Anneliese, Brian, and Carlos among them own all 1500 shares of a small company's stock. At the annual stockholders' meeting, each stockholder's vote counts the same as the number of shares that he or she owns. A measure at the meeting passes if stock holders accounting for 2/3 of all shares support the measure. Because controversial measures about moving manufacturing to Asia and replacing the CEO of the company are to be voted on at the upcoming meeting, Brian and Carlos each buy 100 shares of stock from Anneliese in an effort to gain more influence. Brian finds that he has more influence than before, while Carlos discovers that his vote cannot affect the outcome on any measure, when before it could! How can this happen? I will introduce simple weighted-voting games to model the stockholder scenario, and other voting situations, and power indices to measure the effect voters have on the outcome of yes/no elections. Power indices have been used to analyze the simple weighted-voting game models of the International Monetary Fund [9, 19], the Elec toral College [21], the European Union Council of Ministers [13,18], and the Israeli Knesset [17]. Power index calculations have also been used in the debate on the de sign of institutions, as in articles about the effects of reforms on, and the introduction of new members into, the European Union [32,33]. Counterintuitive results such as Carlos' predicament are often called paradoxes. The literature on power indices is full of paradoxes, as well as real-life institutions that exhibit them. The paradox of redistribution [9, 25], the donor and transfer paradoxes [10], the paradox of quarreling members [15], the paradox of a new member [3, 4], and the paradox of large size [3, 28] capture diverse aspects of counterintuitive be havior in simple weighted-voting games. Using geometry, I will explain and classify the causes of voting power paradoxes. Surprisingly, 3-voter examples are sufficient to understand the geometry. The low dimension and the inherent symmetry of the 3-voter examples are often enough to prove that all power indices are susceptible to a partic ular paradox. Bradberry [2] used the same geometric approach to examine paradoxes of apportionment methods. To place a simple weighted-voting game in a geometric setting, let the weights of the voters (the number of shares in the stockholder example) represent a point in Euclidean space. The voting rule (2/3 for the stockholder game) defines hyperplanes that partition the space into different parts or equivalence classes so that the power of each player is fixed for all games in an equivalence class; for the 3-voter examples, the hyperplanes are lines. Three types of geometric phenomena describe the changes in a game that may result in counterintuitive outcomes. A change in the weights of the