Preface 

xiii  
PART 1 The Mathematics of Social Choice 

2  (146) 

1. The Mathematics of Voting: The Paradoxes of Democracy 


2  (38) 

Preference Ballots and Preference Schedules 


4  (2) 


6  (3) 


9  (1) 

The PluralitywithElimination Method 


10  (6) 

The Method of Pairwise Comparisons 


16  (5) 


21  (5) 

Conclusion: Fairness and Arrow's Impossibility Theorem 


26  (1) 


27  (7) 

Appendix 1: Breaking Ties 


34  (2) 

Appendix 2: A Sampler of Elections in the Real World 


36  (3) 

References and Further Readings 


39  (1) 

2 Weighted Voting Systems: The Power Game 


40  (32) 


42  (3) 


45  (6) 

Applications of the Banzhaf Power Index 


51  (2) 

The ShapleyShubik Power Index 


53  (6) 

Applications of the ShapleyShubik Power Index 


59  (1) 


60  (1) 


61  (9) 

Appendix: Power in the Electoral College 


70  (1) 

References and Further Readings 


71  (1) 

3. Fair Division: The Slice Is Right 


72  (40) 

FairDivision Problems and FairDivision Schemes 


75  (1) 

Two Players: The DividerChooser Method 


76  (2) 


78  (3) 


81  (3) 

The LastDiminisher Method 


84  (5) 

The Method of Sealed Bids 


89  (3) 


92  (3) 


95  (1) 


96  (15) 

References and Further Readings 


111  (1) 

4. The Mathematics of Apportionment: Making the Rounds 


112  (36) 


114  (2) 

A Little Bit of U. S. History 


116  (1) 

The Mathematics of Apportionment: Basic Concepts 


117  (2) 


119  (1) 


120  (1) 


121  (1) 

More Problems with Hamilton's Method 


122  (4) 


126  (2) 

Jefferson's Method and The Quota Rule 


128  (1) 


129  (3) 


132  (1) 

Conclusion: Balinski and Young's Impossibility Theorem 


133  (1) 


134  (7) 

Appendix 1: The HuntingtonHill Method 


141  (3) 

Appendix 2: A Brief History of Apportionment in the United States 


144  (2) 

References and Further Readings 


146  (2) 
PART 2 Management Science 

148  (152) 

5. Euler Circuits: The Circuit Comes to Town 


148  (36) 


150  (4) 


154  (2) 

Graph Concepts and Terminology 


156  (2) 


158  (3) 


161  (3) 


164  (3) 


167  (5) 


172  (1) 


173  (10) 

References and Further Readings 


183  (1) 

6. The TravelingSalesman Problem: Hamilton Joins the Circuit 


184  (40) 

Hamilton Circuits and Hamilton Paths 


188  (1) 


189  (3) 

TravelingSalesman Problems 


192  (2) 

Simple Strategies for Solving TSPs 


194  (3) 

The BruteForce and NearestNeighbor Algorithms 


197  (5) 

The Repetitive NearestNeighbor Algorithm 


202  (1) 

The CheapestLink Algorithm 


203  (5) 


208  (2) 


210  (12) 

References and Further Readings 


222  (2) 

7. The Mathematics of Networks: Connections! 


224  (36) 


226  (4) 


230  (1) 


230  (2) 

The Shortest Distance Between Three Points 


232  (6) 

Shortest Networks Linking More than Three Points 


238  (5) 


243  (2) 


245  (12) 

Appendix: The SoapBubble Solution 


257  (2) 

References and Further Readings 


259  (1) 

8. The Mathematics of Scheduling: Directed Graphs and Critical Paths 


260  (40) 

The Basic Elements of Scheduling 


262  (6) 


268  (2) 

The Priority List Model for Scheduling 


270  (7) 

The DecreasingTime Algorithm 


277  (1) 


278  (4) 

The CriticalPath Algorithm 


282  (2) 

Scheduling with Independent Tasks 


284  (3) 


287  (1) 


288  (9) 

References and Further Readings 


297  (3) 
PART 3 Growth and Symmetry 

300  (124) 

9. Spiral Growth in Nature: Fibonacci Numbers and the Golden Ration 


300  (28) 


302  (3) 

The Equation x(2) = x + 1 and the Golden Ratio 


305  (3) 


308  (7) 


315  (2) 


317  (1) 


318  (8) 

References and Further Readings 


326  (2) 

10. The Mathematics of Population Growth: There Is Strength in Numbers 


328  (30) 

The Dynamics of Population Growth 


330  (3) 


333  (5) 

The Exponential Growth Model 


338  (7) 

The Logistic Growth Model 


345  (5) 


350  (1) 


351  (6) 

References and Further Readings 


357  (1) 


360  (1) 


361  (1) 


362  (2) 


364  (2) 


366  (1) 


366  (1) 


367  (6) 


373  (4) 


377  (1) 


378  (10) 

Appendix: The Seventeen Wallpaper Pattern Types 


388  (3) 

References and Further Readings 


391  (1) 

12. Fractal Geometry: Fractally Speaking 


392  (32) 


394  (6) 


400  (2) 


402  (1) 

The Twisted Sierpinski Gasket 


403  (3) 

Symmetry of Scale in Art and Literature 


406  (1) 


407  (6) 


413  (3) 


416  (5) 

References and Further Readings 


421  (3) 
PART 4 Statistics 

424  (111) 

13. Collecting Statistical Data: Censuses, Surveys, and Studies 


424  (28) 


426  (2) 

Case Study 1: The 1990 U. S. Census 


428  (1) 


429  (1) 

Case Study 2: The 1936 Literary Digest Poll 


430  (2) 

Case Study 3: The 1948 Presidential Election 


432  (2) 


434  (2) 

Case Study 4: Modern Public Opinion Polls: Stratified Samples 


436  (2) 

Sampling: Terminology and Key Concepts 


438  (2) 


440  (1) 

Case Study 5: The 1954 Salk Polio Vaccine Field Trials 


441  (3) 


444  (1) 


445  (6) 

References and Further Readings 


451  (1) 

14. Descriptive Statistics: Graphing and Summarizing Data 


452  (32) 

Graphical Descriptions of Data 


454  (3) 

Variables: Quantitative and Qualitative; Continuous and Discrete 


457  (5) 

Numerical Summaries of Data 


462  (9) 


471  (3) 


474  (1) 


475  (8) 

References and Further Readings 


483  (1) 

15. Chances, Probability, and Odds: Measuring Uncertainty 


484  (26) 

Random Experiments and Sample Spaces 


486  (2) 

Counting: The Multiplication Rule 


488  (2) 

Permutations and Combinations 


490  (6) 


496  (2) 

Probability Spaces with Equally Likely Outcomes 


498  (4) 


502  (1) 


503  (1) 


504  (5) 

References and Further Readings 


509  (1) 

16. Normal Distributions: Everything Is Back to Normal (Almost) 


510  (25) 

Approximately Normal Distributions of Data 


512  (2) 

Normal Curves and Their Properties 


514  (4) 

Normal Curves as Models of RealLife Data Sets 


518  (1) 

Normal Distributions of Random Events 


519  (2) 


521  (5) 


526  (1) 


527  (6) 

References and Further Readings 


533  (2) 
Answers to Selected Problems 

535  (44) 
Index 

579  (8) 
Photo Credits 

587  