In this paper, we propose non-model-based strategies for locally stable convergence to Nash equilibrium in quadratic noncooperative games where acquisition of information (of two different types) incurs delays. Two sets of results are introduced: (a) one, which we call cooperative scenario, where each player employs the knowledge of the functional form of his payoff and knowledge of other players’ actions, but with delays; and (b) the second one, which we term the noncooperative scenario, where the players have access only to their own payoff values, again with delay. Both approaches are based on the extremum seeking perspective, which has previously been reported for real-time optimization problems by exploring sinusoidal excitation signals to estimate the Gradient (first derivative) and Hessian (second derivative) of unknown quadratic functions. In order to compensate distinct delays in the inputs of the players, we have employed predictor feedback. We apply a small-gain analysis as well as averaging theory in infinite dimensions, due to the infinite-dimensional state of the time delays, in order to obtain local convergence results for the unknown quadratic payoffs to a small neighborhood of the Nash equilibrium. We quantify the size of these residual sets and corroborate the theoretical results numerically on an example of a two-player game with delays.

中文翻译：

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两种延迟信息共享方案下二次非合作博弈中的纳什均衡寻求

在本文中，我们提出了非基于模型的策略，用于在二次非合作博弈中局部稳定收敛到纳什均衡，其中（两种不同类型的）信息的获取会导致延迟。引入了两组结果： (a) 一个，我们称之为合作场景，其中每个参与者使用他的收益函数形式的知识和其他参与者的行为的知识，但有延迟；(b) 第二个场景，我们称之为非合作场景，玩家只能访问他们自己的收益值，同样有延迟。这两种方法都基于极值寻找视角，之前已经通过探索正弦激励信号来估计未知二次函数的梯度（一阶导数）和 Hessian（二阶导数）来解决实时优化问题。为了补偿玩家输入的明显延迟，我们采用了预测器反馈。由于时间延迟的无限维状态，我们在无限维中应用小增益分析和平均理论，以获得对纳什均衡小邻域的未知二次收益的局部收敛结果。我们量化了这些残差集的大小，并在一个有延迟的两人游戏的例子上以数值方式证实了理论结果。为了获得纳什均衡小邻域的未知二次收益的局部收敛结果。我们量化了这些残差集的大小，并在一个有延迟的两人游戏示例上以数值方式证实了理论结果。为了获得纳什均衡小邻域的未知二次收益的局部收敛结果。我们量化了这些残差集的大小，并在一个有延迟的两人游戏的例子上以数值方式证实了理论结果。