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.

In later chapters voting and fair division are treated as games. Casino games, poker, auctions, warfare, games with perfect and incomplete information. Dozens and dozens of games. In Surely You’re Joking, Mr. Feynman, the famous physicist explained a trick for always seeming to be the smartest person in the room. Whenever anyone started talking about an abstract concept, he’d think of three examples and modify them accordingly as the speaker elaborated.

Eventually his examples would either lead him to a guess that anticipated the speaker or a counterexample to a hypothesis. At which point he would speak up. (Of course, being the smartest person in the room also works; this was Feynman’s second trick.)

Astoundingly (to me), E. H. Moore, the early 20^th-century mathematician, claimed that he did not think in examples­–he was happier in a mental playground of pure abstraction. I think Moore is the exception that proves the rule; most (had I not read Moore, I would confidently assert all) normal humans understand abstraction through examples. Good mathematics teachers always keep this in mind. Prisner is a very good mathematics teacher.

This textbook is suitable for a liberal-arts math class or first-year seminar or a course for mathematics majors. Prisner developed it for a course that satisfied a general education, quantitative reasoning requirement at his college. As he learned, students interested in economics, political science, computer science, and business can all be brought to a point of deeply understanding and appreciating the power and beauty of mathematics by playing along with him. MAA members can preview samples chapters at maa.org/focus/sample.