Proper Simple Game

A simple game is called proper if the complement of any winning coalition of the game is a losing coalition. This property was first dealt with by John von Neumann and Oskar Morgenstern in their key book of 1944. Proper simple games are among the most widely studied typologies of game and are commonly used to describe voting situations.

Preliminary definitions with a view to introducing the proper simple game concept will be given below, together with an indication of the main peculiarities of these games. This will be followed by an example.

Let N={1,2,…, n) be a set of players. A cooperative game on N is said to be expressed in characteristic function form if a function v (the characteristic function) is given that assigns a win v(S) to every coalition S ⫅N under the condition v{Ø) = 0. The game characterized by v is usually called the game v. A game v is called superadditive if [Page 538]v(SUT) ≥ v(S)+ v (T) for all disjoint coalitions S, T of N. A game v is said to be simple if its characteristic function assumes values only in the set {0,1}: v(S)= 0 or v(S)= 1 for all coalitions S ⫅N . In the first case the coalition is said to be losing; in the second case, winning. All simple games have two properties:

• each coalition is either winning or losing (given that, by definition, neither can be winning and losing at the same time);
• the empty set is losing: v(Ø) = 0 (given that each simple game is a game in characteristic function form).

Further properties that a simple game v may have are:

A. the coalition of all players is winning: v(N) = 1;

B. no losing coalition contains a winning coalition;

C. the complement of any winning coalition is a losing coalition;

D. the complement of any losing coalition is a winning coalition.

Note that, for games satisfying properties A and B, property C is equivalent to

C'. if S and T are two different winning coalitions, then S ∩ T ≠ Ø.

A proper simple game is defined as being a simple game that satisfies properties A, B, and C (or C'). All simple games that do not possess at least one of these properties are known as improper simple games. Property C (and the corresponding property C') is very important in measuring power, given that it prevents any confusion that might result from allowing separate winning coalitions to make simultaneous decisions. It can be proved that every proper simple game is superadditive.

Every proper simple game that also satisfies property D is said to be a strong (or decisive) proper simple game. Property D prevents the paralysis that might result from allowing losing coalitions to obstruct decisions. Thus, strong proper simple games represent efficient group decision rules, whereas other simple games represent procedures that may present contradictions, deadlocks, or both.

One particular category of proper simple games is that of proper weighted majority games. These games help establish the distribution of power between the members of an assembly on the basis of possible majority coalitions that could be formed. Let each ith member (i = 1, …, n) be assigned a nonnegative weight wi that expresses votes, or seats, shares, and so on. Let q be the majority quota that must be achieved for a decision to be made. Let wS be the total weight of the coalition S, for all

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