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Game Theory in Action: An Introduction to Classical and Evolutionary Models

Stephen Schecter and Herbert Gintis
Princeton University Press
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
David S. Mazel
, on

“In his [Henry Kissinger’s, National Security Advisor and later Secretary of State to Richard Nixon and Gerald Ford] view, the most important contribution of the game theory point of view in international relations was that it forced you to make an explicit model of the situation you wanted to understand.” Page 41 of the text.

Schecter and Gintis have written a wonderful introduction to game theory that could have easily been titled “Game Theory by Example.” The authors begin with simple examples and progress to more complicated and illustrative games to show the reader the value of game theory as well as how to construct games for the reader’s own use. This was my first introduction to the topic. And that’s really my point: the text is an excellent introduction for anyone who wants to learn about games and possibly apply them to his own situations.

Let’s take a look at a simple game to give you a feel for the book and introduce Rosenthal’s Centipede Game, a classic. I describe this game in detail to show how well the authors describe not just this game, but all their games and illustrate the level of mathematical expertise (simple calculus at the most) needed to understand this book. Calculus is not needed for this game but we would need it for some other games in the text.

Two players, Mutt and Jeff, have $2.00 each at the start of the game. At each turn, a player can cooperate to receive $1.00 from the game master. Or, a player can defect and take $2.00 from the other player. The game ends when either player defects or when both players have at least $100.00. By the way, note how the game involves money, as many of them do, and hence we begin to see the connection to economics.

Based on the game rules we have full knowledge of the possible game play and can generate a game tree of what can happen at each turn. Let’s say Mutt goes first. The game tree is shown in Figure 1. The current money each play has is given by paired numbers so that (3,2) means Mutt has $3.00 and Jeff has $2.00. From each node, top to bottom on left, each player can cooperate (c) and gain a dollar from the game master, or defect (d) and take $2.00 from the other player. So, let’s play this game.



Figure 1: Game tree for Rosenthal’s Centipede Game (left) and backward induction used to analyze the game (right).

Mutt goes first. Both players have $2.00 so at Mutt’s first turn, top left node, the players’ payoff is (2,2). If Mutt defects at the start, he takes $2.00 from Jeff and the game ends at the right node from the top. The players payoff is now (4, 0); Mutt has $4.00 and Jeff has zero. However, if Mutt cooperates, then he gets $1.00, the players’ payoff is now (3, 2) and Jeff takes his turn. Jeff can defect to give the players (1, 4) and the game ends. Or, Jeff can cooperate to give the players (3,3). If Jeff and Mutt cooperate until the end each will have $100.00 when the game ends.

But let’s apply backward induction and look at the right hand side of Figure 1. At Jeff’s last turn, at the bottom of the game tree, Jeff can cooperate to give a payoff of (100,100). Or, if he wants to beat Mutt and have the most money, he can defect for a payoff of (98, 101). Jeff’s return is greater if he defects so a rational Jeff would defect. Mutt, who can derive the same game tree for his decisions, looks at his last turn when both players have $99.00. Because Mutt expects a rational Jeff to defect at the next turn, Mutt will act rationally and defect at his last turn to give a payoff of (101, 97) ―Mutt ends the game with more money than Jeff. At every turn, working from the end of the game to beginning each player is better off to defect rather than cooperate. That seems a bit odd because to defect right from the start leaves the players with less money than they could have with just a little cooperation.

The authors explain that in experiments people do not behave rationally as described here because: (1) Players care about more than money, (2) They do not analyze the game in this manner, or (3) They do not know the possible outcomes at each step of the game to make such an analysis. Whatever the reason, this simple game and its real-life implementation shows people aren’t necessarily rational (at least as defined here) and, what is more, it shows how economic decisions can be modeled in a logical way and subjected to analysis.

There are many different games in the book and each would be a pleasure to explain and explore. For example, the authors give extensive treatment to the Prisoner’s Dilemma and the Iterated Prisoner’s Dilemma showing how the best strategy is cooperation between prisoners.

The authors usually show a game in its simplest form, and then later in the book use that same game to demonstrate new concepts. For example, Big Monkey and Little Monkey is a game where the two monkeys must each individually decide which one will climb a tree to dislodge fruit. The cost is the energy needed to climb; Big Monkey will take more energy. The payoff is getting the fallen fruit to eat sooner if that monkey does not climb. We see this game at the start of the book, we meet it again to show how to describe games in normal form (a matrix with all the players and corresponding strategies with all payoffs), and then again to show how threats and promises can affect decisions. Threats can be effective as the authors demonstrate with an analysis of the Cuban missile crisis between the United States and Soviet Union, among other games. The authors are happy to discuss theoretical games and show how game theory applies to political and economic issues.

The book details how to understand a Nash equilibrium where players find a strategy that gives each a payoff that is not increased with a change in their individual strategy. The authors show how to analyze probabilistic games where a player’s choice is given by a probability to choose an option instead of a definite choice at each step.

I found almost all the games and examples fun to read and fun to work through the decisions and matrices. Whether the games were about politics, wine merchants and connoisseurs, or simply how to view sex ratios in society, you will find something of interest in this book.

Each chapter has problems for the reader to work and these are fun to read if nothing else. The problems give expanded details to the text and show more examples where the reader can expand his/her learning. The authors have done a wonderful job of teaching by example. If you have any interest in learning about games, this book will teach you.

By the way, the movie A Beautiful Mind is about John Nash, who won a Nobel Prize in economics for his work on game theory. In it, there is a scene where the mathematicians are in a bar and wonder about the best approach to meet an extremely attractive woman or one of her lovely friends. In the movie, John Nash says it is a Nash equilibrium for none of the men to approach the most attractive woman. This is (happily!) not true and you can find out why in this book.

David S. Mazel is a practicing engineer in Washington, DC. He welcomes your thoughts and feedback. He can be reached at mazeld at gmail dot com.