Bilevel programming methods for computing single-leader-multi-follower equilibria in normal-form and polymatrix games

Basilico, Nicola; Coniglio, Stefano; Gatti, Nicola; Marchesi, Alberto

doi:10.1007/s13675-019-00114-8

Cited by 17 publications

(18 citation statements)

References 29 publications

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“…One challenge identified with the present work was tractability and computational cost of estimating the equilibrium of the game, based on empirical data. Therefore, future work will focus on finding more efficient ways to compute the equilibrium [24], including finding an efficient algorithm for game equilibrium estimation, investigation of machine learning techniques to approximate the players' payoff functions and reduction of their strategy space. He has published over 100 papers in top-ranked journals, conferences and edited volumes, in both the areas of Artificial Intelligence and Electrical Engineering.…”

Section: Discussionmentioning

confidence: 99%

“…(1) can be challenging. Moreover, as stated by Basilico et al [24], leader-follower games with multiple followers have not been extensively investigated in the literature, and in addition not many computationally affordable techniques are available for the analysis task. A well-known technique for solving the problem is based on backward induction, i.e., a solution is first derived for the second-level problem by taking as a given the strategy of the leader (process is repeated for all possible σ 1 ), then the leader selects the strategy that maximises her payoff/profit.…”

Section: B Stackelberg-cournot Game Equilibrium Solutionmentioning

confidence: 99%

“…-Second, we develop an empirical solution for equilibrium finding where players' payoffs are directly computed from large-scale datasets of historical observations and simulations that realistically represent operation and control in energy systems. This is an important contribution as according to the literature survey, very few works deal with techniques for solving bi-level optimisation and single-leader-multifollower games [24].…”

Section: Introductionmentioning

confidence: 99%

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Analysis of Strategic Renewable Energy, Grid and Storage Capacity Investments via Stackelberg-Cournot Modelling

et al. 2021

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With increasing decarbonisation and accessibility to our energy systems and markets, there is a need to understand and optimise the value proposition for different stakeholders. Game-theoretic models represent a promising approach to study strategic interactions between self-interested private energy system investors. In this work, we design and evaluate a game-theoretic framework to study strategic interactions between profit-maximising players that invest in network, renewable generation and storage capacity. Specifically, we study the case where grid capacity is developed by a private renewable investor, but line access is shared with competing renewable and storage investors, thus enabling them to export energy and access electricity demand. We model the problem of deducing how much capacity each player should build as a non-cooperative Stackelberg-Cournot game between a dominant player (leader) who builds the power line and renewable generation capacity, and local renewable and storage investors (multiple followers), who react to the installation of the line by increasing their own capacity. Using data-driven analysis and simulations, we developed an empirical search method for estimating the game equilibrium, where the payoffs capture the realistic operation and control of the energy system under study. A practical demonstration of the underlying methodology is shown for a real-world grid reinforcement project in the UK. The methodology provides a realistic mechanism to analyse investor decision-making and investigate feasible tariffs that encourage distributed renewable investment, with sharing of grid access.

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Section: Discussionmentioning

confidence: 99%

Section: B Stackelberg-cournot Game Equilibrium Solutionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Analysis of Strategic Renewable Energy, Grid and Storage Capacity Investments via Stackelberg-Cournot Modelling

et al. 2021

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“…Solving Stackelberg games with multiple followers is challenging in general [5,11]. Previous work often reformulates the followers' best response as stationary and complementarity constraints in the leader's optimization [5,6,9,11,40], casting the entire Stackelberg problem as a single optimization problem. This reformulation approach has achieved significant success in problems with linear or quadratic objectives, assuming a unique equilibrium or a specific equilibrium concept, e.g., followers' optimistic or pessimistic choice of equilibrium [5,6,18].…”

Section: Introductionmentioning

confidence: 99%

Coordinating Followers to Reach Better Equilibria: End-to-End Gradient Descent for Stackelberg Games

Wang¹,

Xu²,

Perrault³

et al. 2021

Preprint

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A growing body of work in game theory extends the traditional Stackelberg game to settings with one leader and multiple followers who play a Nash equilibrium. Standard approaches for computing equilibria in these games reformulate the followers' best response as constraints in the leader's optimization problem. These reformulation approaches can sometimes be effective, but often get trapped in low-quality solutions when followers' objectives are non-linear or non-quadratic. Moreover, these approaches assume a unique equilibrium or a specific equilibrium concept, e.g., optimistic or pessimistic, which is a limiting assumption in many situations. To overcome these limitations, we propose a stochastic gradient descentbased approach, where the leader's strategy is updated by differentiating through the followers' best responses. We frame the leader's optimization as a learning problem against followers' equilibrium, which allows us to decouple the followers' equilibrium constraints from the leader's problem. This approach also addresses cases with multiple equilibria and arbitrary equilibrium selection procedures by back-propagating through a sampled Nash equilibrium. To this end, this paper introduces a novel concept called equilibrium flow to formally characterize the set of equilibrium selection processes where the gradient with respect to a sampled equilibrium is an unbiased estimate of the true gradient. We evaluate our approach experimentally against existing baselines in three Stackelberg problems with multiple followers and find that in each case, our approach is able to achieve higher utility for the leader.Preprint. Under review.

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“…While, with only a few exceptions (see [6,8,13,18,21]), the majority of the gametheoretical investigations on the computation of SEs assumes the presence of a single follower, in this work we address the multi-follower case.…”

Section: Introductionmentioning

confidence: 99%

Computing a Pessimistic Stackelberg Equilibrium with Multiple Followers: The Mixed-Pure Case

2019

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The search problem of computing a Stackelberg (or leader-follower) equilibrium (also referred to as an optimal strategy to commit to) has been widely investigated in the scientific literature in, almost exclusively, the single-follower setting. Although the optimistic and pessimistic versions of the problem, i.e., those where the single follower breaks any ties among multiple equilibria either in favour or against the leader, are solved with different methodologies, both cases allow for efficient, polynomialtime algorithms based on linear programming. The situation is different with multiple followers, where results are only sporadic and depend strictly on the nature of the followers' game. In this paper, we investigate the setting of a normal-form game with a single leader and multiple followers who, after observing the leader's commitment, play a Nash equilibrium. When both leader and followers are allowed to play mixed strategies, the corresponding search problem, both in the optimistic and pessimistic versions, is known to be inapproximable in polynomial time to within any multiplicative polynomial factor unless P = NP. Exact algorithms are known only for the optimistic case. We focus on the case where the followers play pure strategies-a restriction that applies to a number of real-world scenarios and which, in principle, makes the problem easier-under the assumption of pessimism (the optimistic version of the problem can be straightforwardly solved in polynomial time). After casting this search problem (with followers playing pure strategies) as a pessimistic bilevel programming problem, we show that, with two followers, the problem is NP-hard and, with three or more followers, it cannot be approximated in polynomial time to within any multiplicative factor which is polynomial in the size of the normal-form game, nor, assuming utilities in [0, 1], to within any constant additive loss stricly smaller than 1 unless P = NP. This shows that, differently from what happens in the optimistic version, hardness and inapproximability in the pessimistic problem are not due to the adoption of mixed strategies. We then show that the problem admits, in the general case, a supremum but not a maximum, and we propose a single-level mathematical programming reformulation which asks for the maximization of a nonconcave All the authors contributed equally to this manuscript. Extended author information available on the last page of the article quadratic constraints. Since, due to admitting a supremum but not a maximum, only a restricted version of this formulation can be solved to optimality with state-of-theart methods, we propose an exact ad hoc algorithm (which we also embed within a branch-and-bound scheme) capable of computing the supremum of the problem and, for cases where there is no leader's strategy where such value is attained, also an α-approximate strategy where α > 0 is an arbitrary additive loss (at most as large as the supremum). We conclude the paper by evaluating the scalability of our algorithms via computat...

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Bilevel programming methods for computing single-leader-multi-follower equilibria in normal-form and polymatrix games

Cited by 17 publications

References 29 publications

Analysis of Strategic Renewable Energy, Grid and Storage Capacity Investments via Stackelberg-Cournot Modelling

Analysis of Strategic Renewable Energy, Grid and Storage Capacity Investments via Stackelberg-Cournot Modelling

Coordinating Followers to Reach Better Equilibria: End-to-End Gradient Descent for Stackelberg Games

Computing a Pessimistic Stackelberg Equilibrium with Multiple Followers: The Mixed-Pure Case

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