Combining simultaneous and sequential moves, Game Theory

Combining Simultaneous and Sequential Moves

The material in this chapter covers a variety of issues that require some knowledge of the analysis of both sequential- move and simultaneous-move games. Only one section deals with mixed strategies, so it is possible to present most of the material here after having introduced simultaneous-move games but before introducing mixing. We have used this approach in our own teaching.

To address rule changes in games, you will want to draw on examples used in previous class periods. We focus on the different outcomes that can arise when rules are changed. When a game is changed from simultaneous move to sequential move, for example, the change can create a first or second mover advantage. Games like the battle of the two cultures or chicken have first-mover advantages in their sequential move versions; the tennis-point example has a second-mover advantage in its sequential-move version. Other games show no change in equilibrium as a result of the change in rules; games like the prisoners' dilemma, in which both players have dominant strategies, fall into this category.

Similarly, changing a game from sequential play to simul- taneous play can mean that new equilibria arise-either multiple equilibria or equilibria in mixed strategies. Use the sequential-game examples you used to convey the material to show that there might be additional equilibria in the simultaneous-move versions of the game. This works for the tennis-point game if you teach it as a sequential game or for the three-person voting example from Ordeshook.

The most interesting component of the analysis is the representation of sequential-move games in strategic form and the solution of such games from that form. The second (and third) mover's strategies are more complex in sequential games, and the payoff table must have adequate rows (or columns or pages) to accommodate all of the pos- sible contingent strategies available to players. Again, the tennis-point or voting examples can be used to illustrate this idea. One nice exercise is to assert the existence of a new number of Nash equilibria in the strategic form and to show how one or two of these qualify as Nash equilibria; then use successive elim- ination of dominated strategies on the game to arrive at one cell of the table as the single reasonable equilibrium of the game. This helps motivate the idea of subgame perfection. Once you have shown that there may be multiple equilibria but that you can reduce the set of possible equilibria to one by eliminating (weakly) dominated strategies, you can show that the strategies associated with that one equilibrium coin- cide with the strategies found using rollback on the extensive form of the game. Students often have difficulty grasping the idea that the eliminated equilibria are unreasonable be- cause of the strategies associated with them rather than because of the (often) lower payoffs going to the players, so you will want to reinforce this idea as often as possible.

In addition to the examples used in previous chapters, you might want to make use of the game. This game can be played using either simultaneous or sequential moves, and there are several ways in which the sequential-move game can be set up; thus you have an opportunity to discuss rule changes as well as order changes. Also, the Boeing-Airbus example from can be used to show how multiple equilibria can arise when sequential-play games are repre- sented in strategic form. This is another way to show, with a smaller payoff table, that all subgame-perfect equilibria areNash equilibria but that not all Nash equilibria are subgame perfect. This example is also useful for explaining why Boeing's threat to fight if Airbus enters is not credible. This concept will be developed in more detail and used extensively in Chapter 9, so it is useful to introduce it before.

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