Quantum mean-field games
Quantum mean-field games
复制标题
量子平均场博弈
DOI：
10.1214/21-aap1733
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发表时间：
2020
期刊：
影响因子：
--
通讯作者：
V. Kolokoltsov
中科院分区：
文献类型：
--
作者：
V. Kolokoltsov
量子博弈代表了博弈论在世纪的真正分支，与量子计算和量子技术的现代发展紧密相连。到目前为止，这些发展的主要重点是固定或重复的游戏。在作者的前一篇论文中，真正动态的量子博弈论是由玩家在真实的时间内选择策略而建立的。由于直接连续观测已知会破坏量子演化（所谓的量子芝诺悖论），量子动力学博弈的必要新成分代表了非直接观测理论和相应的量子滤波。另一个引人注目的世纪博弈论的分支代表所谓的平均场博弈（MFG），令人印象深刻的和不断增长的发展。
在本文中，我们将合并这两个令人兴奋的新分支博弈论。建立MFG的量子模拟需要完全重建其基础和方法，因为在$N$-粒子量子演化中，粒子在个体动力学中是不分离的，经典MFG理论的关键概念，定义为参与者位置的狄拉克质量之和的经验测量，在量子设置中不适用。
作为初步结果，我们导出了新的非线性随机薛定谔方程，作为连续观测和控制的大量相互作用量子粒子系统的极限，这个结果可能具有独立的价值。然后，我们表明，一个控制量子系统的相互作用的粒子有对应的一个特殊系统的经典相互作用的粒子与相同的限制MFG系统，定义在适当的黎曼流形。这个系统的解决方案，指定近似纳什均衡的$N$代理量子游戏。
Quantum games represent the really 21st century branch of game theory, tightly linked to the modern development of quantum computing and quantum technologies. The main accent in these developments so far was made on stationary or repeated games. In the previous paper of the author the truly dynamic quantum game theory was initiated with strategies chosen by players in real time. Since direct continuous observations are known to destroy quantum evolutions (so-called quantum Zeno paradox) the necessary new ingredient for quantum dynamic games represented the theory of non-direct observations and the corresponding quantum filtering. Another remarkable 21st century branch of game theory represent the so-called mean-field games (MFG), with impressive and ever growing development.
In this paper we are merging these two exciting new branches of game theory. Building a quantum analog of MFGs requires the full reconstruction of its foundations and methodology, because in $N$-particle quantum evolution particles are not separated in individual dynamics and the key concept of the classical MFG theory, the empirical measure defined as the sum of Dirac masses of the positions of the players, is not applicable in quantum setting.
As a preliminary result we derive the new nonlinear stochastic Schrodinger equation, as the limit of continuously observed and controlled system of large number of interacting quantum particles, the result that may have an independent value. We then show that to a control quantum system of interacting particles there corresponds a special system of classical interacting particles with the identical limiting MFG system, defined on an appropriate Riemanian manifold. Solutions of this system are shown to specify approximate Nash equilibria for $N$-agent quantum games.
影响因子：
1.1
作者：
Z. Brzeźniak;A. Millet
通讯作者：
Z. Brzeźniak;A. Millet