MAA Books Beat: Playing to Learn Game Theory

Written by Steve Kennedy, MAA Books Beat is a column written for MAA FOCUS. Playing to Learn Game Theory appears in the June/July 2014 issue.

Playing to Learn Game Theory

by Steve Kennedy

Suppose the graph depicted in figure 1 represents a network of communities. Two doctors are planning to set up practice in one of the nine communities. A practice will attract all the patients in every town closest to it (closest means in terms of number of graph edges joining the vertices). In the case of a distance tie, each doctor will attract half the patients in the town. Where should the doctors locate to ensure their best result? (They are not conspiring; each doctor is acting independently and in ignorance of the other’s choice.)

Of course, neither doctor, presumably desiring to maximize her practice, would choose one of the degree-two vertices. The other six vertices look equally valuable. Suppose Doctor A chooses vertex 2. Were Doctor B to choose any of vertices 3, 5, or 8, each doctor would get 4.5 towns’ worth of business. If Doctor B chooses vertex 6, she “wins” by gaining five towns to A’s four. (If she chooses vertex 7, the payoffs are reversed and A wins five to four.)

If we conceive of the competition as a game between the two doctors, then this game has no Nash equilibrium. Each doctor can guarantee herself only four-ninths of the available customers.

The example comes from the new MAA ebook, Game Theory through Examples by Erich Prisner. The Doctor Locator Game is one of scores of clever, rich, interesting games described and analyzed in this lively text. There are also scores of apps, linked to the etext, so that the reader can play the games, the best way to understand them. In the doctor game, the reader can, with two mouse clicks, choose vertices to represent the doctors’ choices and immediately see a color-coded graph giving the division and the count.

The example games introduce and exposit all the important concepts of game theory. The Doctor Locator Game, on a different graph, introduces the idea of a Nash equilibrium and explains how to find one. The obvious next question of their existence is answered by the above example.

New eBook: Game Theory through Examples

Game Theory through Examples

by Erich Prisner

Classroom Resource Materials

Game Theory through Examples is a thorough introduction to elementary game theory, covering finite games with complete information.

The core philosophy underlying this volume is that abstract concepts are best learned when encountered first (and repeatedly) in concrete settings. Thus, the essential ideas of game theory are here presented in the context of actual games, real games much more complex and rich than the typical toy examples. All the fundamental ideas are here: Nash equilibria, backward induction, elementary probability, imperfect information, extensive and normal form, mixed and behavioral strategies. The active-learning, example-driven approach makes the text suitable for a course taught through problem solving. Students will be thoroughly engaged by the extensive classroom exercises, compelling homework problems and nearly sixty projects in the text. Also available are approximately eighty Java applets and three dozen Excel spreadsheets in which students can play games and organize information in order to acquire a gut feeling to help in the analysis of the games. Mathematical exploration is a deep form of play, that maxim is embodied in this book.

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Coming Soon to the MAA eBooks Store

Game Theory through Examples
Erich Prisner
Classroom Resource Materials

This book is a thorough introduction to elementary game theory, covering finite games with complete information. The core philosophy underlying this volume is that abstract concepts are best learned when encountered first (and repeatedly) in concrete settings. Thus, the essential ideas of game theory are here presented in the context of actual games, real games much more complex and rich than the typical toy examples. All the fundamental ideas are here: Nash equilibria, backward induction, elementary probability, imperfect information, extensive and normal form, mixed and behavioral strategies. The active-learning, example-driven approach makes the text suitable for a course taught through problem solving. Students will be thoroughly engaged by the extensive classroom exercises, compelling homework problems, and nearly sixty projects in the text. Also available are approximately eighty Java applets and three dozen Excel spreadsheets in which students can play games and organize information in order to acquire a gut feeling to help in the analysis of the games. Mathematical exploration is a deep form of play; that maxim is embodied in this book.

Game Theory through Examples is a lively introduction to this appealing theory. Assuming only high school prerequisites makes the volume especially suitable for a liberal arts or general education spirit-of-mathematics course. It could also serve as the active-learning supplement to a more abstract text in an upper-division game theory course.

375 pp., 2014
eISBN: 978-1-61444-115-1
Catalog Code: GTE
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