## Voting Games: Part I
What is the intuitive meaning of a veto player? This is someone without whose approval one can not win. This idea can be captured by saying that a player is a veto player in a game if that player is a member of every minimal winning coalition. Can a voting game with two dictators (weird thought, right?) have a veto player? If one is not content with the relationship between definitions one has made, one must either change the definition or restrict the kinds of games one wants to deal with.
Another common term in voting is the idea of a blocking coalition. The intuitive idea here is that one has a collection of players who are not winning yet can prevent any other collection of players from being a winning coalition. Based on this intuition one can formalize the definition of a blocking coalition. After defining these words one can try to find interesting examples or prove theorems involving these concepts. The important thing to notice is that mathematics often provides unexpected insights by examining situations that seem artificial. Another reason for sometimes allowing general structures that may not occur in the real world is that it is often easier to prove general assertions, which may allow for things that one may not see in practice, than to prove assertions restricted only to the cases one sees in practice! Although this is not used in common parlance, if a voting game has a player who is not the member of any minimal winning coalition, the player is known as a
Given a weighted voting game in this form it is a straightforward process to write out the winning coalitions and minimal winning coalitions for the game.
which has three players who cast 4, 2, and 1 vote(s), respectively. If we set the quota Q at the "majority" level, namely, 4, then we have the following collection of minimal winning coalitions: {1}! So the game has a dictator in the sense that there is no minimal winning coalition with more than one player, and that player 1 by him/herself constitutes a minimal winning coalition. Players 2 and 3 are dummies in this game because they are not members of any minimal winning coalition. Even though it is not perhaps of much "political interest," it is of mathematical interest to note that as the value of Q varies from 1 to 7 one gets a different set of minimal winning coalitions each time (and one might even want to consider the "degenerate" game where the quota is zero!). |
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