Bribe Index

A bribe index is an idea introduced by Peter Morriss, following a remark made by Brian Barry that the powerful people in a society must include those whom the U.S. Central Intelligence Agency (CIA) would want to bribe. The intuitive idea is that the extent of their power can be gauged by the amount that they can demand as a bribe.

Morriss applied this idea to the development of power indices measuring the power of votes. This discussion abstracts from any consideration the morality or immorality of taking or accepting bribes; all it considers is how much one could charge for one's vote if votes were for sale. The same idea has been applied to considering what the market value would be of shares in a company, if shares were bought and sold only for purposes of altering the policies of that company.

Consider, then, voters who each have different numbers of votes—such as parties in a parliament, states in the U.S. Electoral College, or shareholders at an annual general meeting who each have one vote per share. They are about to make a decision on some matter of public policy. You do not have a vote, but stand to gain $100 if policy X is chosen over the voters' own preferred outcome. If you know that voter A by himself or herself could deliver policy X, then you would be willing to pay voter A as much as $100 to vote for it. If voter B could increase the probability of policy X being chosen from nothing to 50%, then you would be willing to pay B as much as $50. That shows that A's votes have twice the value to you that B's have, which is an indication of A's greater voting power.

Morriss showed how some of the standard power indices can be understood as bribe indices. The Banzhaf index is produced under the constraint that you can bribe any, but only one, voter. But that is only part of the story: for, if you had bribed B first, then you would still want to raise your 50% to 100% by buying up the votes of some others. When you can bribe as many voters as you wish, until your preferred outcome is guaranteed, we get the Shapley—Shubik Index. So the Shapley—Shubik Index measures the relative worth of votes, and hence their relative power.

This has subsequently been shown mathematically by Dan Felsenthal and Moshé Machover. They define P-power as the proportion of a fixed purse (the spoils of victory) each voter could obtain for himself or herself. This is the same as what the voter could deliver to an outside briber and is represented by the Shapley—Shubik Index.