Games with Sequential Moves

Most students find the idea of rollback very simple and natural, even without drawing or understanding trees. Of course, they start by being able to do only one or two steps. A very simple way to get them to think about carrying the process all the way back to the beginning is to get pairs of students in class to play a game that requires rollback rea- soning.  The discussion that follows the play of the game can build up to the general notion of rollback. With this under- stood, the class will be much more receptive to the formalism of trees as a systematic way of practicing rollback.

Once you have succeeded in introducing the idea of roll- back and in motivating the usefulness of trees, it is also important to focus on specifics. It is nice to have a sample game (similar in size and format to the Senate race game in the chapter) that you can use to illustrate a variety of con- cepts and can refer to continually during the first part of the term. One possibility is the two by two tennis-point game. While the game most logically fits the description of a simultaneous- move game (with no equilibrium in pure strategies), you can argue that you want to analyze the case in which the player about to hit the ball makes a particular move that indicates in which direction the ball will go; her opponent can read the body language and respond accordingly. Then the game is sequential and amenable to rollback analysis. You could also use a baseball example where the pitcher "tips" his pitch and the batter can be ready for it,When you first illustrate the tree for a specific game like the sequential-move version of the tennis-point game, you will want to identify and label the various components of the tree: initial node, decision nodes, terminal nodes, and branches. You will also want to introduce the important components of analysis that are considered in any game: players, actions, payoffs, strategies, outcomes, and equilibria. In your simple tree, you will easily be able to identify players, actions, payoffs, and outcomes. Strategies, especially for sequential games, and equilibria take more effort.

One of the hardest ideas for students to grasp is the game- theoretic concept of a strategy as a complete plan of action. We have exhorted many classes of students to think of a strategy as something you can write down on a piece of paper and give to your mother so that she can play the game for you; then you have to write down instructions for mom in such a way that she knows what you want her to do no matter what happens in the game before it is your turn to move. And that means no matter what happens-that strategy has to cover every possible contingency. Most students can relate to the need for explaining everything down to the last detail to their mothers. In your sample sequential-move game, you can then show the number of strategies available to the first mover and the larger number available to the second mover; you will want to show how to construct the second mover's contingent strategies and how to describe them.

Once you have discussed the issues surrounding contin- gent strategies, you can derive the rollback equilibrium and show which strategies are used in equilibrium. To help stu- dents appreciate how quickly contingent strategies increase in complexity, you can go on to an example of a three-or- more player game. The text abbreviates the contingent strat-egies in this game; for example, Big Giant's strategy "try for Urban Mall if Frieda's does and try for Urban Mall if Frieda's tries for Rural Mall" is abbreviated (Urban, Urban) or (UU). Students who have difficulty with the idea of strat- egies as complete plans of action may be more comfortable with the description of this strategy (and others) in the nota- tion (if U, then U; if R, then U). If you are looking for an example that is similar but not identical to ours, Peter Orde- shook provides a comprehensive analysis of a three-person voting example involving a roll-call (sequential) vote on a pay raise in Game Theory and Political Theory [London/New York: Cambridge University Press (1986)]; the tree for his game is the same size and shape as that for our three-store-mall game.

You may want to have some discussion with your students about the increase in complexity that arises when the number of moves, in addition to the number of players, is increased. Most students will have at least heard of the chess-playing computer, Deep Blue, and may be interested in what game theory has to tell them about the ongoing chess-based saga of human versus machine. Others may want to pursue the dis- crepancies between the predictions of rollback-like imme- diate pickup in the pile of dimes game-and outcomes in actual play, particularly if they had the opportunity to play a game of this type themselves. Our students have found it interesting to see the tree for the centipede game at this point, and you can use it to show how a full tree is sometimes not necessary for the complete analysis of the game.