Munoz-Garcia Felix - Strategy and Game Theory Practice Exercises with Answers

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Felix Munoz-Garcia

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This chapter first analyzes how to represent games in normal form (using matrices) and in extensive form (using game trees). We afterwards describe how to systematically detect strictly dominated strategies , i.e., strategies that a player would not use regardless of the action chosen by his opponents.

In particular, we say that player finds strategy as strictly dominated by another strategy if for every strategy profile , where represents the profile of strategies selected by player s opponents, i.e., a vector with components. In words, strategy strictly dominates if it yields a strictly higher utility than strategy regardless of the strategy profile that player i s rival choose.

Since we can anticipate that strictly dominated strategies will not be selected by rational players, we apply the Iterative Deletion of Strictly Dominated Strategies (IDSDS) to predict players behavior. We elaborate on the application of IDSDS in games with two and more players, and in games where players are allowed to choose between two strategies, between more than two strategies, or a continuum of strategies. In some games, we will show that the application of IDSDS is powerful, as it rules out dominated strategies and leaves us with a relatively precise equilibrium prediction, i.e., only one or two strategy profiles surviving the application of IDSDS. In other games, however, we will see that IDSDS does not have a bite because no strategies are dominated; that is, a strategy does not provide a strictly lower payoff to player regardless of the strategy profile selected by his opponents (it can provide a higher payoff under some of his opponents strategies). In this case, we wont be able to offer an equilibrium prediction, other than to say that the entire game is our most precise equilibrium prediction! In subsequent chapters, however, we explore other solution concepts that provide more precise predictions that IDSDS.

Finally, we study the deletion of weakly dominated strategies does not necessarily lead to the same equilibrium outcomes as IDSDS, and its application is in fact sensible to deletion order. We can apply the above definition of strictly dominated strategies to define weakly dominated strategies. Specifically, we say that strategy weakly dominates if

Represent the extensive-form game depicted in Fig. using its normal-form (matrix) representation.