A geometric approach to differential game theory is illustrated. The parallel pursuit is considered as a two-player zero-sum differential game. The optimal strategies of each player is designed based on Riemann-Finsler geometry. Our approach incorporates a closed loop optimal control and the presentation is familiar wi…
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48 results for “optimal differential game”
Deep fictitious play converges to Nash equilibrium in stochastic differential games.
problem Finding Nash equilibrium in large stochastic differential games.
method Decouples the game into sub-optimization problems and solves each player's optimal strategy with deep BSDE method.
result Deep fictitious play converges to the true Nash equilibrium.
GT-DDP optimizer trains residual networks using game theory.
problem Applying DDP to residual networks.
method Game-theoretic DDP optimizer for residual networks.
result Improved training convergence and variance reduction.
Transforms game optimization dynamics into frequency domain for precise hyperparameter analysis.
problem Analyzing convergence of hyperparameters in game optimization.
method Frequency-domain framework using High-Resolution Differential Equations (HRDEs) and Laplace transforms.
result Derives precise convergence criteria for the Lookahead algorithm.
The paper solves investment problems with uncertain factors using game theory.
problem Optimal forward investment in an incomplete market with model uncertainty.
method Combining stochastic differential games and ergodic BSDE approach.
result Representation of robust forward performance processes in factor form.
This paper studies the optimal extraction and taxation of nonrenewable natural resources. It is well known that the market values of the main strategic resources such as oil, natural gas, uranium, copper,..., etc, fluctuate randomly following global and seasonal macroeconomic parameters, these values are modeled using …
The paper solves TIC LQ control problems using stochastic differential games.
problem Time-inconsistent linear-quadratic stochastic control problems.
method Stochastic differential games, spike variation approach.
result Achieves Nash equilibrium for TIC problems, demonstrating impact of ambiguity aversion.
The study examines how brokers' identity affects their trading strategies on the Toronto Stock Exchange.
problem Impact of anonymous trading on brokers' optimal execution strategies.
method Formulated a stochastic differential game and mean-field game to analyze the optimal execution problem of anonymous and identity-revealed trading.
result Obtained a closed-form solution for the optimal strategy under Almgren-Chris price impact framework.
Paper solves complex game theory problems with new equations.
problem Zero-sum stochastic games with non-Markovian switching.
method New multidimensional SRE and BSDE solutions.
result Existence and uniqueness of SRE solutions.
Games generalize the single-objective optimization paradigm by introducing different objective functions for different players. Differentiable games often proceed by simultaneous or alternating gradient updates. In machine learning, games are gaining new importance through formulations like generative adversarial netwo…
This paper optimizes trading strategies to minimize risk and maximize profit while accounting for market uncertainty.
problem Optimizing trading strategies to minimize risk and maximize profit while accounting for market uncertainty.
method Relative entropy-regularized robust optimal control problem, modeled as a stochastic differential game.
result Analytical expressions for optimal strategy and trajectory are derived under specific assumptions.
This paper uses recent results on continuous-time finite-horizon optimal switching problems with negative switching costs to prove the existence of a saddle point in an optimal stopping (Dynkin) game. Sufficient conditions for the game's value to be continuous with respect to the time horizon are obtained using recent …
Solves a game between brokers and informed traders using stochastic differential equations.
problem Optimizing wealth in a game between brokers and informed traders with private signals.
method Closed-form solutions to a mean-field game using forward-backward SDEs.
result Optimal trading strategies for both brokers and informed traders are found.
New algorithm improves on static methods in Active Simple Hypothesis Testing.
problem Optimizing Active Simple Hypothesis Testing with active sampling.
method Game-theoretic formulation, differential games, PDEs, Blackwell Approachability.
result Proposes an efficient algorithm that outperforms static methods in ASHT.
This paper presents a novel approach to numerically solve stochastic differential games for nonlinear systems. The proposed approach relies on the nonlinear Feynman-Kac theorem that establishes a connection between parabolic deterministic partial differential equations and forward-backward stochastic differential equat…
Trading with the Crowdq-fin.TR
Game theory models how agents trade in a risky asset considering price impact and a common signal.
problem Modeling how financial agents liquidate assets in a risky market with price impact and a common signal.
method Formulated and solved a multi-player stochastic differential game and mean field game.
result Equilibrium strategies reveal how agents adjust the predictive trading signal to price impact.
Mean field game theory studies the behavior of a large number of interacting individuals in a game theoretic setting and has received a lot of attention in the past decade (Lasry and Lions, Japanese journal of mathematics, 2007). In this work, we derive mean field game partial differential equation systems from determi…
Paper resolves ambiguity in non-convex bilevel optimization problems.
problem Ambiguity in bilevel optimization with non-convex lower-level objectives.
method Introduces selection maps to define critical points and resolves ambiguity.
result Validates new analytical tools in Morse theory for implicit differentiation.
Algorithm identifies correct hypothesis from alternatives in bandit problems.
problem Efficiently identifying the correct hypothesis from a finite set of alternatives in structured stochastic multi-armed bandits.
method Frank-Wolfe Self-Play (FWSP) reformulates the game as a saddle-point problem, using a differential-inclusion argument to prove convergence.
result Convergence of the game value for best-arm identification in linear bandits, with uniform global convergence to the optimal value.
The extragradient method accelerates convergence in complex game dynamics.
problem Complex interactions in game dynamics cause simple methods to diverge, necessitating more sophisticated approaches.
method A polynomial-based analysis to identify three scenarios for accelerated convergence of the momentum extragradient method.
result The momentum extragradient method achieves faster convergence under specific eigenvalue conditions.
A new method uses Mean Field Games to optimize mixture models of Bernoulli and categorical distributions.
problem Optimizing parameters of finite mixture models of Bernoulli and categorical distributions.
method Mean Field Games theory applied to multi-population systems.
result The Mean Field Games approach provides a method to compute mixture model parameters.
This paper studies the valuation and optimal strategy of convertible bonds as a Dynkin game by using the reflected backward stochastic differential equation method and the variational inequality method. We first reduce such a Dynkin game to an optimal stopping time problem with state constraint, and then in a Markovian…
New algorithm solves non-convex, non-differentiable min-max games.
problem Limited theoretical understanding of non-smooth min-max games.
method Proximal gradient descent-ascent algorithm for convex-strongly convex games.
result Algorithm converges to ε-Nash equilibrium with polynomial gradient evaluations.
We consider risk-averse agents who compete for liquidity in an Almgren--Chriss market impact model. Mathematically, this situation can be described by a Nash equilibrium for a certain linear-quadratic differential game with state constraints. The state constraints enter the problem as terminal boundary conditions f…
New algorithms solve nonconvex-nonconcave minimax optimization problems.
problem Solving minimax optimization problems in machine learning.
method Two novel Newton-type algorithms for nonconvex-nonconcave minimax optimization.
result Proved local convergence at strict local minimax points.
This paper introduces a new class of Dynkin games, where the two players are allowed to make their stopping decisions at a sequence of exogenous Poisson arrival times. The value function and the associated optimal stopping strategy are characterized by the solution of a backward stochastic differential equation. The pa…
Last-Iterate Convergence of Saddle-Point Optimizers via High-Resolution Differential Equationsmath.OC
New ODE models show saddle-point optimization methods converge differently, with last-iterate convergence for OGDA.
problem Analyzing convergence properties of saddle-point optimization methods.
method High-Resolution Differential Equations (HRDEs) to design differential equation models for saddle-point optimization methods.
result HRDEs reveal last-iterate convergence for Optimistic Gradient Descent Ascent (OGDA) in bilinear games.
Financial markets are often driven by latent factors which traders cannot observe. Here, we address an algorithmic trading problem with collections of heterogeneous agents who aim to perform optimal execution or statistical arbitrage, where all agents filter the latent states of the world, and their trading actions hav…
In this paper, we apply the idea of fictitious play to design deep neural networks (DNNs), and develop deep learning theory and algorithms for computing the Nash equilibrium of asymmetric -player non-zero-sum stochastic differential games, for which we refer as \emph{deep fictitious play}, a multi-stage learning pro…
In this paper, we consider a risk-based optimal investment problem of an insurer in a regime-switching jump diffusion model with noisy memory. Using the model uncertainty modeling, we formulate the investment problem as a zero-sum, stochastic differential delay game between the insurer and the market, with a convex ris…
Proposes a game-theoretic framework for ML trust regulation.
problem Lack of coordination between ML model builders and regulators.
method Formulates trustworthy ML as a multi-objective multi-agent optimization problem and introduces regulation games and ParetoPlay.
result Enables efficient enforcement of ML model specifications without discouraging participation.
In this paper we propose and analyze a class of -player stochastic games that include finite fuel stochastic games as a special case. We first derive sufficient conditions for the Nash equilibrium (NE) in the form of a verification theorem. The associated Quasi-Variational-Inequalities include an essential game comp…
Study on optimal trading in a finite population with market frictions and asymmetric information.
problem Optimal trading in a finite population with market frictions and asymmetric information.
method Investigates stochastic differential games with asymmetric information and market frictions, proving existence and uniqueness of Nash and Stackelberg-Nash equilibria.
result Existence and uniqueness of Nash and Stackelberg-Nash equilibria in both unconstrained and constrained trading scenarios.
In this article we consider a game theoretic approach to the Risk-Sensitive Benchmarked Asset Management problem (RSBAM) of Davis and Lleo \cite{DL}. In particular, we consider a stochastic differential game between two players, namely, the investor who has a power utility while the second player represents the market …
Recent successes of game-theoretic formulations in ML have caused a resurgence of research interest in differentiable games. Overwhelmingly, that research focuses on methods and upper bounds on their speed of convergence. In this work, we approach the question of fundamental iteration complexity by providing lower boun…
This paper investigates a hybrid stochastic differential reinsurance and investment game between one reinsurer and two insurers, including a stochastic Stackelberg differential subgame and a non-zero-sum stochastic differential subgame. The reinsurer, as the leader of the Stackelberg game, can price reinsurance premium…
Deep learning is built on the foundational guarantee that gradient descent on an objective function converges to local minima. Unfortunately, this guarantee fails in settings, such as generative adversarial nets, that exhibit multiple interacting losses. The behavior of gradient-based methods in games is not well under…
AutoBayes simplifies variational inference by composing models and optimizing them.
problem Generalized variational inference complexities and optimization challenges.
method Compositional framework exploiting chain rules for automatic differentiation.
result Optimized models and parameterized statistical games can be locally optimized.
The paper analyzes arbitrage opportunities in a large investor market with common stock noises.
problem Identifying arbitrage opportunities in a market with many competitive investors.
method Stochastic differential games and mean-field systems to study market dynamics and optimal arbitrage.
result Optimal arbitrage is characterized by a solution to a Cauchy PDE involving volatility terms.
Paper proves existence and uniqueness of solutions to nonlocal systems, generalizing stochastic game theory.
problem Time inconsistency in stochastic differential games.
method Proves existence and uniqueness of solutions to nonlocal fully-nonlinear parabolic systems.
result Generalizes stochastic game theory to include time-inconsistent preferences.
We consider differentiable games where the goal is to find a Nash equilibrium. The machine learning community has recently started using variants of the gradient method (GD). Prime examples are extragradient (EG), the optimistic gradient method (OG) and consensus optimization (CO), which enjoy linear convergence in cas…
Study shows how multiple traders can trade together without excessive price impact.
problem Coordination issues in trading to exploit a common signal.
method Closed-loop Nash competition model for stochastic differential games.
result Excessive trading reduced but not significantly for practical parameters.
This paper presents OptNet, a network architecture that integrates optimization problems (here, specifically in the form of quadratic programs) as individual layers in larger end-to-end trainable deep networks. These layers encode constraints and complex dependencies between the hidden states that traditional convoluti…
Investors' strategic trading affects asset prices, modeled as a game.
problem Investors' trading rates influence asset prices in dynamic markets.
method Model as a non-zero sum singular stochastic differential game, establishing equivalence between best-response and auxiliary control problems.
result Unique Nash equilibrium is deterministic with a closed-form solution.
Paper studies optimal tracking portfolio in mean field game of large fund competition.
problem Optimal tracking portfolio in large fund competition with relative performance benchmark.
method Formulated mean field game problem, established existence of mean field equilibrium using PDE approach, constructed approximate Nash equilibrium.
result Existence of mean field equilibrium and consistency condition verified.
This paper analyzes a game between insurer and reinsurer under ambiguity and risk aversion, optimizing reinsurance and investment strategies.
problem Optimizing reinsurance and investment strategies in a game between insurer and reinsurer under ambiguity and risk aversion.
method Stackelberg game, -maxmin mean-variance criterion, Heston's stochastic volatility, Hamilton-Jacobi-Bellman equations, Riccati differential equations.
result Excess-of-loss reinsurance is optimal for the insurer, and the equilibrium strategies are determined by specific equations.
Although recent work in AI has made great progress in solving large, zero-sum, extensive-form games, the underlying assumption in most past work is that the parameters of the game itself are known to the agents. This paper deals with the relatively under-explored but equally important "inverse" setting, where the param…
We study the competition of two strategic agents for liquidity in the benchmark portfolio tracking setup of Bank, Soner, Voß (2017). Specifically, both agents track their own stochastic running trading targets while interacting through common aggregated temporary and permanent price impact à la Almgren and Chriss (2001…