 A common situation is one or more people have veto power. They can cause a measure to fail by voting against it. In this case, the person who can veto a measure has power unavailable to the others. Suppose there are three voters, A, B, and C, who can pass a measure by a simple majority. However, a fourth person D can veto. Let's find the winning coalitions and the critical voters. They'll make a table. In order to pass a measure, at least two of A, B, and C must support it. Since D can veto it, D must also support it. So first, a measure passes if all voters support it. So that's the winning coalition A, B, C, and D. Next, if D and two of A, B, and C support it, the measure passes. So those are the coalitions A, B, and D, A, C, and D, and B, C, and D. So let's find our critical voters. Who's critical in the coalition A, B, C, and D? If D leaves, so they veto the measure, the measure fails, so D is critical. If A leaves, the remaining numbers are B, C, and D, and we see this is still winning, so A is not critical. And the same is true for B and C. So the only critical voter is D. In the coalition A, B, and D, D is still critical. If A leaves, the remaining numbers are B and D, which is not one of our winning coalitions. So A is critical, and the same argument can be made for B. So in the coalition A, B, and D, the critical voters are A, B, and D. And we can make a similar argument for the coalition A, C, D, and B, C, D, which gives us the critical voters. And now we count. A, B, and C are each critical in two coalitions, meanwhile D is critical in four coalitions. So in some sense, D has twice the power of A, B, or C. While we could use the numbers 2, 2, 2, and 4 as a measure of the voting power of A, B, C, and D, it's convenient to normalize these by dividing by their sum, 10. This gives the Bonzov index as a measure of voting power. D was critical voter four times, so they have Bonzov index four-tenths. A, B, and C were critical voters two times, so their Bonzov indices are all two-tenths. Now in most situations where somebody has veto power, there's also a way to override the veto. So I suppose D's veto can be overridden if the other three unanimously support a measure. Let's find the Bonzov indices for A, B, C, and D. So again, we'll make a table that lists the winning coalitions and identifies the critical voters. And all the winning coalitions from before remain winning, but there's one more. Suppose A, B, and C pass a measure while D opposes it and vetoes it. Since the decision to pass a measure was unanimous, this will override D's veto. So A, B, C is a winning coalition. Now D is no longer critical in the coalition of all four voters. The original winning coalitions still have the same critical voters. And in the new coalition, losing any voter would turn the coalition into a losing coalition. So every voter is critical. And now we count. D is critical in three coalitions. Meanwhile, A, B, and C are each critical in three coalitions. So we should divide each of these by the sum, 3 plus 3 plus 3 plus 3. So we can calculate the Bonzov indices. D is the critical voter 3 out of 12 times. So their Bonzov index is 312. A, B, and C are each critical voters 3 out of 12 times. So their Bonzov index are 312. And notice they all have equal power. So even though D has veto power, D's veto power is more apparent than real.