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.
