Mathematical Sciences: Research into Multilinear Extensions of Games and Applications

游戏的多线性扩展(MLE)最初是作为一种计算游戏价值的技术引入的，在过去的二十年里已经被证明有许多其他的应用。在这个项目中，我们将更详细地研究它的数学性质，以期应用于计算大型博弈的对称值和非对称值，导出博弈的其他(非对称)值，为没有边付的博弈开发解决方案的概念，以及计算简化博弈。MLE也可以用来解决在某些类型的游戏的当前表示中发现的某些问题。其中包括非原子博弈和空间博弈，前者被用来代表纯粹竞争的经济形势，后者经常被用来模拟政治局势中的投票和联盟组建。在这两种情况下，当前的表示都无法允许代理部分的一些随机行为；MLE(具有概率解释)可以弥补这一缺陷。

The multilinear extension (MLE) of a game, first introduced as a technique for calculating the value of a game, has over the past two decades been shown to have numerous other applications. In this project its mathematical properties will be studied in some detail, with a view towards such applications as the computation of both symmetric and asymmetric values of large games, deriving other (asymmetric) values to games, developing solution concepts for games without side payments, and computing reduced games. MLE may also be used to resolve certain problems which are found in the current representation of some types of games. These include non-atomic games, which are used to represent economic situations of pure competition, and spatial games, which are frequently used to model voting and coalition formation in political situations. In both of these cases, the current representation fails to allow for some stochastic behavior on the part of agents; the MLE (which has a probabilistic interpretation) can remedy this drawback.