2. Excursions in Modern Mathematics, 7e: 2.Conclusion - 2Copyright © 2010 Pearson Education, Inc. 2 The Mathematics of Power CONCLUSION

3. Excursions in Modern Mathematics, 7e: 2.Conclusion - 3Copyright © 2010 Pearson Education, Inc. In any society, no matter how democratic, some individuals and groups have more power than others. This is simply a consequence of the fact that individuals and groups are not all equal. Diversity is the inherent reason the concept of power exists. Themes of Chapter 2

4. Excursions in Modern Mathematics, 7e: 2.Conclusion - 4Copyright © 2010 Pearson Education, Inc. Power itself comes in many different forms. We often hear cliché such as “In strength lies power” or “Money is power” (and the newer cyber version, “Information is power”). We discussed the notion of power as it applies to formal voting situations called weighted voting systems and saw how mathematical methods allow us to measure the power of an individual or group by means of a power index. Themes of Chapter 2

5. Excursions in Modern Mathematics, 7e: 2.Conclusion - 5Copyright © 2010 Pearson Education, Inc. In particular, we looked at two different kinds of power indexes: the Banzhaf power index and the Shapley-Shubik power index. These indexes provide two different ways to measure power, and, while they occasionally agree, they often differ significantly. Of the two, which one better measures real power? Themes of Chapter 2

6. Excursions in Modern Mathematics, 7e: 2.Conclusion - 6Copyright © 2010 Pearson Education, Inc. Unfortunately, there is no simple answer. Both of them are useful, and in some sense the choice is subjective. Perhaps the best way to evaluate them is to think of them as being based on a slightly different set of assumptions. The idea behind the Banzhaf interpretation of power is that players are free to come and go, negotiating their allegiance for power (somewhat like professional athletes since the advent of free agency). Themes of Chapter 2

7. Excursions in Modern Mathematics, 7e: 2.Conclusion - 7Copyright © 2010 Pearson Education, Inc. Underlying the Shapley-Shubik interpretation of power is the assumption that when a player joins a coalition, he or she is making a commitment to stay. In the latter case a player’s power is generated by his ability to be in the right place at the right time. Themes of Chapter 2

8. Excursions in Modern Mathematics, 7e: 2.Conclusion - 8Copyright © 2010 Pearson Education, Inc. In practice the choice of which method to use for measuring power is based on which of the assumptions better fits the specifics of the situation. Contrary to what we’ve often come to expect, mathematics does not give us the answer, just the tools that might help us make an informed decision. Themes of Chapter 2