Coalition theory is the theory of how and why coalitions form.
Long-standing NomicPlayers will have noticed that the ability to cast more votes does not always equal more power. Consider the following 3 player game:
- A has 49 votes
- B has 49 votes
- C has 2 votes
A Majority vote (51/100) is required to pass a proposal.
Question: Which player holds the most power?
Answer: They are all the same.
None of the players has a majority of votes on their own, so they need to form a coalition to have their way. Any two players can join together to form a majority of the vote. So C has just as much power as A and B, even though it has far fewer votes.
Now consider the 4 player game:
- A has 51 votes
- B has 51 votes
- C has 51 votes
- D has 47 votes
Again a majority vote (101/200) is needed to pass a proposal.
Question: What is D's power?
Answer: Zero.
Any winning coalition has to contain two of A, B or C in order to control a majority of votes. D alone with A (or B, or C) only controls 98 votes, and so is not a winning coalition. But in a coalition with two other players (such as {A,B,D}) D is completely superfluous. The coalition could win just as easily without them. D has nothing to bargain with, and is completely without power in this game, even though it has only slightly fewer votes than its companions.
The mathematician L. S. Shapley invented a formula known as the ShapleyValue which estimates the "real" power of a player in a situation such as the above.