A single-semester elective course in game theory would be an attractive feature of any undergraduate mathematics program. Students would get to use the various mathematical skills they have acquired in a thought-provoking applied context. The book under review is intended as a textbook for such a course. I would have mixed feelings about adopting it as a primary text.
Certainly the choice of topics and overall organization is good. There are three main parts, Decisions (Chapters 1, 2, and 3), Interaction (Chapters 4, 5, 6, and 7), and Evolution (Chapters 8 and 9). As the author correctly says in the introduction, the three parts represent considerable breadth, as there are entire textbooks on each one of the three topics. The final part consists of appendices including 29 pages of worked-out solutions to the exercises. The author reasonably recommends that Chapters 1, 2, 4, 5, 6, 7 would be the best support for a short course focused on classical game theory, while Chapters 1, 2, 4, 8, 9 would likewise be appropriate for a short course focused on evolutionary game theory.
Decisions. Decision theory, in the context of game theory, can be viewed as the theory of one-player games. The idea in treating decision theory first is that it allows the introduction of some of the fundamental ideas of game theory in a simpler context.
Mostly, the author succeeds in communicating the fundamental ideas: utility functions, maximization of utility, all-at-once versus sequential decision making, and so on. However one place that the presentation is awkward is in its treatment of probabilistically mixing strategies. Students should be clearly informed that the situation is highly degenerate in the one-player setting and becomes interesting only in the multi-player setting.
Interaction. Interaction refers mostly to interactions between two "rational" players. Mathematically speaking, Player 1 chooses a strategy s1 from a given set of possibilities while Player 2 chooses a strategy s2 from a typically different set of possibilities. Player 1 then receives payoff π1(s1,s2) while Player 2 receives payoff π2(s1,s2), the πi being given functions. Game theory singles out certain strategy pairs (s1,s2 ) as Nash equilibria, namely those for which neither player can improve his or her payoff by a unilateral change of strategy.
The theory works best when the given game has exactly one Nash equilibrium. Then, in situations when game theory is used descriptively, game theory predicts that Player i will play strategy si. In situations when game theory is used prescriptively, game theory recommends strategy si to Player i as the best course of action. Under very general conditions, there is always at least one Nash equilibrium. However, in many games of interest there is more than one; the situation is then immediately more problematic.
The choice of topics — simultaneous games, sequential games, continuous games, and repeated games — is very good. However I find the presentation a little rushed.
For example, consider just Chapter 4 on simultaneous games. I would rather see more attention on the zero-sum case, π1 = –π2, where the Nash equilibria form one connected family, all with the same payoffs. For general-sum games, I would like to see the simple general linear-algebra formula for Nash equilibria (take adjoints of minors…) as a culmination of the special cases discussed. I find the sentence "The extension of the theory to games with more than two players is straightforward" misleading. In fact, while definitions do extend straightforwardly, the theory certainly becomes more complicated. For example, the definitive uniqueness result for two-player zero-sum games evaporates into bewildering coalition possibilities for three-player zero-sum games.
Evolution. In classical game theory, hypercalculating players choose to be at a Nash equilibrium. In evolutionary game theory, perhaps even non-thinking players evolve to a Nash equilibrium. They stay there if a certain stability condition is satisfied. It is a good idea to present students with this complementary and more recent perspective.
Conclusion. The book's biggest drawback is that it treats game theory as a purely mathematical subject. It should be viewed as just as important to give students an intuitive feel for how game theory is applied, even though this requires both author and students to step well outside mathematics.
On the plus side, it should be emphasized that game theory gives many general insights into real world situations. For example, one gets firm quantitative mathematical support for the intuitive ideas that a legal system which can enforce contracts is beneficial to society, that conflict situations often do not have a single "most just" resolution, and that repeated interactions can give rise to cooperation even among distrustful parties.
On the minus side, it should be emphasized that applying game theory to specific real world situations is not at all like applying calculus to physics. One cannot expect to simply perform the right calculations and get the indisputably right answer to several decimal places. Instead, one works with models which are always inadequate in some way. Particularities of the situation always play a prominent role. Quantitative predictions coming from mathematical game theory should always be viewed with some suspicion: are they robust with respect to changing the model? do they agree with intuition? do they accord with observation? Students should be given a refined sensitivity to the limitations of game theory.
I have yet to see my ideal introductory game theory text. It would do justice to the mathematician's viewpoint, clearly distinguishing models from real world situations, and clearly stating general theorems about models. However it would also expose students to something we often do not emphasize: mathematics is just one ingredient, often a secondary ingredient, in understanding real world situations.
David Roberts is an associate professor of mathematics at the University of Minnesota, Morris.
Part I: Decisions. Simple Decision Models.- Simple Decision Processes.- Markov Decision Processes. Part II: Interaction. Static Games.- Finite Dynamic Games.- Games with Continuous Strategy Sets.- Infinite Dynamic Games. Part III: Evolution. Population Games.- Replicator Dynamics. Part IV: Appendices.- Constrained Optimisation.- Dynamical Systems.- Solutions to Exercises.- Further Reading.- Bibliography.- Index.