A two-house legislature can often be modelled as a proper simple game whose outcome depends on whether a coalition wins, blocks or loses in two smaller proper simple games. It is shown that there are exactly ﬁve ways to combine the smaller games into a larger one. This paper focuses on one of the rules, lexicographic composition , where a coalition wins in G 1 => G 2 when it either wins in G 1 , or blocks in G 1 and wins in G 2 . It is the most decisive of the ﬁve. A lexicographically decomposable game is one that can be represented in this way using components whose player sets partition the whole set. Games with veto players are not decomposable, and anonymous games are decomposable if and only if they are decisive and have two or more players. If a player’s beneﬁt is assessed by any semi-value, then for two isomorphic games a player is better oﬀ from having a role in the ﬁrst game than having the same role in the second. Lexicographic decomposability is sometimes compatible with equality of roles. A relaxation of it is suggested for its practical beneﬁts.
Peleg, Bezalel, "Lexicographic Composition of Simple Games" (2006). Cowles Foundation Discussion Papers. 1847.