The Hamiltonian of Poly-matrix Zero-sum Games

• URL: http://arxiv.org/abs/2505.12609v2
• Date: Fri, 23 May 2025 02:13:05 GMT
• Title: The Hamiltonian of Poly-matrix Zero-sum Games
• Authors: Toshihiro Ota, Yuma Fujimoto,
• Abstract summary: We identify the Hamiltonian function that generates the dynamics of poly-matrix zero-sum games.<n>We reveal the symmetries of our Hamiltonian and derive the associated conserved quantities.<n>Our results highlight the potential of Hamiltonian dynamics in uncovering the structural properties of learning dynamics in games.
• Score: 0.276240219662896
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Understanding a dynamical system fundamentally relies on establishing an appropriate Hamiltonian function and elucidating its symmetries. By formulating agents' strategies and cumulative payoffs as canonically conjugate variables, we identify the Hamiltonian function that generates the dynamics of poly-matrix zero-sum games. We reveal the symmetries of our Hamiltonian and derive the associated conserved quantities, showing how the conservation of probability and the invariance of the Fenchel coupling are intrinsically encoded within the system. Furthermore, we propose the dissipation FTRL (DFTRL) dynamics by introducing a perturbation that dissipates the Fenchel coupling, proving convergence to the Nash equilibrium and linking DFTRL to last-iterate convergent algorithms. Our results highlight the potential of Hamiltonian dynamics in uncovering the structural properties of learning dynamics in games, and pave the way for broader applications of Hamiltonian dynamics in game theory and machine learning.

Related papers

• Weak coupling limit for quantum systems with unbounded weakly commuting system operators [50.24983453990065]
  This work is devoted to a rigorous analysis of the weak coupling limit (WCL) for the reduced dynamics of an open infinite-dimensional quantum system interacting with electromagnetic field or a reservoir formed by Fermi or Bose particles.<n>We derive in the weak coupling limit the reservoir statistics, which is determined by whose terms in the multi-point correlation functions of the reservoir are non-zero in the WCL.<n>We prove that the resulting reduced system dynamics converges to unitary dynamics with a modified Hamiltonian which can be interpreted as a Lamb shift to the original Hamiltonian.
  arXiv  Detail & Related papers  (2025-05-13T05:32:34Z)
• Lie symmetries and ghost-free representations of the Pais-Uhlenbeck model [44.99833362998488]
  We investigate the Pais-Uhlenbeck (PU) model, a paradigmatic example of a higher time-derivative theory.<n>Exploiting Lie symmetries in conjunction with the model's Bi-Hamiltonian structure, we construct distinct Poisson bracket formulations.<n>Our approach yields a unified framework for interpreting and stabilising higher time-derivative dynamics.
  arXiv  Detail & Related papers  (2025-05-09T15:16:40Z)
• General Hamiltonian description of nonreciprocal interactions [0.0]
  In a vast class of systems, interactions do not stem from a potential, and are in general nonreciprocal.<n>Here, we overcome these limitations by constructing a Hamiltonian that includes auxiliary degrees of freedom.<n>We show that Glauber dynamics based on the constrained Hamiltonian reproduces the steady states of the original Langevin dynamics.
  arXiv  Detail & Related papers  (2025-05-08T13:45:31Z)
• Truncated Gaussian basis approach for simulating many-body dynamics [0.0]
  The approach constructs an effective Hamiltonian within a reduced subspace, spanned by fermionic Gaussian states, and diagonalizes it to obtain approximate eigenstates and eigenenergies.<n> Symmetries can be exploited to perform parallel computation, enabling to simulate systems with much larger sizes.<n>For quench dynamics we observe that time-evolving wave functions in the truncated subspace facilitates the simulation of long-time dynamics.
  arXiv  Detail & Related papers  (2024-10-05T15:47:01Z)
• Quantum Simulation of Nonlinear Dynamical Systems Using Repeated Measurement [42.896772730859645]
  We present a quantum algorithm based on repeated measurement to solve initial-value problems for nonlinear ordinary differential equations. We apply this approach to the classic logistic and Lorenz systems in both integrable and chaotic regimes.
  arXiv  Detail & Related papers  (2024-10-04T18:06:12Z)
• Fourier Neural Operators for Learning Dynamics in Quantum Spin Systems [77.88054335119074]
  We use FNOs to model the evolution of random quantum spin systems. We apply FNOs to a compact set of Hamiltonian observables instead of the entire $2n$ quantum wavefunction.
  arXiv  Detail & Related papers  (2024-09-05T07:18:09Z)
• Fourier Neural Differential Equations for learning Quantum Field Theories [57.11316818360655]
  A Quantum Field Theory is defined by its interaction Hamiltonian, and linked to experimental data by the scattering matrix. In this paper, NDE models are used to learn theory, Scalar-Yukawa theory and Scalar Quantum Electrodynamics. The interaction Hamiltonian of a theory can be extracted from network parameters.
  arXiv  Detail & Related papers  (2023-11-28T22:11:15Z)
• Coarse-Graining Hamiltonian Systems Using WSINDy [0.0]
  We show that WSINDy can successfully identify a reduced Hamiltonian system in the presence of large intrinsics. WSINDy naturally preserves the Hamiltonian structure by restricting to a trial basis of Hamiltonian vector fields. We also provide a contribution to averaging theory by proving that first-order averaging at the level of vector fields preserves Hamiltonian structure in nearly-periodic Hamiltonian systems.
  arXiv  Detail & Related papers  (2023-10-09T17:20:04Z)
• Separable Hamiltonian Neural Networks [1.8674308456443722]
  Hamiltonian neural networks (HNNs) are state-of-the-art models that regress the vector field of a dynamical system. We propose separable HNNs that embed additive separability within HNNs using observational, learning, and inductive biases.
  arXiv  Detail & Related papers  (2023-09-03T03:54:43Z)
• Capturing dynamical correlations using implicit neural representations [85.66456606776552]
  We develop an artificial intelligence framework which combines a neural network trained to mimic simulated data from a model Hamiltonian with automatic differentiation to recover unknown parameters from experimental data. In doing so, we illustrate the ability to build and train a differentiable model only once, which then can be applied in real-time to multi-dimensional scattering data.
  arXiv  Detail & Related papers  (2023-04-08T07:55:36Z)
• Decimation technique for open quantum systems: a case study with driven-dissipative bosonic chains [62.997667081978825]
  Unavoidable coupling of quantum systems to external degrees of freedom leads to dissipative (non-unitary) dynamics. We introduce a method to deal with these systems based on the calculation of (dissipative) lattice Green's function. We illustrate the power of this method with several examples of driven-dissipative bosonic chains of increasing complexity.
  arXiv  Detail & Related papers  (2022-02-15T19:00:09Z)
• Machine Learning S-Wave Scattering Phase Shifts Bypassing the Radial Schr\"odinger Equation [77.34726150561087]
  We present a proof of concept machine learning model resting on a convolutional neural network capable to yield accurate scattering s-wave phase shifts. We discuss how the Hamiltonian can serve as a guiding principle in the construction of a physically-motivated descriptor.
  arXiv  Detail & Related papers  (2021-06-25T17:25:38Z)
• Adding machine learning within Hamiltonians: Renormalization group transformations, symmetry breaking and restoration [0.0]
  We include the predictive function of a neural network, designed for phase classification, as a conjugate variable coupled to an external field within the Hamiltonian of a system. Results show that the field can induce an order-disorder phase transition by breaking or restoring the symmetry. We conclude by discussing how the method provides an essential step toward bridging machine learning and physics.
  arXiv  Detail & Related papers  (2020-09-30T18:44:18Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.