A recursive sequence of sums of consecutive embedded coalitions

Abstract

The set of players in a cooperative game may be divided into various coalitions forming partitions with different coalition structures. The well-known Bell (1934) number is used to obtain the number of partitions in a n-person cooperative game. The number of embedded coalitions in a partition is the number of subsets formed in that partition. The total number of embedded coalitions in a n-person game is the sum of the numbers of embedded coalitions in different partitions of the game. This article presents a recursive sequence yielding the total sum of the embedded coalitions from a 1-person game to a n-person game.