Scenario-Game ADMM: A Parallelized Scenario-Based Solver for Stochastic Noncooperative Games

arXiv:2304.01945 · eess.SY, cs.SY · Submitted 2023-04-04 · Read on arXiv

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Introduction to the show: ident: Robotics Radio. Generated commentary on the latest robotics and control papers.

Rosa: I'm Rosa, and with me are Dev and Taro, guest researcher.

Dev: Today's paper: "Scenario-Game ADMM: A Parallelized Scenario-Based Solver for Stochastic Noncooperative Games".

Rosa: The gist: This work proposes a new sample-based approximation to solve stochastic, general-sum,

Dev: First, who's behind it and why it matters.

Title and authors: Rosa: So, we just looked at the setup for this new paper by Li and colleagues—it’s about taking these tough stochastic games, where players have uncertain goals and chance constraints, and they’re proposing a way to solve them using sampling and a decentralized algorithm.

Dev: Right. Basically, they're approximating the original problem by creating a bunch of different scenarios or samples theta j. Instead of dealing with the whole mess at once, you break it down into smaller games for each sample and then use this decentralized ADMM thing to find a solution that works across all those samples.

Rosa: It’s smart because it uses the idea from standard sample average approximation to get an accurate expected value, but they've added a specific way to make sure the chance constraints actually hold true for those sampled scenarios with high probability <ref:2304.01945#pg1>.

Dev: That feasibility part is key. They show that if you take enough samples, you can guarantee that your solution respects those original constraints in a statistically reliable way, which is something you don't get from just solving one static version of the game.

Rosa: The big technical takeaway here is how they handle the complexity; because they break the game into smaller games with fewer constraints per sample, it makes the whole process scalable and computationally manageable <ref:2304.01945#pg2>.

Dev: Computationally, that’s good for us. It means we can tackle problems with a huge number of coupled inequality constraints—like thirty-five thousand of them—that other methods just couldn't even compile or run on because they got overwhelmed by the scale <ref:2304.01945#pg3>.

Rosa: And their validation experiments show this approach is faster than some state-of-the-art baseline algorithms for solving those problems, especially when you use a parallel version of the method <ref:2304.01945#pg3>.

Dev: Faster is good for latency and loop rates, Rosa. We’re talking about real time decision-making in complex systems where we need results quickly <ref:2304.01945#pg3>.

Rosa: And they proved it converges to the generalized Nash equilibrium—that it actually finds a stable solution to the game—by showing a specific Lyapunov function keeps dropping with every iteration of their algorithm <ref:2304.01945#pg3>.

Dev: That convergence proof is solid, but I always look at the assumptions they made, like that requirement for strong monotonicity on their operator F(w), because real-world dynamics can be much messier than what's assumed in those theorems <ref:2304.01945#pg3>.

Rosa: Exactly. So this paper gives us a robust framework for handling large-scale uncertainty in game theory, but it also sets up the next challenge: figuring out exactly how good the solution from their approximation is when you go back to the original, unsimplified problem <ref:2304.01945#pg1>.

Dev: The next step we need to watch is how they extend that sample complexity analysis; we need to know precisely how many samples are actually required before this method becomes too slow or inaccurate for our specific applications <ref:2304.01945#pg2>.

The paper's summary: Rosa: We just talked about how they break down games into scenarios and use ADMM to solve them in parallel—now we’re looking at what they say is actually better about this approach compared to older methods.

Dev: They highlight that the main improvement is handling those huge constraint sets effectively; they show this method can handle problems with thirty-five thousand coupled inequality constraints where previous tools just wouldn't compile or run <ref:2304.01945#pg3>.

Rosa: It’s like they found a way to manage massive amounts of data and constraints without having to solve one gigantic, impossibly complex system all at once <ref:2304.01945#pg3>.

Dev: Yeah, that’s because they decompose the problem into smaller games with fewer constraints for each sample, which makes it computationally feasible on a real machine <ref:2304.01945#pg2>.

Rosa: The feasibility guarantee is another big point; they show that by sampling enough scenarios, you actually get a reliable chance that your solution meets the original physical constraints <ref:2304.01945#pg1>.

Dev: That moves it from just an approximation to something with some level of statistical confidence, which is important for deployment where we need reliability <ref:2304.01945#pg2>.

Rosa: And they’re showing speed improvements too; their parallelized version runs faster than a centralized solver when you use a moderate number of scenarios, like ten to one hundred <ref:2304.01945#pg3>.

Dev: Speed matters for latency and real-time control loops, so being twice as fast over that range is pretty significant for keeping our systems responsive <ref:2304.01945#pg3>.

Rosa: But they do have some caveats; they state that the accuracy depends heavily on the sample size, and if you need high accuracy, you’re going to need a lot more samples, which means more constraints to manage <ref:2304.01945#pg1>.

Dev: That means there's a trade-off: you get better results for a larger system, but the computational cost of generating those samples and running ADMM scales with that size <ref:2304.01945#pg2>.

Rosa: So they’re offering a method that is scalable in terms of constraints and reasonably fast in terms of computation, as long as you can tolerate the sample size trade-off <ref:2304.01945#pg3>.

Dev: The implication for us is that we might be able to tackle more realistic, complex coordination problems than we could with the older ADMM and interior point methods because this structure allows it to scale <ref:2304.01945#pg2>.

Rosa: What they’re setting up for next is analyzing those convergence proofs under different conditions—specifically, how well the equilibrium of this simplified scenario game actually matches what you get from solving the original, full chance-constrained stochastic game <ref:2304.01945#pg1>.

The paper's improvements: Rosa: So we’ve covered how this paper by Li and colleagues proposes using scenario sampling to approximate complex games and then uses a parallel ADMM method to solve them efficiently while keeping those chance constraints in check across many scenarios.

Dev: That’s right, it’s about breaking down hard stochastic problems into smaller, manageable pieces that can be solved simultaneously across multiple samples <ref:2304.01945#pg3>.

Taro: I think the biggest takeaway for autonomy researchers is seeing how robust this system is when things go wrong; it gives us a statistical way to be confident that our AI decisions won't violate safety constraints under uncertainty.

Rosa: It certainly offers a more reliable path forward than just relying on one single, perfect solution to the whole complex game at once <ref:2304.01945#pg1>.

Dev: Yeah, and the convergence proof they established means we know that if we run this algorithm long enough, it will settle down to a stable result for us <ref:2304.01945#pg3>.

Taro: It’s interesting how they are framing the future work around checking how well this approximate solution actually matches what you would get from solving the original chance-constrained stochastic game <ref:2304.01945#pg1>.

Rosa: That’s the next big question—does this approximation actually capture the real behavior of a complex system under those tough conditions <ref:2304.01945#pg1>.

Dev: We’ll need to see if that gap between the approximate game and the original problem is small enough for real-world control systems <ref:2304.01945#pg3>.

Taro: I think it opens up a new avenue for designing autonomous agents in uncertain environments, provided we can manage that sample size trade-off effectively <ref:2304.01945#pg1>.

Rosa: Alright, so on a final note, this paper "Scenario-Game ADMM: A Parallelized Scenario-Based Solver for Stochastic Noncooperative Games" shows a promising path toward solving these high-dimensional games with constraints using scalable decentralized optimization <ref:2304.01945#pg1>.

Conclusion: Dev: So, we've covered how this paper by Li and colleagues proposes using scenario sampling to approximate complex games and then uses a parallel ADMM method to solve them efficiently while keeping those chance constraints in check across many scenarios.

Taro: That’s right, it’s about breaking down hard stochastic problems into smaller, manageable pieces that can be solved simultaneously across multiple samples <ref:2304.01945#pg3>.

Rosa: It means we have a way to tackle those massive constraint sets in coordination problems that were previously too big for our standard solvers to handle at all <ref:2304.01945#pg3>.

Dev: Exactly, and the speed gains are real when you’re dealing with loop rates and latency issues because this decomposition lets us use parallelism effectively <ref:2304.01945#pg3>.

Rosa: It certainly offers a more reliable path forward than just relying on one single, perfect solution to the whole complex game at once <ref:2304.01945#pg1>.

Dev: Yeah, and the convergence proof they established means we know that if we run this algorithm long enough, it will settle down to a stable result for us <ref:2304.01945#pg3>.

Rosa: That’s the next big question—does this approximation actually capture the real behavior of a complex system under those tough conditions <ref:2304.01945#pg1>.

Dev: We’ll need to see if that gap between the approximate game and the original problem is small enough for real-world control systems <ref:2304.01945#pg3>.

Taro: I think it opens up a new avenue for designing autonomous agents in uncertain environments, provided we can manage that sample size trade-off effectively <ref:2304.01945#pg1>.

University of California, Berkeley · Delft University of Technology · University of Texas, Austin

eess.SY, cs.SY

Submitted: 2023-04-04

Updated: 2026-10-09

Comments: We corrected a typo and clarified an implicit assumption in Lemma 5. The overall proof strategy remains unchanged and is adapted from the pioneering work on analyzing ADMM convergence using integral quadratic constraints (IQCs) [Nishihara et al., 2015]

License: http://creativecommons.org/licenses/by-nc-nd/4.0/

Importance score: 88/100

The gist: The gist: This work proposes a new sample-based approximation to solve stochastic, general-sum, pure Nash games with expected-value objectives and chance constraints by decomposing the game into

Key concepts

Sample-Based Approximation (SAA): This technique replaces the complex expected value in a stochastic game with an average calculated from a set of sampled scenarios. This turns the original hard problem into a deterministic one that is easier to solve, while still maintaining accuracy related to the true expectation.
Chance Constraints: These are constraints that must be satisfied only with high probability, rather than being certain. The paper deals with games where players must obey these probabilistic limitations on their actions within the uncertain environment.
Scenario-Game ADMM (SG-ADMM): This is a decentralized optimization algorithm designed to find the Nash equilibrium of the approximated game. It works by solving several smaller, scenario-specific games in parallel and then using consensus steps via ADMM to synchronize solutions across all scenarios efficiently.

Terminology

Accuracy — 2× in this paper · explained in 1 episode
Nash Equilibrium — 1× in this paper · explained in 2 episodes
Sample Complexity — 1× in this paper · explained in 20 episodes

Summary

The gist: This work proposes a new sample-based approximation to solve stochastic, general-sum, pure Nash games with expected-value objectives and chance constraints by decomposing the game into smaller games and employing a decentralized ADMM algorithm.

Problem Formulation

The paper addresses constrained stochastic general-sum games where each player seeks to minimize their expected objective subject to chance constraints on the original problem (2a)-(2b). The authors approximate this stochastic game by constructing a deterministic problem (3a)-(3b) using a set of sampled scenarios θj, replacing the expected value with its empirical mean and enforcing the original constraint for all sampled scenarios. This approximation is valid under Assumption 1, which requires the constraint function hi(x; θ) to be convex in x and satisfy Slater’s condition.

Sample Complexity Analysis

The sample complexity of this scenario-game approximation scheme is characterized through Proposition 1. This proposition states that for a sufficiently large sample size S, the following statements hold simultaneously for each player i ∈ [N]: 1) the objective approximation error satisfies 1/S PS j=1 fi(x; θj)−Eθ[fi(x; θ)] ≤ ε˜, for all x ∈ HS; and 2) the chance constraint holds with high probability, Pθ(hi(x; θ) ≤ 0) ≥ 1 − ϵ, for all x ∈ HS with probability at least 1 − δ.

Scenario-Game ADMM Algorithm

To compute the generalized Nash equilibrium (GNE) of the approximated game (3), the authors propose a decentralized ADMM algorithm called Scenario-Game ADMM (SG-ADMM). The algorithm involves several key steps outlined in Algorithm 1: 1. For each scenario j = 1 to S in parallel, solve for auxiliary variables wj i(k + 1) by minimizing Lj i(w j, x(k), λi(k)) subject to hi(w j; θj) ≤ 0. 2. Update the decision variables xi(k + 1) for each player i by taking a weighted average of the auxiliary variables and Lagrange multipliers. 3. Update the dual variables λj i(k + 1) using the vanishing gradient condition derived from Algorithm 1.

Convergence Proofs

The convergence of Algorithm 1 to a GNE is established by showing that a Lyapunov function V(k) defined as (1/ρ)∥λ(k)−λ∗∥2 + ρ∥M(x(k) − x∗)∥2 monotonically decreases with each iteration. Theorem 1 proves this asymptotic convergence, showing that limk→∞ V (k) = 0. Furthermore, under Assumption 2, which requires the objective function fi(x; θ) to be differentiable and the operator F(w) to be L-Lipschitz continuous and m-strongly monotone, a stronger convergence rate is established in Theorem 2.

Empirical Validation

Empirical results demonstrate that the SG-ADMM method can handle a large number of constraints with faster convergence than a state-of-the-art baseline algorithm based on ADMM and interior point method. Specifically, in experiments with 1000 sampled scenarios, the ScenarioGame ADMM converges, whereas the ACVI fails to compile due to the scale of having 35000 coupled inequality constraints. The parallelized version of the method is observed to be more than 2x faster than a centralized PATH solver for scenario sizes S ∈ [10, 100]. This suggests that Scenario-Game ADMM can solve game problems with a large number of constraints within a reasonable amount of time.

Conclusion and Future Work

The paper introduces a new sample-based approximation for stochastic games, characterizes its sample complexity and feasibility guarantees, and proposes an efficient decentralized ADMM solver. Future work should focus on extending the results on sample complexity and analyzing how well equilibria of the scenario game approximate solutions to the original chance-constrained stochastic game.

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Scenario-Game ADMM: A Parallelized Scenario-Based Solver for Stochastic Noncooperative Games

Jingqi Li1, Chih-Yuan Chiu1, Lasse Peters2, Fernando Palafox∗3, Mustafa Karabag∗3, Javier Alonso-Mora2, Somayeh Sojoudi1, Claire Tomlin1, and David Fridovich-Keil3

Abstract— Decision-making in multi-player games can be extremely challenging, particularly under uncertainty. In this work, we propose a new sample-based approximation to a class of stochastic, general-sum, pure Nash games, where each player has an expected-value objective and a set of chance constraints. This new approximation scheme inherits the accuracy of objective approximation from the established sample average approximation (SAA) method and enjoys a feasibility guarantee derived from the scenario optimization literature. We characterize the sample complexity of this new game-theoretic approximation scheme, and observe that high accuracy usually requires a large number of samples, which results in a large number of sampled constraints. To accommodate this, we decompose the approximated game into a set of smaller games with few constraints for each sampled scenario, and propose a decentralized, consensus-based ADMM algorithm to efficiently compute a generalized Nash equilibrium (GNE) of the approximated game. We prove the convergence of our algorithm to a GNE and empirically demonstrate superior performance relative to a recent baseline algorithm based on ADMM and interior point method.

I. INTRODUCTION

Stochastic game theory [1] provides a principled mathematical foundation for modeling interactions between multiple self-interested players in uncertain environments, and has applications in traffic control [2], multi-robot coordination [3], and human-robot interaction [4]. In this framework, each player selects actions to optimize their own objective, obey a set of constraints, and reason about the strategic response of other players. Uncertainty in the players’ potentially conflicting objectives and coupled constraints makes these problems extremely challenging to solve.

Classical results in stochastic games are often derived under strong assumptions regarding the problem structure and the distribution of the underlying random process. In the class of linear quadratic Gaussian games, necessary and sufficient conditions for the existence of Nash equilibria are characterized in [5]. It is also shown that in N-player noncooperative stochastic games, the convexity of player-specific objectives and convex, compact strategy sets are sufficient for the existence of the Nash equilibria [6]. However, for general stochastic games, it is NP-hard to determine the existence of Nash equilibria [7]. Moreover, computing a Nash equilibrium can also be a hard problem [8], partially due to the complexity of solving the nonlinear equations induced by the Nash equilibrium condition.

∗Equal contribution. The authors are with: 1University of California, Berkeley, 2Delft University of Technology, 3University of Texas, Austin.

Correspondence to jingqili@berkeley.edu.

II. RELATED WORK

A. Stochastic Games

Originally due to Shapley [1], the field of stochastic game theory has expanded to model uncertainties in players’ objectives [16], constraints [17], and in the case of dynamic games, underlying state dynamics [18–20]. Exact generalized Nash equilibrium solutions to stochastic constrained games can be obtained by solving their equivalent stochastic variational inequality problems [11, 21]. Under an appropriate constraint qualification, the well-known Karush–Kuhn–Tucker (KKT) conditions must be satisfied for all players at a generalized Nash equilibrium [21, 22]. We focus on games with monotone objective pseudogradients [23] and convex constraints, where solutions can be found in polynomial time [24].

B. ADMM for Games

We are ultimately interested in decentralized methods [25, 26] for identifying generalized Nash equilibria, because they can often exploit computational parallelism for efficiency gains compared with centralized method. In particular, the alternating direction method of multipliers (ADMM) [27, 28] is an appealing approach for efficient decentralized computation. The ADMM enjoys convergence guarantees for convex problems [29, 30], convex-concave saddle point problems [31, 32] and monotone variational inequality problems [33, 34]. Recent work [34] has adopted an interior-point method to ensure constraint feasibility, thereby outperforming projection-based consensus ADMM methods [35–37]. Our algorithm differs from prior work [28–40] in that we decompose the objective and constraints over scenarios. For each scenario, we solve an N-player game with relatively few constraints, and then synchronize across scenarios via ADMM. Unlike prior methods, we do not require constraint projection or an interior-point method in the consensus step. Moreover, we can handle nonlinear coupled constraints, while prior works [23, 41] consider affine constraints.

Improvements for AI systems

The proposed Scenario-Game ADMM (SG-ADMM) can solve large-scale decentralized constrained optimization problems by decomposing them into smaller games with few constraints per scenario, enabling parallel—and hence more efficient—computation of equilibrium solutions. This allows AI systems to efficiently compute a generalized Nash equilibrium (GNE) for complex stochastic games, such as those arising in multi-robot coordination or traffic control.
The method provides a mechanism to handle large numbers of constraints effectively, as the authors show that the algorithm converges even when replacing linear dynamics with nonlinear unicycle dynamics, and can solve problems with 35000 coupled inequality constraints in total where prior methods like ACVI fails to compile due to the scale of problem.
The system can achieve high-probability feasibility for chance constraints by ensuring that the set of sampled scenarios, HS, satisfies conditions derived from Proposition 1, which guarantees P(hi(x; θ) ≤ 0) ≥ 1 − ϵ with high probability based on the sample size S.

Abstract

Decision-making in multi-player games can be extremely challenging, particularly under uncertainty. In this work, we propose a new sample-based approximation to a class of stochastic, general-sum, pure Nash games, where each player has an expected-value objective and a set of chance constraints. This new approximation scheme inherits the accuracy of objective approximation from the established sample average approximation (SAA) method and enjoys a feasibility guarantee derived from the scenario optimization literature. We characterize the sample complexity of this new game-theoretic approximation scheme, and observe that high accuracy usually requires a large number of samples, which results in a large number of sampled constraints. To accommodate this, we decompose the approximated game into a set of smaller games with few constraints for each sampled scenario, and propose a decentralized, consensus-based ADMM algorithm to efficiently compute a generalized Nash equilibrium (GNE) of the approximated game. We prove the convergence of our algorithm to a GNE and empirically demonstrate superior performance relative to a recent baseline algorithm based on ADMM and interior point method.

Scenario-Game ADMM: A Parallelized Scenario-Based Solver for Stochastic Noncooperative Games

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