Simple Games

In a simple game, every coalition is either winning or losing. The concept was first explicitly introduced by John von Neumann and Oskar Morgenstern in their key work from 1944, though clearly the basic idea goes back much farther. Simple games are among the most widely studied types of games, and the terminology associated with simple games is especially useful in describing voting rules.

A simple game may be described by the list of coalitions (subsets of players) that are winning, where this list satisfies these conditions:

• 1.
  the empty coalition (of no players) is losing;
• 2.
  the grand coalition (of all players) is winning; and
• 3.

  if coalition S is winning, any more inclusive coalition (i.e., any superset of S) is also winning.

  A coalition S a minimal winning coalition if (a) S is winning, but (b) the loss of any member converts S into a losing coalition. A dummy is a player who does not belong to any minimal winning coalition. A dictator is a single-player (minimal) winning coalition. Given the superset property above, a simple game can be more concisely described by adding a description of its minimal winning coalitions to the list of conditions.

  [Page 608]A simple game is proper if:

• 4.

  the complement of every winning coalition is losing.

  (Improper games, in which disjoint coalitions may both be winning, have analytical utility.)

  A simple game is strong if it is proper and also:

• 5.
  the complement of every losing coalition is winning.

Complementary losing coalitions are sometimes called blocking coalitions. A one-player blocking coalition is said to have veto power. Strong simple games have no blocking coalitions.

A simple game is weak if there is at least one player who belongs to every winning coalition. Each such player has veto power. If the intersection of all winning coalitions is itself a winning coalition, that intersection is the unique minimal winning coalition and all players outside the intersection are dummies. If the winning intersection contains several players, there is a bargaining game among these players. If the winning intersection contains a single player, that single player is a dictator.

A weighted majority game is a simple game such that every player can be assigned some numerical weight, and a numerical quota can be fixed such that a coalition is winning if and only if the total weight of its members equals or exceeds the quota. A voting rule not ordinarily described as a weighted voting system may in fact be equivalent to a weighted majority game (e.g., that of the United Nations Security Council). However, not every simple game is a weighted majority game or equivalent to one. A weighted majority game is homogeneous if it is possible to assign weights and a quota such that every minimal winning coalition has a total weight equal to the quota.

Solution concepts introduced for cooperative games in characteristic function form are also valid for simple games. In particular, the Shapley value becomes the Shapley-Shubik power index in the case of simple games.