one approach is to incorporate more advanced approximation techniques that can handle a wider range of set geometries. This may involve using more sophisticated cutting plane methods or branching strategies that can effectively approximate non-convex sets. Another approach is to leverage machine learning or artificial intelligence algorithms to assist in the approximation process. These techniques can learn patterns from the data and guide the refinement of the approximations in a more efficient and effective manner, even for sets that are not easily computationally convexifiable. Furthermore, exploring hybrid methods that combine convex optimization with non-convex optimization techniques can help in relaxing the computational convexifiability assumption. By integrating different optimization approaches, the algorithm can adapt to a broader range of set complexities and improve its convergence properties.

What are the potential applications of the Cut-and-Play algorithm beyond the two families of nonconvex games presented in the paper?

The Cut-and-Play algorithm has the potential for various applications beyond the specific families of nonconvex games discussed in the paper. Some potential applications include: Economic Modeling: The algorithm can be applied to model and analyze complex economic systems where decision-makers interact in non-cooperative environments. This can be useful in understanding market dynamics, pricing strategies, and competition scenarios. Supply Chain Optimization: Cut-and-Play can be utilized in optimizing supply chain networks involving multiple stakeholders with conflicting objectives. By computing equilibria in supply chain games, the algorithm can help in decision-making processes and resource allocation. Network Routing and Traffic Management: The algorithm can be employed in optimizing traffic flow, routing decisions, and resource allocation in communication networks, transportation systems, and logistics networks. By computing equilibria in network games, it can enhance efficiency and performance. Environmental Resource Management: Cut-and-Play can be used in modeling and optimizing environmental resource allocation games, such as water distribution, energy management, and pollution control. By computing equilibria, the algorithm can aid in sustainable resource utilization and conservation efforts. Healthcare Decision-Making: The algorithm can be applied in healthcare systems to model strategic interactions among healthcare providers, insurers, and patients. By computing equilibria in healthcare games, it can assist in optimizing resource allocation, treatment strategies, and healthcare policies. Overall, the Cut-and-Play algorithm's versatility and ability to handle complex nonconvex games make it applicable in various domains where decision-making involves strategic interactions and optimization challenges.