The main topic of this paper are algorithms for computing Nash equilibria. We cast our particular methods as instances of a general algorithmic abstraction, namely, a method we call {\em algorithmic boosting}, which is also relevant to other fixed-point computation problems. Algorithmic boosting is the principle of computing fixed points by taking (long-run) averages of iterated maps and it is a generalization of exponentiation. We first define our method in the setting of nonlinear maps. Secondly, we restrict attention to convergent linear maps (for computing dominant eigenvectors, for example, in the PageRank algorithm) and show that our algorithmic boosting method can set in motion {\em exponential speedups in the convergence rate}. Thirdly, we show that algorithmic boosting can convert a (weak) non-convergent iterator to a (strong) convergent one. We also consider a {\em variational approach} to algorithmic boosting providing tools to convert a non-convergent continuous flow to a convergent one. Then, by embedding the construction of averages in the design of the iterated map, we constructively prove the existence of Nash equilibria (and, therefore, Brouwer fixed points). We then discuss implementations of averaging and exponentiation, an important matter even for the scalar case. We finally discuss a relationship between dominant (PageRank) eigenvectors and Nash equilibria.

翻译：本文主要研究计算纳什均衡的算法。我们将提出的特定方法归入一种通用的算法抽象，即称为“算法加速”的方法，该方法也适用于其他不动点计算问题。算法加速是通过迭代映射的（长期）平均值来计算不动点的原理，是指数运算的推广。我们首先在非线性映射的背景下定义该方法。其次，我们关注收敛的线性映射（例如，在PageRank算法中用于计算主特征向量），并证明我们的算法加速方法能够实现收敛速度的指数级加速。第三，我们展示算法加速可以将（弱）非收敛迭代器转化为（强）收敛迭代器。我们还考虑了算法加速的变分方法，为将非收敛连续流转化为收敛连续流提供了工具。然后，通过将平均值的构造嵌入到迭代映射的设计中，我们构造性地证明了纳什均衡的存在性（从而也证明了布劳威尔不动点的存在性）。接着，我们讨论平均值和指数运算的实现，即使在标量情况下这也是一个重要问题。最后，我们探讨了（PageRank）主特征向量与纳什均衡之间的关系。