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，它被证明有许多其他的应用。在这个项目中，它的数学性质将被详细研究，着眼于诸如计算大型游戏的对称和非对称值，推导游戏的其他（非对称）值，为没有侧支付的游戏开发解决方案概念以及计算简化游戏等应用。MLE也可以用于解决某些类型游戏的当前表示中发现的某些问题。这包括非原子游戏（用于表现纯粹竞争的经济状况）和空间游戏（经常用于模拟政治局势中的投票和联盟形成）。在这两种情况下，当前的表示都不能考虑agent的一些随机行为；MLE（具有概率解释）可以弥补这个缺点。

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.