Voting Games

Game Theory · Axiom Academy

Understanding power dynamics and coalition formation in weighted voting systems A weighted voting game is specified by a quota and player weights, written as: where q is the quota (votes needed to pass a motion), and w i is the weight (number of votes) of player i . 2. Winning and Losing Coalitions A coalition is any subset of players. Coalitions are classified based on whether their combined weights meet or exceed the quota: For the system [7; 4, 3, 2, 1], let's examine some coalitions: P1, P2 has weight 4 + 3 = 7 (winning) P1, P3 has weight 4 + 2 = 6 (losing) P2, P3, P4 has weight 3 + 2 + 1 = 6 (losing) P1, P2, P3 has weight 4 + 3 + 2 = 9 (winning) A player's power doesn't come from their vote weight alone, but from their ability to change outcomes: Consider coalition P1, P2, P3 with weight 9 in system [7; 4, 3, 2, 1]: Remove P1: P2, P3 has weight 5 < 7 (P1 is pivotal) Remove P2: P1, P3 has weight 6 < 7 (P2 is pivotal) Remove P3: P1, P2 has weight 7 = 7 (P3 is NOT pivotal) The Shapley-Shubik index measures voting power by counting how often a player is pivotal across all possible orderings of voters. List all n! permutations (orderings) of the n players For each permutation, identify the pivotal player : the first player in the ordering whose addition causes the cumulative weight to reach or exceed the quota Count how many times each player is pivotal Divide each count by n! to get the power index

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