COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 1559

Lexicographic Composition of Simple Games

Barry O'Neill
University of California, Los Angeles

Bezalel Peleg
Hebrew University of Jerusalem

February 2006

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 G₁ => G₂ when it either wins in G₁, or blocks in G₁ and wins in G₂. 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.

JEL Classification: C71, D71

Keywords: Simple games, Voting, Game composition, Game decomposition, Semi-value, Decisiveness, Fairness