 An important question we might want to ask is, how much power does a voter have? To begin with, we'll make a few simplifying assumptions. First, we'll assume that voters are deciding whether to approve or reject a measure. They can only vote to approve or vote to reject. In other words, there's two alternatives. The measure passes if it meets a specified quota, and the measure fails if it doesn't meet quota. A common quota is simple majority, more than half the voters. However, there are many other possibilities. For example, a supermajority is when more than a majority is required. For example, passing an amendment to the Constitution requires approval by three-fourths of the states. We could go even further, and we might even require unanimity. All voters must approve a measure, otherwise it fails. For example, a guilty verdict usually requires a unanimous decision. For example, suppose there are fifty voters, how many are needed to form a majority, and to form a supermajority of two-thirds. So half of fifty is twenty-five. But twenty-five is only half, so at least twenty-six are needed to form a majority. Two-thirds of fifty is thirty-three and a third. And since you can't have one-third of a voter, and thirty-three voters is less than two-thirds, you need at least thirty-four to form a supermajority of two-thirds. We can begin with the idea that if a voter's decision doesn't matter, then they don't have power. So if we want to measure a voter's power, let's see how often their vote matters. To decide whether their decision matters, we can consider coalitions of voters. A group of voters is called a winning coalition if they can pass a measure by voting for it. A group is a blocking coalition if they can stop a measure by voting against it. And it's a losing coalition if they can neither pass nor block a measure. So let's consider a voter in a winning or blocking coalition. A voter defects if they leave the coalition. A critical voter is one whose defection turns a winning or blocking coalition into a losing coalition. So suppose there are three voters, A, B, and C, and unanimity is required. In the winning coalitions, and identify the critical voters. Since unanimity is required, the only winning coalition is all three voters, and the departure of any voter turns it into a losing coalition. So A, B, and C are each critical voters in the winning coalition. Since there's only one winning coalition in unanimity, then every voter is critical one time. Meanwhile, if majority or supermajority is required, then a winning coalition either has more than enough votes, so no single voter is critical, or has exactly enough votes, so every voter is critical. So unanimity, majority, or supermajority systems, every voter has equal power. But what about other systems? Let's look at those next.