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Register a new userMethods for finding leader--follower equilibria with multiple followers
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Nicola Gatti
Publication date
2017
fields
Informatics Engineering
and research's language is
English
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The concept of leader--follower (or Stackelberg) equilibrium plays a central role in a number of real--world applications of game theory. While the case with a single follower has been thoroughly investigated, results with multiple followers are only sporadic and the problem of designing and evaluating computationally tractable equilibrium-finding algorithms is still largely open. In this work, we focus on the fundamental case where multiple followers play a Nash equilibrium once the leader has committed to a strategy---as we illustrate, the corresponding equilibrium finding problem can be easily shown to be $mathcal{FNP}$--hard and not in Poly--$mathcal{APX}$ unless $mathcal{P} = mathcal{NP}$ and therefore it is one among the hardest problems to solve and approximate. We propose nonconvex mathematical programming formulations and global optimization methods to find both exact and approximate equilibria, as well as a heuristic black box algorithm. All the methods and formulations that we introduce are thoroughly evaluated computationally.
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The search problem of computing a textit{leader-follower equilibrium} has been widely investigated in the scientific literature in, almost exclusively, the single-follower setting. Although the textit{optimistic} and textit{pessimisti
In this paper, we employ a hypergame framework to analyze the single-leader-multiple-followers (SLMF) Stackelberg security game with two typical misinformed situations: misperception and deception. We provide a stability criterion with the help of hyper Nash equilibrium (HNE) to analyze both strategic stability and cognitive stability of equilibria in SLMF games with misinformation. To this end, we find mild stable conditions such that the equilibria with misperception and deception can derive HNE. Moreover, we analyze the robustness of the equilibria to reveal whether the players have the ability to keep their profits.
We study the problem of computing correlated strategies to commit to in games with multiple leaders and followers. To the best of our knowledge, this problem is widely unexplored so far, as the majority of the works in the literature focus on games with a single leader and one or more followers. The fundamental ingredient of our model is that a leader can decide whether to participate in the commitment or to defect from it by taking on the role of follower. This introduces a preliminary stage where, before the underlying game is played, the leaders make their decisions to reach an agreement on the correlated strategy to commit to. We distinguish three solution concepts on the basis of the constraints that they enforce on the agreement reached by the leaders. Then, we provide a comprehensive study of the properties of our solution concepts, in terms of existence, relation with other solution concepts, and computational complexity.
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 leaders 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 descent--based approach, where the leaders strategy is updated by differentiating through the followers best responses. We frame the leaders optimization as a learning problem against followers equilibrium, which allows us to decouple the followers equilibrium constraints from the leaders 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.
The features of animal population dynamics, for instance, flocking and migration, are often synchronized for survival under large-scale climate change or perceived threats. These coherent phenomena have been explained using synchronization models. However, such models do not take into account asynchronous and adaptive updating of an individuals status at each time. Here, we modify the Kuramoto model slightly by classifying oscillators as leaders or followers, according to their angular velocity at each time, where individuals interact asymmetrically according to their leader/follower status. As the angular velocities of the oscillators are updated, the leader and follower status may also be reassigned. Owing to this adaptive dynamics, oscillators may cooperate by taking turns acting as a leader or follower. This may result in intriguing patterns of synchronization transitions, including hybrid phase transitions, and produce the leader-follower switching pattern observed in bird migration patterns.
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