Definition 2.2.1 Dictator
A player is a dictator if their weight is equal to or greater than the quota.
Section 2.2 A Look at Power
¶Consider the voting system [10: 11, 3, 2]. Notice that in this system, player 1 can reach quota without the support of any other player. When this happens, we say that player 1 is a dictator.
Recall that in Section 1 we specified that the quota must be more than half the number of votes. This means that any dictator must have more than half the votes and so the other players combined will have less than half. So the other players combined cannot reach the quota. Thus a dictator can block any proposal from passing; the other players cannot reach quota without the dictator.
In the voting system [8: 6, 3, 2], no player is a dictator. However, in this system, the quota can only be reached if player 1 is in support of the proposal; player 2 and 3 cannot reach quota without player 1’s support. In this case, player 1 is said to have veto power. Notice that player 1 is not a dictator, since player 1 would still need player 2 or 3’s support to reach quota.
Definition 2.2.2 Veto Power
A player has veto power if their support is necessary for the quota to be reached. It is possible for more than one player to have veto power, or for no player to have veto power.
Based on our comments above, we see a dictator automatically has veto power.
With the system [10: 7, 6, 2], the only way the quota can be met is with the support of both players 1 and 2, so both players 1 and 2 have veto power. Interestingly, the vote of player 3 is irrelevant to the outcome in all situations. Player 3 is said to be a dummy voter, meaning they have no influence in the outcome.
Definition 2.2.3 Dummy Voter
A player is a dummy voter if their vote is never essential for a group to reach quota.
Example 2.2.4
In the voting system [16: 7, 6, 3, 3, 2], are any players dictators? Do any have veto power? Are any dummies?
No player can reach quota alone, so there are no dictators.
Without player 1, the rest of the players’ weights add to 14, which doesn’t reach quota, so player 1 has veto power. Likewise, without player 2, the rest of the players’ weights add to 15, which doesn’t reach quota, so player 2 also has veto power.
Since player 1 and 2 can reach quota with either player 3 or player 4’s support, neither player 3 or player 4 have veto power. However they cannot reach quota with player 5’s support alone, so player 5 has no influence on the outcome and is a dummy.
Example 2.2.6
Try it Now 2.2.8
In the voting system \([q : 10, 5, 3]\text{,}\) which players are dictators, have veto power, and are dummies if the quota is 10? 12? 16?
To better define power, we need to introduce the idea of a coalition. A coalition is a group of players voting the same way. In the example above, \(\{P_1, P_2, P_4\}\) would represent the coalition of players 1, 2 and 4. This coalition has a combined weight of 7+6+3 = 16, which meets quota, so this would be a winning coalition.
A player is said to be critical in a coalition if them leaving the coalition would change it from a winning coalition to a losing coalition. In the coalition \(\{P_1, P_2, P_4\}\text{,}\) every player is critical. In the coalition \(\{P_3, P_4, P_5\}\text{,}\) no player is critical, since it wasn’t a winning coalition to begin with. In the coalition \(\{P_1, P_2, P_3, P_4, P_5\}\text{,}\) only players 1 and 2 are critical; any other player could leave the coalition and it would still meet quota.
Definition 2.2.9 Coalitions and Critical Players
A coalition is any group of players voting the same way.
A coalition is a winning coalition if the coalition has enough weight to meet quota.
A player is critical in a coalition if them leaving the coalition would change it from a winning coalition to a losing coalition.
Example 2.2.10
In the Scottish Parliament in 2009 there were 5 political parties: 47 representatives for the Scottish National Party, 46 for the Labour Party, 17 for the Conservative Party, 16 for the Liberal Democrats, and 2 for the Scottish Green Party. Typically all representatives from a party vote as a block, so the parliament can be treated like the weighted voting system: [65 : 47, 46, 17, 16, 2]
Consider the coalition \(\{P_1, P_3, P_4\}\text{.}\) No two players alone could meet the quota, so all three players are critical in this coalition.
In the coalition \(\{P_1, P_3, P_4, P_5\}\text{,}\) any player except \(P_1\) could leave the coalition and it would still meet quota, so only \(P_1\) is critical in this coalition.
Notice that a player with veto power will be critical in every winning coalition, since removing their support would prevent a proposal from passing.
Likewise, a dummy will never be critical, since their support will never change a losing coalition to a winning one.
Definition 2.2.12 Dictators, Veto, and Dummies and Critical Players
A player is a dictator if the single-player coalition containing them is a winning coalition.
A player has veto power if they are critical in every winning coalition.
A player is a dummy if they are not critical in any winning coalition.