You are here

Chance, Strategy, and Choice

Samuel Bruce Smith
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Mathematical Textbooks
[Reviewed by
Adam Graham-Squire
, on

Chance, Strategy and Choice fits an important niche for general audience textbooks about games, elections, and other introductory material related to social choice theory. While it covers the basic topics thoroughly, its more advanced material also encompasses simple proof writing as well as proofs of more complicated theorems such as Arrow’s Impossibility Theorem. Thus the textbook is more terse and rigorous than the typical “Topics in Contemporary Math” (that is, math for liberal arts majors) textbook, but is not as theoretical as upper-level material in social choice theory. Topics covered include basic probability (including Bayes’ Law), fairness in elections, as well as gambling, partial-conflict and takeaway games, among others.

Smith’s stated goal is flexibility. That is, he intends the introductory material to be straightforward and interesting enough to be used as the main text for a general education math course, but the later material challenging enough that the text could also be used in a course for sophomore math majors. Overall, the textbook seems to meet those goals, though this flexibility carries some drawbacks. In particular, the book does not have quite as many examples and exercises as some other books do, and it is possible that an instructor would want to supplement with material from other sources if teaching first-year or general education students. Similarly, some of the proofs in the later material seem not as challenging as one might want in a proof-intensive course for math majors.

There are a number of aspects of Chance, Strategy and Choice that are excellent. It is quite readable and approachable, with many real-world examples and applications. The book is broken up into three parts: First Notions, Basic Theory, and Special Topics. At the end of each part Smith gives web resources or suggestions for further reading, which could be useful for designing projects for either a general education course or one for majors.

One of my favorite features of the book is that it does an excellent job of integrating the topics of games and elections to illustrate the interconnections between the different areas of social choice theory, often through illustrative examples. In particular, the 2000 presidential election in Florida is used to exemplify connections between games and elections. Thus game theory, tree diagrams, and social welfare are tied together under the umbrella of social choice theory as opposed to being viewed as disparate topics.

Adam Graham-Squire is an Assistant Professor of Mathematics at High Point University in High Point, North Carolina. His research interests include voting theory, recreational mathematics, algebraic geometry, and the scholarship of teaching and learning. Email at

1. Introduction
2. Games and elections
3. Chance
4. Strategy
5. Choice
6. Strategy and choice
7. Choice and chance
8. Chance and strategy
9. Nash equilibria
10. Proofs and counterexamples
11. Laws of probability
12. Fairness in elections
13. Weighted voting
14. Gambling games
15. Zero-sum games
16. Partial conflict games
17. Take-away games
18. Fairness and impossibility
19. Paradoxes and puzzles in probability
20. Combinatorial games
21. Borda versus Condorcet
22. The Sprague–Grundy theorem
23. Arrow's impossibility theorem.